Properties

Label 32-273e16-1.1-c1e16-0-3
Degree $32$
Conductor $9.519\times 10^{38}$
Sign $1$
Analytic cond. $260037.$
Root an. cond. $1.47645$
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  − 8·3-s − 5·4-s + 28·9-s − 12·11-s + 40·12-s + 4·13-s + 13·16-s + 10·17-s − 2·23-s + 46·25-s − 48·27-s + 12·29-s + 6·32-s + 96·33-s − 140·36-s + 18·37-s − 32·39-s + 18·41-s − 10·43-s + 60·44-s − 104·48-s + 4·49-s − 80·51-s − 20·52-s + 12·53-s − 60·59-s − 6·61-s + ⋯
L(s)  = 1  − 4.61·3-s − 5/2·4-s + 28/3·9-s − 3.61·11-s + 11.5·12-s + 1.10·13-s + 13/4·16-s + 2.42·17-s − 0.417·23-s + 46/5·25-s − 9.23·27-s + 2.22·29-s + 1.06·32-s + 16.7·33-s − 23.3·36-s + 2.95·37-s − 5.12·39-s + 2.81·41-s − 1.52·43-s + 9.04·44-s − 15.0·48-s + 4/7·49-s − 11.2·51-s − 2.77·52-s + 1.64·53-s − 7.81·59-s − 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 7^{16} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(260037.\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{273} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{16} \cdot 7^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3072363367\)
\(L(\frac12)\) \(\approx\) \(0.3072363367\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T + T^{2} )^{8} \)
7 \( ( 1 - T^{2} + T^{4} )^{4} \)
13 \( 1 - 4 T - 25 T^{2} + 60 T^{3} + 491 T^{4} - 22 p T^{5} - 6835 T^{6} - 2154 T^{7} + 98049 T^{8} - 2154 p T^{9} - 6835 p^{2} T^{10} - 22 p^{4} T^{11} + 491 p^{4} T^{12} + 60 p^{5} T^{13} - 25 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
good2 \( 1 + 5 T^{2} + 3 p^{2} T^{4} - 3 p T^{5} + 9 T^{6} - 3 p^{3} T^{7} - 7 p^{2} T^{8} - 21 p T^{9} - 41 p T^{10} - 3 p^{3} T^{11} - 53 T^{12} + 3 p^{4} T^{13} + 35 p^{2} T^{14} + 9 p^{3} T^{15} + 545 T^{16} + 9 p^{4} T^{17} + 35 p^{4} T^{18} + 3 p^{7} T^{19} - 53 p^{4} T^{20} - 3 p^{8} T^{21} - 41 p^{7} T^{22} - 21 p^{8} T^{23} - 7 p^{10} T^{24} - 3 p^{12} T^{25} + 9 p^{10} T^{26} - 3 p^{12} T^{27} + 3 p^{14} T^{28} + 5 p^{14} T^{30} + p^{16} T^{32} \)
5 \( 1 - 46 T^{2} + 1059 T^{4} - 652 p^{2} T^{6} + 188481 T^{8} - 1738054 T^{10} + 2642519 p T^{12} - 84381032 T^{14} + 457280329 T^{16} - 84381032 p^{2} T^{18} + 2642519 p^{5} T^{20} - 1738054 p^{6} T^{22} + 188481 p^{8} T^{24} - 652 p^{12} T^{26} + 1059 p^{12} T^{28} - 46 p^{14} T^{30} + p^{16} T^{32} \)
11 \( 1 + 12 T + 116 T^{2} + 816 T^{3} + 5175 T^{4} + 27642 T^{5} + 133718 T^{6} + 564486 T^{7} + 194121 p T^{8} + 6857574 T^{9} + 17694434 T^{10} + 25673940 T^{11} - 64847759 T^{12} - 775227828 T^{13} - 4289274536 T^{14} - 18189427152 T^{15} - 65192876099 T^{16} - 18189427152 p T^{17} - 4289274536 p^{2} T^{18} - 775227828 p^{3} T^{19} - 64847759 p^{4} T^{20} + 25673940 p^{5} T^{21} + 17694434 p^{6} T^{22} + 6857574 p^{7} T^{23} + 194121 p^{9} T^{24} + 564486 p^{9} T^{25} + 133718 p^{10} T^{26} + 27642 p^{11} T^{27} + 5175 p^{12} T^{28} + 816 p^{13} T^{29} + 116 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \)
