Properties

Label 40-2151e20-1.1-c1e20-0-0
Degree $40$
Conductor $4.496\times 10^{66}$
Sign $1$
Analytic cond. $4.99287\times 10^{24}$
Root an. cond. $4.14437$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 3·4-s + 16·5-s − 4·7-s − 32·8-s + 64·10-s + 12·11-s − 4·13-s − 16·14-s − 15·16-s + 24·17-s − 4·19-s − 48·20-s + 48·22-s + 12·23-s + 89·25-s − 16·26-s + 12·28-s + 24·29-s − 4·31-s + 116·32-s + 96·34-s − 64·35-s − 10·37-s − 16·38-s − 512·40-s + 66·41-s + ⋯
L(s)  = 1  + 2.82·2-s − 3/2·4-s + 7.15·5-s − 1.51·7-s − 11.3·8-s + 20.2·10-s + 3.61·11-s − 1.10·13-s − 4.27·14-s − 3.75·16-s + 5.82·17-s − 0.917·19-s − 10.7·20-s + 10.2·22-s + 2.50·23-s + 89/5·25-s − 3.13·26-s + 2.26·28-s + 4.45·29-s − 0.718·31-s + 20.5·32-s + 16.4·34-s − 10.8·35-s − 1.64·37-s − 2.59·38-s − 80.9·40-s + 10.3·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 239^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 239^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(3^{40} \cdot 239^{20}\)
Sign: $1$
Analytic conductor: \(4.99287\times 10^{24}\)
Root analytic conductor: \(4.14437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2151} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 3^{40} \cdot 239^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4462.776796\)
\(L(\frac12)\) \(\approx\) \(4462.776796\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( ( 1 - T )^{20} \)
good2 \( 1 - p^{2} T + 19 T^{2} - 7 p^{3} T^{3} + 21 p^{3} T^{4} - 51 p^{3} T^{5} + 61 p^{4} T^{6} - 515 p^{2} T^{7} + 4261 T^{8} - 2015 p^{2} T^{9} + 3745 p^{2} T^{10} - 6483 p^{2} T^{11} + 44291 T^{12} - 17831 p^{2} T^{13} + 113903 T^{14} - 43313 p^{2} T^{15} + 32889 p^{3} T^{16} - 47989 p^{3} T^{17} + 140921 p^{2} T^{18} - 100023 p^{3} T^{19} + 1149537 T^{20} - 100023 p^{4} T^{21} + 140921 p^{4} T^{22} - 47989 p^{6} T^{23} + 32889 p^{7} T^{24} - 43313 p^{7} T^{25} + 113903 p^{6} T^{26} - 17831 p^{9} T^{27} + 44291 p^{8} T^{28} - 6483 p^{11} T^{29} + 3745 p^{12} T^{30} - 2015 p^{13} T^{31} + 4261 p^{12} T^{32} - 515 p^{15} T^{33} + 61 p^{18} T^{34} - 51 p^{18} T^{35} + 21 p^{19} T^{36} - 7 p^{20} T^{37} + 19 p^{18} T^{38} - p^{21} T^{39} + p^{20} T^{40} \)
5 \( 1 - 16 T + 167 T^{2} - 1308 T^{3} + 8509 T^{4} - 47832 T^{5} + 239566 T^{6} - 1088384 T^{7} + 4549173 T^{8} - 17665336 T^{9} + 64237876 T^{10} - 220040912 T^{11} + 713472877 T^{12} - 2198069812 T^{13} + 1290863177 p T^{14} - 724279896 p^{2} T^{15} + 48628028733 T^{16} - 125200815528 T^{17} + 309388474326 T^{18} - 734346929584 T^{19} + 1674963515326 T^{20} - 734346929584 p T^{21} + 309388474326 p^{2} T^{22} - 125200815528 p^{3} T^{23} + 48628028733 p^{4} T^{24} - 724279896 p^{7} T^{25} + 1290863177 p^{7} T^{26} - 2198069812 p^{7} T^{27} + 713472877 p^{8} T^{28} - 220040912 p^{9} T^{29} + 64237876 p^{10} T^{30} - 17665336 p^{11} T^{31} + 4549173 p^{12} T^{32} - 1088384 p^{13} T^{33} + 239566 p^{14} T^{34} - 47832 p^{15} T^{35} + 8509 p^{16} T^{36} - 1308 p^{17} T^{37} + 167 p^{18} T^{38} - 16 p^{19} T^{39} + p^{20} T^{40} \)
7 \( 1 + 4 T + 69 T^{2} + 278 T^{3} + 2455 T^{4} + 9692 T^{5} + 59998 T^{6} + 227852 T^{7} + 1127804 T^{8} + 582068 p T^{9} + 17266759 T^{10} + 59007926 T^{11} + 222453408 T^{12} + 717528026 T^{13} + 2461135993 T^{14} + 7487645376 T^{15} + 23695220523 T^{16} + 67960532252 T^{17} + 200206171917 T^{18} + 540721681082 T^{19} + 1491428149122 T^{20} + 540721681082 p T^{21} + 200206171917 p^{2} T^{22} + 67960532252 p^{3} T^{23} + 23695220523 p^{4} T^{24} + 7487645376 p^{5} T^{25} + 2461135993 p^{6} T^{26} + 717528026 p^{7} T^{27} + 222453408 p^{8} T^{28} + 59007926 p^{9} T^{29} + 17266759 p^{10} T^{30} + 582068 p^{12} T^{31} + 1127804 p^{12} T^{32} + 227852 p^{13} T^{33} + 59998 p^{14} T^{34} + 9692 p^{15} T^{35} + 2455 p^{16} T^{36} + 278 p^{17} T^{37} + 69 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \)
11 \( 1 - 12 T + 175 T^{2} - 1400 T^{3} + 12212 T^{4} - 75300 T^{5} + 498937 T^{6} - 2550272 T^{7} + 14146937 T^{8} - 62853996 T^{9} + 309147235 T^{10} - 1236175720 T^{11} + 508703984 p T^{12} - 20626631332 T^{13} + 87786271209 T^{14} - 302493740192 T^{15} + 111193453285 p T^{16} - 3975269376708 T^{17} + 15368186308140 T^{18} - 4320605757052 p T^{19} + 176371874481210 T^{20} - 4320605757052 p^{2} T^{21} + 15368186308140 p^{2} T^{22} - 3975269376708 p^{3} T^{23} + 111193453285 p^{5} T^{24} - 302493740192 p^{5} T^{25} + 87786271209 p^{6} T^{26} - 20626631332 p^{7} T^{27} + 508703984 p^{9} T^{28} - 1236175720 p^{9} T^{29} + 309147235 p^{10} T^{30} - 62853996 p^{11} T^{31} + 14146937 p^{12} T^{32} - 2550272 p^{13} T^{33} + 498937 p^{14} T^{34} - 75300 p^{15} T^{35} + 12212 p^{16} T^{36} - 1400 p^{17} T^{37} + 175 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \)
13 \( 1 + 4 T + 121 T^{2} + 446 T^{3} + 7455 T^{4} + 25236 T^{5} + 310172 T^{6} + 964456 T^{7} + 9792518 T^{8} + 28039018 T^{9} + 250656019 T^{10} + 663997532 T^{11} + 5433074506 T^{12} + 13402447616 T^{13} + 102735172405 T^{14} + 237709361562 T^{15} + 1727736839161 T^{16} + 290303898680 p T^{17} + 26126383818923 T^{18} + 54128648624898 T^{19} + 357035253022638 T^{20} + 54128648624898 p T^{21} + 26126383818923 p^{2} T^{22} + 290303898680 p^{4} T^{23} + 1727736839161 p^{4} T^{24} + 237709361562 p^{5} T^{25} + 102735172405 p^{6} T^{26} + 13402447616 p^{7} T^{27} + 5433074506 p^{8} T^{28} + 663997532 p^{9} T^{29} + 250656019 p^{10} T^{30} + 28039018 p^{11} T^{31} + 9792518 p^{12} T^{32} + 964456 p^{13} T^{33} + 310172 p^{14} T^{34} + 25236 p^{15} T^{35} + 7455 p^{16} T^{36} + 446 p^{17} T^{37} + 121 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \)
17 \( 1 - 24 T + 471 T^{2} - 6492 T^{3} + 78327 T^{4} - 794496 T^{5} + 7311346 T^{6} - 60127352 T^{7} + 458355425 T^{8} - 3217667596 T^{9} + 21235745972 T^{10} - 131365057008 T^{11} + 771474972163 T^{12} - 4295477760168 T^{13} + 22862506359093 T^{14} - 116255339979500 T^{15} + 567819129849789 T^{16} - 2663034016611488 T^{17} + 12033725103325618 T^{18} - 52367906886202820 T^{19} + 219916888871300182 T^{20} - 52367906886202820 p T^{21} + 12033725103325618 p^{2} T^{22} - 2663034016611488 p^{3} T^{23} + 567819129849789 p^{4} T^{24} - 116255339979500 p^{5} T^{25} + 22862506359093 p^{6} T^{26} - 4295477760168 p^{7} T^{27} + 771474972163 p^{8} T^{28} - 131365057008 p^{9} T^{29} + 21235745972 p^{10} T^{30} - 3217667596 p^{11} T^{31} + 458355425 p^{12} T^{32} - 60127352 p^{13} T^{33} + 7311346 p^{14} T^{34} - 794496 p^{15} T^{35} + 78327 p^{16} T^{36} - 6492 p^{17} T^{37} + 471 p^{18} T^{38} - 24 p^{19} T^{39} + p^{20} T^{40} \)