17 \( 1 - 10 T + 19 T^{2} - 26 T^{3} + 602 T^{4} + 360 T^{5} + 1083 T^{6} - 115270 T^{7} + 133867 T^{8} - 128682 T^{9} + 9371074 T^{10} - 6070456 T^{11} + 49129107 T^{12} - 1064307190 T^{13} + 520807897 T^{14} + 1031803334 T^{15} + 52537385156 T^{16} + 1031803334 p T^{17} + 520807897 p^{2} T^{18} - 1064307190 p^{3} T^{19} + 49129107 p^{4} T^{20} - 6070456 p^{5} T^{21} + 9371074 p^{6} T^{22} - 128682 p^{7} T^{23} + 133867 p^{8} T^{24} - 115270 p^{9} T^{25} + 1083 p^{10} T^{26} + 360 p^{11} T^{27} + 602 p^{12} T^{28} - 26 p^{13} T^{29} + 19 p^{14} T^{30} - 10 p^{15} T^{31} + p^{16} T^{32} \)
19 \( 1 + 3 p T^{2} + 44 p T^{4} + 1044 T^{5} + 4361 T^{6} + 84024 T^{7} + 549609 T^{8} + 119484 p T^{9} + 16912000 T^{10} + 22130100 T^{11} + 177486614 T^{12} + 363421944 T^{13} + 4812750550 T^{14} + 19707369300 T^{15} + 154354952440 T^{16} + 19707369300 p T^{17} + 4812750550 p^{2} T^{18} + 363421944 p^{3} T^{19} + 177486614 p^{4} T^{20} + 22130100 p^{5} T^{21} + 16912000 p^{6} T^{22} + 119484 p^{8} T^{23} + 549609 p^{8} T^{24} + 84024 p^{9} T^{25} + 4361 p^{10} T^{26} + 1044 p^{11} T^{27} + 44 p^{13} T^{28} + 3 p^{15} T^{30} + p^{16} T^{32} \)
23 \( 1 + 2 T - 83 T^{2} - 466 T^{3} + 5 p^{2} T^{4} + 29774 T^{5} - 14548 T^{6} - 973432 T^{7} - 1637158 T^{8} + 23171840 T^{9} + 72664926 T^{10} - 518691346 T^{11} - 2569182466 T^{12} + 10085274924 T^{13} + 3706027531 p T^{14} - 96609881634 T^{15} - 2241487551239 T^{16} - 96609881634 p T^{17} + 3706027531 p^{3} T^{18} + 10085274924 p^{3} T^{19} - 2569182466 p^{4} T^{20} - 518691346 p^{5} T^{21} + 72664926 p^{6} T^{22} + 23171840 p^{7} T^{23} - 1637158 p^{8} T^{24} - 973432 p^{9} T^{25} - 14548 p^{10} T^{26} + 29774 p^{11} T^{27} + 5 p^{14} T^{28} - 466 p^{13} T^{29} - 83 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \)
29 \( 1 - 12 T - 30 T^{2} + 544 T^{3} + 2655 T^{4} - 16348 T^{5} - 133128 T^{6} + 175374 T^{7} + 3789124 T^{8} + 5548096 T^{9} - 65751966 T^{10} - 101865324 T^{11} + 611902738 T^{12} - 3114218388 T^{13} - 23508631999 T^{14} + 130226259950 T^{15} + 574276170378 T^{16} + 130226259950 p T^{17} - 23508631999 p^{2} T^{18} - 3114218388 p^{3} T^{19} + 611902738 p^{4} T^{20} - 101865324 p^{5} T^{21} - 65751966 p^{6} T^{22} + 5548096 p^{7} T^{23} + 3789124 p^{8} T^{24} + 175374 p^{9} T^{25} - 133128 p^{10} T^{26} - 16348 p^{11} T^{27} + 2655 p^{12} T^{28} + 544 p^{13} T^{29} - 30 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \)
31 \( 1 - 274 T^{2} + 36117 T^{4} - 3064024 T^{6} + 189436692 T^{8} - 9208910404 T^{10} + 373115821036 T^{12} - 13241404537070 T^{14} + 427330255514908 T^{16} - 13241404537070 p^{2} T^{18} + 373115821036 p^{4} T^{20} - 9208910404 p^{6} T^{22} + 189436692 p^{8} T^{24} - 3064024 p^{10} T^{26} + 36117 p^{12} T^{28} - 274 p^{14} T^{30} + p^{16} T^{32} \)
37 \( 1 - 18 T + 289 T^{2} - 3258 T^{3} + 32236 T^{4} - 270726 T^{5} + 2055931 T^{6} - 13610730 T^{7} + 80763947 T^{8} - 388056396 T^{9} + 1325635264 T^{10} + 662912808 T^{11} - 63555541467 T^{12} + 20044337646 p T^{13} - 6357657815939 T^{14} + 1251003029070 p T^{15} - 297328284382140 T^{16} + 1251003029070 p^{2} T^{17} - 6357657815939 p^{2} T^{18} + 20044337646 p^{4} T^{19} - 63555541467 p^{4} T^{20} + 662912808 p^{5} T^{21} + 1325635264 p^{6} T^{22} - 388056396 p^{7} T^{23} + 80763947 p^{8} T^{24} - 13610730 p^{9} T^{25} + 2055931 p^{10} T^{26} - 270726 p^{11} T^{27} + 32236 p^{12} T^{28} - 3258 p^{13} T^{29} + 289 p^{14} T^{30} - 18 p^{15} T^{31} + p^{16} T^{32} \)
41 \( 1 - 18 T + 399 T^{2} - 5238 T^{3} + 72411 T^{4} - 794616 T^{5} + 8736332 T^{6} - 84303900 T^{7} + 798553439 T^{8} - 6925634478 T^{9} + 58629668183 T^{10} - 463918738998 T^{11} + 3579009525658 T^{12} - 26080614077958 T^{13} + 185108988031523 T^{14} - 1247974765127814 T^{15} + 8190746268637604 T^{16} - 1247974765127814 p T^{17} + 185108988031523 p^{2} T^{18} - 26080614077958 p^{3} T^{19} + 3579009525658 p^{4} T^{20} - 463918738998 p^{5} T^{21} + 58629668183 p^{6} T^{22} - 6925634478 p^{7} T^{23} + 798553439 p^{8} T^{24} - 84303900 p^{9} T^{25} + 8736332 p^{10} T^{26} - 794616 p^{11} T^{27} + 72411 p^{12} T^{28} - 5238 p^{13} T^{29} + 399 p^{14} T^{30} - 18 p^{15} T^{31} + p^{16} T^{32} \)
43 \( 1 + 10 T - 3 p T^{2} - 1894 T^{3} + 4313 T^{4} + 147352 T^{5} + 144304 T^{6} - 6240298 T^{7} - 12096040 T^{8} + 152892562 T^{9} - 159762796 T^{10} - 1678954346 T^{11} + 59685156704 T^{12} - 342841132 p T^{13} - 4421471559689 T^{14} + 475015735130 T^{15} + 217901059192953 T^{16} + 475015735130 p T^{17} - 4421471559689 p^{2} T^{18} - 342841132 p^{4} T^{19} + 59685156704 p^{4} T^{20} - 1678954346 p^{5} T^{21} - 159762796 p^{6} T^{22} + 152892562 p^{7} T^{23} - 12096040 p^{8} T^{24} - 6240298 p^{9} T^{25} + 144304 p^{10} T^{26} + 147352 p^{11} T^{27} + 4313 p^{12} T^{28} - 1894 p^{13} T^{29} - 3 p^{15} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \)
47 \( 1 - 530 T^{2} + 134385 T^{4} - 21717804 T^{6} + 2518916564 T^{8} - 224376191240 T^{10} + 16076999249512 T^{12} - 958624162474622 T^{14} + 48644296651156844 T^{16} - 958624162474622 p^{2} T^{18} + 16076999249512 p^{4} T^{20} - 224376191240 p^{6} T^{22} + 2518916564 p^{8} T^{24} - 21717804 p^{10} T^{26} + 134385 p^{12} T^{28} - 530 p^{14} T^{30} + p^{16} T^{32} \)
53 \( ( 1 - 6 T + 256 T^{2} - 1200 T^{3} + 27982 T^{4} - 101214 T^{5} + 1836448 T^{6} - 5401950 T^{7} + 97827967 T^{8} - 5401950 p T^{9} + 1836448 p^{2} T^{10} - 101214 p^{3} T^{11} + 27982 p^{4} T^{12} - 1200 p^{5} T^{13} + 256 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
59 \( 1 + 60 T + 2147 T^{2} + 56820 T^{3} + 1228503 T^{4} + 22742856 T^{5} + 371591538 T^{6} + 5461218480 T^{7} + 73200936512 T^{8} + 903602333664 T^{9} + 10349334811196 T^{10} + 110590752542850 T^{11} + 1107278449703020 T^{12} + 10420641109533570 T^{13} + 92395865156502677 T^{14} + 773049846143404830 T^{15} + 6108986809346342393 T^{16} + 773049846143404830 p T^{17} + 92395865156502677 p^{2} T^{18} + 10420641109533570 p^{3} T^{19} + 1107278449703020 p^{4} T^{20} + 110590752542850 p^{5} T^{21} + 10349334811196 p^{6} T^{22} + 903602333664 p^{7} T^{23} + 73200936512 p^{8} T^{24} + 5461218480 p^{9} T^{25} + 371591538 p^{10} T^{26} + 22742856 p^{11} T^{27} + 1228503 p^{12} T^{28} + 56820 p^{13} T^{29} + 2147 p^{14} T^{30} + 60 p^{15} T^{31} + p^{16} T^{32} \)