19 \( 1 + 4 T + 194 T^{2} + 884 T^{3} + 19630 T^{4} + 94782 T^{5} + 1369253 T^{6} + 6689522 T^{7} + 73143332 T^{8} + 351913306 T^{9} + 3157064983 T^{10} + 14721110858 T^{11} + 113722242476 T^{12} + 508725538078 T^{13} + 3491150313113 T^{14} + 14878625374302 T^{15} + 92619672231067 T^{16} + 374009574294482 T^{17} + 2142751616638105 T^{18} + 8155786815099566 T^{19} + 43450233432678348 T^{20} + 8155786815099566 p T^{21} + 2142751616638105 p^{2} T^{22} + 374009574294482 p^{3} T^{23} + 92619672231067 p^{4} T^{24} + 14878625374302 p^{5} T^{25} + 3491150313113 p^{6} T^{26} + 508725538078 p^{7} T^{27} + 113722242476 p^{8} T^{28} + 14721110858 p^{9} T^{29} + 3157064983 p^{10} T^{30} + 351913306 p^{11} T^{31} + 73143332 p^{12} T^{32} + 6689522 p^{13} T^{33} + 1369253 p^{14} T^{34} + 94782 p^{15} T^{35} + 19630 p^{16} T^{36} + 884 p^{17} T^{37} + 194 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \)
23 \( 1 - 12 T + 320 T^{2} - 3398 T^{3} + 50336 T^{4} - 474288 T^{5} + 5173763 T^{6} - 43568738 T^{7} + 390011978 T^{8} - 2962370036 T^{9} + 22954408997 T^{10} - 158703804576 T^{11} + 1096390535614 T^{12} - 6953804653556 T^{13} + 43589230261903 T^{14} - 255103393012768 T^{15} + 1466829954581925 T^{16} - 7952165486071986 T^{17} + 42225706627301545 T^{18} - 212491564882351294 T^{19} + 1045977734839799140 T^{20} - 212491564882351294 p T^{21} + 42225706627301545 p^{2} T^{22} - 7952165486071986 p^{3} T^{23} + 1466829954581925 p^{4} T^{24} - 255103393012768 p^{5} T^{25} + 43589230261903 p^{6} T^{26} - 6953804653556 p^{7} T^{27} + 1096390535614 p^{8} T^{28} - 158703804576 p^{9} T^{29} + 22954408997 p^{10} T^{30} - 2962370036 p^{11} T^{31} + 390011978 p^{12} T^{32} - 43568738 p^{13} T^{33} + 5173763 p^{14} T^{34} - 474288 p^{15} T^{35} + 50336 p^{16} T^{36} - 3398 p^{17} T^{37} + 320 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \)
29 \( 1 - 24 T + 596 T^{2} - 328 p T^{3} + 144167 T^{4} - 1777028 T^{5} + 20646280 T^{6} - 211441520 T^{7} + 2051403443 T^{8} - 18220750268 T^{9} + 154333505866 T^{10} - 1221945354872 T^{11} + 9278561538147 T^{12} - 66676162788092 T^{13} + 461476019914776 T^{14} - 3045466763210300 T^{15} + 19414511057459821 T^{16} - 118539822246893732 T^{17} + 700415369742476966 T^{18} - 3973128101053688396 T^{19} + 21828835037749606690 T^{20} - 3973128101053688396 p T^{21} + 700415369742476966 p^{2} T^{22} - 118539822246893732 p^{3} T^{23} + 19414511057459821 p^{4} T^{24} - 3045466763210300 p^{5} T^{25} + 461476019914776 p^{6} T^{26} - 66676162788092 p^{7} T^{27} + 9278561538147 p^{8} T^{28} - 1221945354872 p^{9} T^{29} + 154333505866 p^{10} T^{30} - 18220750268 p^{11} T^{31} + 2051403443 p^{12} T^{32} - 211441520 p^{13} T^{33} + 20646280 p^{14} T^{34} - 1777028 p^{15} T^{35} + 144167 p^{16} T^{36} - 328 p^{18} T^{37} + 596 p^{18} T^{38} - 24 p^{19} T^{39} + p^{20} T^{40} \)
31 \( 1 + 4 T + 328 T^{2} + 1162 T^{3} + 53290 T^{4} + 5502 p T^{5} + 5771233 T^{6} + 17005562 T^{7} + 471296525 T^{8} + 1299418250 T^{9} + 31021389021 T^{10} + 81044143238 T^{11} + 1713566621204 T^{12} + 4277676172346 T^{13} + 81518883496948 T^{14} + 195159558209108 T^{15} + 3396509725060321 T^{16} + 7788281977911780 T^{17} + 125251572254324642 T^{18} + 273570360580223152 T^{19} + 4111579076938534830 T^{20} + 273570360580223152 p T^{21} + 125251572254324642 p^{2} T^{22} + 7788281977911780 p^{3} T^{23} + 3396509725060321 p^{4} T^{24} + 195159558209108 p^{5} T^{25} + 81518883496948 p^{6} T^{26} + 4277676172346 p^{7} T^{27} + 1713566621204 p^{8} T^{28} + 81044143238 p^{9} T^{29} + 31021389021 p^{10} T^{30} + 1299418250 p^{11} T^{31} + 471296525 p^{12} T^{32} + 17005562 p^{13} T^{33} + 5771233 p^{14} T^{34} + 5502 p^{16} T^{35} + 53290 p^{16} T^{36} + 1162 p^{17} T^{37} + 328 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \)
37 \( 1 + 10 T + 422 T^{2} + 4002 T^{3} + 88016 T^{4} + 796764 T^{5} + 12178997 T^{6} + 105480366 T^{7} + 1262413564 T^{8} + 10453259090 T^{9} + 104649324529 T^{10} + 826598908964 T^{11} + 7217078030706 T^{12} + 54228731978928 T^{13} + 11476270925311 p T^{14} + 3026867846350722 T^{15} + 21663805424880339 T^{16} + 146110564550052756 T^{17} + 968302232724717753 T^{18} + 6160317253473298046 T^{19} + 38133385128873714268 T^{20} + 6160317253473298046 p T^{21} + 968302232724717753 p^{2} T^{22} + 146110564550052756 p^{3} T^{23} + 21663805424880339 p^{4} T^{24} + 3026867846350722 p^{5} T^{25} + 11476270925311 p^{7} T^{26} + 54228731978928 p^{7} T^{27} + 7217078030706 p^{8} T^{28} + 826598908964 p^{9} T^{29} + 104649324529 p^{10} T^{30} + 10453259090 p^{11} T^{31} + 1262413564 p^{12} T^{32} + 105480366 p^{13} T^{33} + 12178997 p^{14} T^{34} + 796764 p^{15} T^{35} + 88016 p^{16} T^{36} + 4002 p^{17} T^{37} + 422 p^{18} T^{38} + 10 p^{19} T^{39} + p^{20} T^{40} \)
41 \( 1 - 66 T + 2596 T^{2} - 74562 T^{3} + 1727001 T^{4} - 33829678 T^{5} + 578608610 T^{6} - 8819809788 T^{7} + 121670488142 T^{8} - 1536073013090 T^{9} + 17905237878984 T^{10} - 194027361710104 T^{11} + 1965564348808620 T^{12} - 18697586823015100 T^{13} + 167629481625536632 T^{14} - 1420535158825491298 T^{15} + 11405706309882991249 T^{16} - 86927268451922624206 T^{17} + \)\(62\!\cdots\!22\)\( T^{18} - \)\(43\!\cdots\!08\)\( T^{19} + \)\(28\!\cdots\!30\)\( T^{20} - \)\(43\!\cdots\!08\)\( p T^{21} + \)\(62\!\cdots\!22\)\( p^{2} T^{22} - 86927268451922624206 p^{3} T^{23} + 11405706309882991249 p^{4} T^{24} - 1420535158825491298 p^{5} T^{25} + 167629481625536632 p^{6} T^{26} - 18697586823015100 p^{7} T^{27} + 1965564348808620 p^{8} T^{28} - 194027361710104 p^{9} T^{29} + 17905237878984 p^{10} T^{30} - 1536073013090 p^{11} T^{31} + 121670488142 p^{12} T^{32} - 8819809788 p^{13} T^{33} + 578608610 p^{14} T^{34} - 33829678 p^{15} T^{35} + 1727001 p^{16} T^{36} - 74562 p^{17} T^{37} + 2596 p^{18} T^{38} - 66 p^{19} T^{39} + p^{20} T^{40} \)
43 \( 1 - 8 T + 420 T^{2} - 3394 T^{3} + 89976 T^{4} - 731240 T^{5} + 13089015 T^{6} - 105952562 T^{7} + 1452628796 T^{8} - 11569122020 T^{9} + 3045786445 p T^{10} - 1012996432268 T^{11} + 9965251073880 T^{12} - 73982409873400 T^{13} + 655515452043497 T^{14} - 4630279425385904 T^{15} + 37849911969540907 T^{16} - 253085296224759006 T^{17} + 1936122002875875053 T^{18} - 12231650975187520690 T^{19} + 88168387757602099408 T^{20} - 12231650975187520690 p T^{21} + 1936122002875875053 p^{2} T^{22} - 253085296224759006 p^{3} T^{23} + 37849911969540907 p^{4} T^{24} - 4630279425385904 p^{5} T^{25} + 655515452043497 p^{6} T^{26} - 73982409873400 p^{7} T^{27} + 9965251073880 p^{8} T^{28} - 1012996432268 p^{9} T^{29} + 3045786445 p^{11} T^{30} - 11569122020 p^{11} T^{31} + 1452628796 p^{12} T^{32} - 105952562 p^{13} T^{33} + 13089015 p^{14} T^{34} - 731240 p^{15} T^{35} + 89976 p^{16} T^{36} - 3394 p^{17} T^{37} + 420 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \)