61 \( 1 + 6 T - 220 T^{2} - 784 T^{3} + 30106 T^{4} + 53102 T^{5} - 2574980 T^{6} + 275382 T^{7} + 138814805 T^{8} - 367535886 T^{9} - 2221856224 T^{10} + 38330286782 T^{11} - 426984284742 T^{12} - 2226959954260 T^{13} + 55614486923360 T^{14} + 54818705405186 T^{15} - 4112834430862772 T^{16} + 54818705405186 p T^{17} + 55614486923360 p^{2} T^{18} - 2226959954260 p^{3} T^{19} - 426984284742 p^{4} T^{20} + 38330286782 p^{5} T^{21} - 2221856224 p^{6} T^{22} - 367535886 p^{7} T^{23} + 138814805 p^{8} T^{24} + 275382 p^{9} T^{25} - 2574980 p^{10} T^{26} + 53102 p^{11} T^{27} + 30106 p^{12} T^{28} - 784 p^{13} T^{29} - 220 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \)
67 \( 1 + 18 T + 479 T^{2} + 6678 T^{3} + 108133 T^{4} + 1337460 T^{5} + 17295516 T^{6} + 197474154 T^{7} + 2220303852 T^{8} + 23372750982 T^{9} + 237751690728 T^{10} + 2331719404974 T^{11} + 21962942757900 T^{12} + 202347331161216 T^{13} + 1774415311859627 T^{14} + 15378613593108510 T^{15} + 126252360314432713 T^{16} + 15378613593108510 p T^{17} + 1774415311859627 p^{2} T^{18} + 202347331161216 p^{3} T^{19} + 21962942757900 p^{4} T^{20} + 2331719404974 p^{5} T^{21} + 237751690728 p^{6} T^{22} + 23372750982 p^{7} T^{23} + 2220303852 p^{8} T^{24} + 197474154 p^{9} T^{25} + 17295516 p^{10} T^{26} + 1337460 p^{11} T^{27} + 108133 p^{12} T^{28} + 6678 p^{13} T^{29} + 479 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \)
71 \( 1 - 6 T + 140 T^{2} - 768 T^{3} - 2287 T^{4} + 8160 T^{5} - 494046 T^{6} - 1443108 T^{7} + 83333171 T^{8} - 706659516 T^{9} + 4733384180 T^{10} - 5514749520 T^{11} - 453096953179 T^{12} + 2391950553354 T^{13} - 2078788698780 T^{14} - 171499252930908 T^{15} + 3485124294651225 T^{16} - 171499252930908 p T^{17} - 2078788698780 p^{2} T^{18} + 2391950553354 p^{3} T^{19} - 453096953179 p^{4} T^{20} - 5514749520 p^{5} T^{21} + 4733384180 p^{6} T^{22} - 706659516 p^{7} T^{23} + 83333171 p^{8} T^{24} - 1443108 p^{9} T^{25} - 494046 p^{10} T^{26} + 8160 p^{11} T^{27} - 2287 p^{12} T^{28} - 768 p^{13} T^{29} + 140 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \)
73 \( 1 - 522 T^{2} + 143519 T^{4} - 27285404 T^{6} + 4006904037 T^{8} - 482127894730 T^{10} + 49234172397455 T^{12} - 4360305841766668 T^{14} + 339029076370008793 T^{16} - 4360305841766668 p^{2} T^{18} + 49234172397455 p^{4} T^{20} - 482127894730 p^{6} T^{22} + 4006904037 p^{8} T^{24} - 27285404 p^{10} T^{26} + 143519 p^{12} T^{28} - 522 p^{14} T^{30} + p^{16} T^{32} \)
79 \( ( 1 + 2 T + 69 T^{2} + 132 T^{3} + 7640 T^{4} - 51688 T^{5} + 352216 T^{6} - 4388422 T^{7} + 25667720 T^{8} - 4388422 p T^{9} + 352216 p^{2} T^{10} - 51688 p^{3} T^{11} + 7640 p^{4} T^{12} + 132 p^{5} T^{13} + 69 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