47 \( 1 - 28 T + 849 T^{2} - 16180 T^{3} + 301169 T^{4} - 4502640 T^{5} + 64505650 T^{6} - 808244438 T^{7} + 205870090 p T^{8} - 105483338936 T^{9} + 1100394351507 T^{10} - 10695345394662 T^{11} + 99789795127138 T^{12} - 880496393462378 T^{13} + 7487529739715921 T^{14} - 1295079538207828 p T^{15} + 10190683555569159 p T^{16} - 3632409673357944206 T^{17} + 26770613437771336185 T^{18} - \)\(19\!\cdots\!28\)\( T^{19} + \)\(13\!\cdots\!50\)\( T^{20} - \)\(19\!\cdots\!28\)\( p T^{21} + 26770613437771336185 p^{2} T^{22} - 3632409673357944206 p^{3} T^{23} + 10190683555569159 p^{5} T^{24} - 1295079538207828 p^{6} T^{25} + 7487529739715921 p^{6} T^{26} - 880496393462378 p^{7} T^{27} + 99789795127138 p^{8} T^{28} - 10695345394662 p^{9} T^{29} + 1100394351507 p^{10} T^{30} - 105483338936 p^{11} T^{31} + 205870090 p^{13} T^{32} - 808244438 p^{13} T^{33} + 64505650 p^{14} T^{34} - 4502640 p^{15} T^{35} + 301169 p^{16} T^{36} - 16180 p^{17} T^{37} + 849 p^{18} T^{38} - 28 p^{19} T^{39} + p^{20} T^{40} \)
53 \( 1 - 28 T + 938 T^{2} - 18442 T^{3} + 378762 T^{4} - 5962960 T^{5} + 94710173 T^{6} - 1270256994 T^{7} + 16970282392 T^{8} - 200597278700 T^{9} + 2348743217301 T^{10} - 24972357258736 T^{11} + 262300866907868 T^{12} - 2541153574269820 T^{13} + 24288646349890515 T^{14} - 216208404248231248 T^{15} + 1897552581217418135 T^{16} - 294373076102604918 p T^{17} + \)\(12\!\cdots\!17\)\( T^{18} - \)\(96\!\cdots\!06\)\( T^{19} + \)\(72\!\cdots\!60\)\( T^{20} - \)\(96\!\cdots\!06\)\( p T^{21} + \)\(12\!\cdots\!17\)\( p^{2} T^{22} - 294373076102604918 p^{4} T^{23} + 1897552581217418135 p^{4} T^{24} - 216208404248231248 p^{5} T^{25} + 24288646349890515 p^{6} T^{26} - 2541153574269820 p^{7} T^{27} + 262300866907868 p^{8} T^{28} - 24972357258736 p^{9} T^{29} + 2348743217301 p^{10} T^{30} - 200597278700 p^{11} T^{31} + 16970282392 p^{12} T^{32} - 1270256994 p^{13} T^{33} + 94710173 p^{14} T^{34} - 5962960 p^{15} T^{35} + 378762 p^{16} T^{36} - 18442 p^{17} T^{37} + 938 p^{18} T^{38} - 28 p^{19} T^{39} + p^{20} T^{40} \)
59 \( 1 - 54 T + 1990 T^{2} - 54558 T^{3} + 1249763 T^{4} - 24633246 T^{5} + 432525466 T^{6} - 6866987704 T^{7} + 100145027132 T^{8} - 1353336607042 T^{9} + 17096157307596 T^{10} - 203015977440828 T^{11} + 2278528073674030 T^{12} - 24258920104019028 T^{13} + 245873534212687652 T^{14} - 2378087698058768354 T^{15} + 21999528509475746611 T^{16} - \)\(19\!\cdots\!50\)\( T^{17} + \)\(16\!\cdots\!72\)\( T^{18} - \)\(13\!\cdots\!12\)\( T^{19} + \)\(10\!\cdots\!98\)\( T^{20} - \)\(13\!\cdots\!12\)\( p T^{21} + \)\(16\!\cdots\!72\)\( p^{2} T^{22} - \)\(19\!\cdots\!50\)\( p^{3} T^{23} + 21999528509475746611 p^{4} T^{24} - 2378087698058768354 p^{5} T^{25} + 245873534212687652 p^{6} T^{26} - 24258920104019028 p^{7} T^{27} + 2278528073674030 p^{8} T^{28} - 203015977440828 p^{9} T^{29} + 17096157307596 p^{10} T^{30} - 1353336607042 p^{11} T^{31} + 100145027132 p^{12} T^{32} - 6866987704 p^{13} T^{33} + 432525466 p^{14} T^{34} - 24633246 p^{15} T^{35} + 1249763 p^{16} T^{36} - 54558 p^{17} T^{37} + 1990 p^{18} T^{38} - 54 p^{19} T^{39} + p^{20} T^{40} \)
61 \( 1 + 4 T + 456 T^{2} + 3142 T^{3} + 110958 T^{4} + 996310 T^{5} + 20047273 T^{6} + 194829434 T^{7} + 2976967633 T^{8} + 28337965318 T^{9} + 370874447173 T^{10} + 3371088599202 T^{11} + 39306620378196 T^{12} + 341602227423346 T^{13} + 3617449600788988 T^{14} + 30009208002740356 T^{15} + 293794801623845769 T^{16} + 2315223379708574408 T^{17} + 21225757678378652842 T^{18} + \)\(15\!\cdots\!60\)\( T^{19} + \)\(13\!\cdots\!74\)\( T^{20} + \)\(15\!\cdots\!60\)\( p T^{21} + 21225757678378652842 p^{2} T^{22} + 2315223379708574408 p^{3} T^{23} + 293794801623845769 p^{4} T^{24} + 30009208002740356 p^{5} T^{25} + 3617449600788988 p^{6} T^{26} + 341602227423346 p^{7} T^{27} + 39306620378196 p^{8} T^{28} + 3371088599202 p^{9} T^{29} + 370874447173 p^{10} T^{30} + 28337965318 p^{11} T^{31} + 2976967633 p^{12} T^{32} + 194829434 p^{13} T^{33} + 20047273 p^{14} T^{34} + 996310 p^{15} T^{35} + 110958 p^{16} T^{36} + 3142 p^{17} T^{37} + 456 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \)
67 \( 1 - 12 T + 672 T^{2} - 8828 T^{3} + 241796 T^{4} - 3227676 T^{5} + 60912420 T^{6} - 787498812 T^{7} + 11830743454 T^{8} - 144411963284 T^{9} + 1857838661472 T^{10} - 21175564171364 T^{11} + 242729267995938 T^{12} - 2574307165468076 T^{13} + 26891902526026632 T^{14} - 265379077064149756 T^{15} + 2559258758618361809 T^{16} - 23531494893611487992 T^{17} + \)\(21\!\cdots\!72\)\( T^{18} - \)\(18\!\cdots\!36\)\( T^{19} + \)\(15\!\cdots\!64\)\( T^{20} - \)\(18\!\cdots\!36\)\( p T^{21} + \)\(21\!\cdots\!72\)\( p^{2} T^{22} - 23531494893611487992 p^{3} T^{23} + 2559258758618361809 p^{4} T^{24} - 265379077064149756 p^{5} T^{25} + 26891902526026632 p^{6} T^{26} - 2574307165468076 p^{7} T^{27} + 242729267995938 p^{8} T^{28} - 21175564171364 p^{9} T^{29} + 1857838661472 p^{10} T^{30} - 144411963284 p^{11} T^{31} + 11830743454 p^{12} T^{32} - 787498812 p^{13} T^{33} + 60912420 p^{14} T^{34} - 3227676 p^{15} T^{35} + 241796 p^{16} T^{36} - 8828 p^{17} T^{37} + 672 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \)
71 \( 1 - 36 T + 1283 T^{2} - 29888 T^{3} + 666071 T^{4} - 12025428 T^{5} + 208123038 T^{6} - 3143223336 T^{7} + 45801835926 T^{8} - 603852133360 T^{9} + 7731190266857 T^{10} - 91448885645304 T^{11} + 1056318912940690 T^{12} - 11421795948673232 T^{13} + 121124358672464675 T^{14} - 1212518784542386408 T^{15} + 11939560203410869801 T^{16} - \)\(11\!\cdots\!56\)\( T^{17} + \)\(10\!\cdots\!59\)\( T^{18} - \)\(89\!\cdots\!84\)\( T^{19} + \)\(77\!\cdots\!50\)\( T^{20} - \)\(89\!\cdots\!84\)\( p T^{21} + \)\(10\!\cdots\!59\)\( p^{2} T^{22} - \)\(11\!\cdots\!56\)\( p^{3} T^{23} + 11939560203410869801 p^{4} T^{24} - 1212518784542386408 p^{5} T^{25} + 121124358672464675 p^{6} T^{26} - 11421795948673232 p^{7} T^{27} + 1056318912940690 p^{8} T^{28} - 91448885645304 p^{9} T^{29} + 7731190266857 p^{10} T^{30} - 603852133360 p^{11} T^{31} + 45801835926 p^{12} T^{32} - 3143223336 p^{13} T^{33} + 208123038 p^{14} T^{34} - 12025428 p^{15} T^{35} + 666071 p^{16} T^{36} - 29888 p^{17} T^{37} + 1283 p^{18} T^{38} - 36 p^{19} T^{39} + p^{20} T^{40} \)
73 \( 1 - 14 T + 826 T^{2} - 9564 T^{3} + 335790 T^{4} - 3351378 T^{5} + 90698961 T^{6} - 799223310 T^{7} + 18397870006 T^{8} - 145485150576 T^{9} + 2992300152627 T^{10} - 21493589059926 T^{11} + 405814103710966 T^{12} - 2673143788371274 T^{13} + 47032057457578709 T^{14} - 286333220976644320 T^{15} + 4731341921881303625 T^{16} - 26798331358296675684 T^{17} + \)\(41\!\cdots\!05\)\( T^{18} - \)\(22\!\cdots\!58\)\( T^{19} + \)\(32\!\cdots\!88\)\( T^{20} - \)\(22\!\cdots\!58\)\( p T^{21} + \)\(41\!\cdots\!05\)\( p^{2} T^{22} - 26798331358296675684 p^{3} T^{23} + 4731341921881303625 p^{4} T^{24} - 286333220976644320 p^{5} T^{25} + 47032057457578709 p^{6} T^{26} - 2673143788371274 p^{7} T^{27} + 405814103710966 p^{8} T^{28} - 21493589059926 p^{9} T^{29} + 2992300152627 p^{10} T^{30} - 145485150576 p^{11} T^{31} + 18397870006 p^{12} T^{32} - 799223310 p^{13} T^{33} + 90698961 p^{14} T^{34} - 3351378 p^{15} T^{35} + 335790 p^{16} T^{36} - 9564 p^{17} T^{37} + 826 p^{18} T^{38} - 14 p^{19} T^{39} + p^{20} T^{40} \)
79 \( 1 + 12 T + 723 T^{2} + 6920 T^{3} + 240590 T^{4} + 1834630 T^{5} + 49558135 T^{6} + 292402914 T^{7} + 7198168790 T^{8} + 30620334064 T^{9} + 803636157986 T^{10} + 2075276056300 T^{11} + 73749662530108 T^{12} + 54428998676968 T^{13} + 5795003615480006 T^{14} - 9209538749964068 T^{15} + 395018804581850025 T^{16} - 1926705412472571790 T^{17} + 24490517296400064638 T^{18} - \)\(21\!\cdots\!10\)\( T^{19} + \)\(16\!\cdots\!68\)\( T^{20} - \)\(21\!\cdots\!10\)\( p T^{21} + 24490517296400064638 p^{2} T^{22} - 1926705412472571790 p^{3} T^{23} + 395018804581850025 p^{4} T^{24} - 9209538749964068 p^{5} T^{25} + 5795003615480006 p^{6} T^{26} + 54428998676968 p^{7} T^{27} + 73749662530108 p^{8} T^{28} + 2075276056300 p^{9} T^{29} + 803636157986 p^{10} T^{30} + 30620334064 p^{11} T^{31} + 7198168790 p^{12} T^{32} + 292402914 p^{13} T^{33} + 49558135 p^{14} T^{34} + 1834630 p^{15} T^{35} + 240590 p^{16} T^{36} + 6920 p^{17} T^{37} + 723 p^{18} T^{38} + 12 p^{19} T^{39} + p^{20} T^{40} \)
83 \( 1 - 20 T + 899 T^{2} - 12692 T^{3} + 347542 T^{4} - 3697412 T^{5} + 11997 p^{2} T^{6} - 681269044 T^{7} + 14533473373 T^{8} - 94082789544 T^{9} + 2121154931807 T^{10} - 10774163281424 T^{11} + 3260241204390 p T^{12} - 1053661915792016 T^{13} + 30618531576935381 T^{14} - 88866409913361448 T^{15} + 3124573030683088211 T^{16} - 6890672675223799932 T^{17} + \)\(29\!\cdots\!72\)\( T^{18} - \)\(54\!\cdots\!60\)\( T^{19} + \)\(25\!\cdots\!66\)\( T^{20} - \)\(54\!\cdots\!60\)\( p T^{21} + \)\(29\!\cdots\!72\)\( p^{2} T^{22} - 6890672675223799932 p^{3} T^{23} + 3124573030683088211 p^{4} T^{24} - 88866409913361448 p^{5} T^{25} + 30618531576935381 p^{6} T^{26} - 1053661915792016 p^{7} T^{27} + 3260241204390 p^{9} T^{28} - 10774163281424 p^{9} T^{29} + 2121154931807 p^{10} T^{30} - 94082789544 p^{11} T^{31} + 14533473373 p^{12} T^{32} - 681269044 p^{13} T^{33} + 11997 p^{16} T^{34} - 3697412 p^{15} T^{35} + 347542 p^{16} T^{36} - 12692 p^{17} T^{37} + 899 p^{18} T^{38} - 20 p^{19} T^{39} + p^{20} T^{40} \)
89 \( 1 - 130 T + 8825 T^{2} - 414178 T^{3} + 15070372 T^{4} - 452642462 T^{5} + 11673799205 T^{6} - 265539724938 T^{7} + 5429746128360 T^{8} - 101210503021582 T^{9} + 1737870152815856 T^{10} - 27709971135264698 T^{11} + 412845780145886968 T^{12} - 5775620068807121266 T^{13} + 76163521218608495288 T^{14} - \)\(94\!\cdots\!26\)\( T^{15} + \)\(11\!\cdots\!19\)\( T^{16} - \)\(12\!\cdots\!20\)\( T^{17} + \)\(13\!\cdots\!90\)\( T^{18} - \)\(13\!\cdots\!28\)\( T^{19} + \)\(13\!\cdots\!68\)\( T^{20} - \)\(13\!\cdots\!28\)\( p T^{21} + \)\(13\!\cdots\!90\)\( p^{2} T^{22} - \)\(12\!\cdots\!20\)\( p^{3} T^{23} + \)\(11\!\cdots\!19\)\( p^{4} T^{24} - \)\(94\!\cdots\!26\)\( p^{5} T^{25} + 76163521218608495288 p^{6} T^{26} - 5775620068807121266 p^{7} T^{27} + 412845780145886968 p^{8} T^{28} - 27709971135264698 p^{9} T^{29} + 1737870152815856 p^{10} T^{30} - 101210503021582 p^{11} T^{31} + 5429746128360 p^{12} T^{32} - 265539724938 p^{13} T^{33} + 11673799205 p^{14} T^{34} - 452642462 p^{15} T^{35} + 15070372 p^{16} T^{36} - 414178 p^{17} T^{37} + 8825 p^{18} T^{38} - 130 p^{19} T^{39} + p^{20} T^{40} \)
97 \( 1 + 2 T + 1052 T^{2} + 1902 T^{3} + 550771 T^{4} + 880866 T^{5} + 190872032 T^{6} + 269889568 T^{7} + 49240745326 T^{8} + 63273807542 T^{9} + 10102006273412 T^{10} + 12455293874620 T^{11} + 1721622731133922 T^{12} + 2164502060133612 T^{13} + 251452323307664780 T^{14} + 334371974309806370 T^{15} + 32189102325156748385 T^{16} + 45052957228976334886 T^{17} + \)\(36\!\cdots\!40\)\( T^{18} + \)\(51\!\cdots\!08\)\( T^{19} + \)\(37\!\cdots\!18\)\( T^{20} + \)\(51\!\cdots\!08\)\( p T^{21} + \)\(36\!\cdots\!40\)\( p^{2} T^{22} + 45052957228976334886 p^{3} T^{23} + 32189102325156748385 p^{4} T^{24} + 334371974309806370 p^{5} T^{25} + 251452323307664780 p^{6} T^{26} + 2164502060133612 p^{7} T^{27} + 1721622731133922 p^{8} T^{28} + 12455293874620 p^{9} T^{29} + 10102006273412 p^{10} T^{30} + 63273807542 p^{11} T^{31} + 49240745326 p^{12} T^{32} + 269889568 p^{13} T^{33} + 190872032 p^{14} T^{34} + 880866 p^{15} T^{35} + 550771 p^{16} T^{36} + 1902 p^{17} T^{37} + 1052 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.88079864176721001076213495850, −1.83923177796702803955643412343, −1.80957575256963323366859808270, −1.79993504459734983700415049360, −1.75565091585114491665158062310, −1.62903003934374518836550486256, −1.60018050156410494012015757311, −1.51550702145875351847690420241, −1.31132000116664843546780904914, −1.29627644274867005905248642541, −1.16810420926851387049553973493, −1.14735846984620016865425758151, −1.12771464448261886233080464413, −1.06305757238960019216776506424, −0.880813547125327509298831050375, −0.794218483406619714272094512578, −0.792898479980978233185509339486, −0.77244657914673040894494676809, −0.72691599840968540756370118854, −0.72256515204836132870876839576, −0.70196685524487497770624422735, −0.58186875671990591098887996610, −0.54318857227819321416259959152, −0.33604071870634662206544211115, −0.19056569335131045504513235218, 0.19056569335131045504513235218, 0.33604071870634662206544211115, 0.54318857227819321416259959152, 0.58186875671990591098887996610, 0.70196685524487497770624422735, 0.72256515204836132870876839576, 0.72691599840968540756370118854, 0.77244657914673040894494676809, 0.792898479980978233185509339486, 0.794218483406619714272094512578, 0.880813547125327509298831050375, 1.06305757238960019216776506424, 1.12771464448261886233080464413, 1.14735846984620016865425758151, 1.16810420926851387049553973493, 1.29627644274867005905248642541, 1.31132000116664843546780904914, 1.51550702145875351847690420241, 1.60018050156410494012015757311, 1.62903003934374518836550486256, 1.75565091585114491665158062310, 1.79993504459734983700415049360, 1.80957575256963323366859808270, 1.83923177796702803955643412343, 1.88079864176721001076213495850

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.