83 \( 1 - 644 T^{2} + 196750 T^{4} - 37604546 T^{6} + 5016497429 T^{8} - 497461264520 T^{10} + 38941859150187 T^{12} - 2681450738964230 T^{14} + 199363988119754490 T^{16} - 2681450738964230 p^{2} T^{18} + 38941859150187 p^{4} T^{20} - 497461264520 p^{6} T^{22} + 5016497429 p^{8} T^{24} - 37604546 p^{10} T^{26} + 196750 p^{12} T^{28} - 644 p^{14} T^{30} + p^{16} T^{32} \)
89 \( 1 - 78 T + 3089 T^{2} - 82758 T^{3} + 1668380 T^{4} - 26650626 T^{5} + 347602269 T^{6} - 3782008038 T^{7} + 35295549365 T^{8} - 300505093068 T^{9} + 2640212739056 T^{10} - 27045035291412 T^{11} + 313827593009978 T^{12} - 3653732141307708 T^{13} + 39865368362424882 T^{14} - 405209739836645736 T^{15} + 3905545894953676584 T^{16} - 405209739836645736 p T^{17} + 39865368362424882 p^{2} T^{18} - 3653732141307708 p^{3} T^{19} + 313827593009978 p^{4} T^{20} - 27045035291412 p^{5} T^{21} + 2640212739056 p^{6} T^{22} - 300505093068 p^{7} T^{23} + 35295549365 p^{8} T^{24} - 3782008038 p^{9} T^{25} + 347602269 p^{10} T^{26} - 26650626 p^{11} T^{27} + 1668380 p^{12} T^{28} - 82758 p^{13} T^{29} + 3089 p^{14} T^{30} - 78 p^{15} T^{31} + p^{16} T^{32} \)
97 \( 1 + 54 T + 1691 T^{2} + 38826 T^{3} + 726927 T^{4} + 11679996 T^{5} + 164560562 T^{6} + 2046300702 T^{7} + 22297305816 T^{8} + 207529152282 T^{9} + 1520366742020 T^{10} + 5897275482636 T^{11} - 59843577620612 T^{12} - 1850325328520130 T^{13} - 29377896049023875 T^{14} - 365498679079522632 T^{15} - 3867774748390671767 T^{16} - 365498679079522632 p T^{17} - 29377896049023875 p^{2} T^{18} - 1850325328520130 p^{3} T^{19} - 59843577620612 p^{4} T^{20} + 5897275482636 p^{5} T^{21} + 1520366742020 p^{6} T^{22} + 207529152282 p^{7} T^{23} + 22297305816 p^{8} T^{24} + 2046300702 p^{9} T^{25} + 164560562 p^{10} T^{26} + 11679996 p^{11} T^{27} + 726927 p^{12} T^{28} + 38826 p^{13} T^{29} + 1691 p^{14} T^{30} + 54 p^{15} T^{31} + p^{16} T^{32} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.31731039768377144785294118965, −3.21395782472208381261761294549, −3.05453953878134623888870215124, −3.02866207562938846791269847982, −2.97811001665262521775832877720, −2.96937396874208553017894898956, −2.95372017102436677146215278842, −2.87088073098854890033205704952, −2.75750080339041911404455371521, −2.67010648588185920453983947084, −2.49904813453131525438500628654, −2.36916759424727980800318820373, −2.35573200798291593166441127509, −2.27668666330297373085927859866, −1.73099028008793298113723508082, −1.71586109997505953043835547531, −1.42803683863734388698153027173, −1.34155023466227341585228271325, −1.28555986199851768307966132343, −1.10124861593729235968586562643, −0.841896706745642262145486861644, −0.822248566157718432086431418861, −0.76256831023569140720676677006, −0.54162034020040983132923793345, −0.33118687026746805255249679418, 0.33118687026746805255249679418, 0.54162034020040983132923793345, 0.76256831023569140720676677006, 0.822248566157718432086431418861, 0.841896706745642262145486861644, 1.10124861593729235968586562643, 1.28555986199851768307966132343, 1.34155023466227341585228271325, 1.42803683863734388698153027173, 1.71586109997505953043835547531, 1.73099028008793298113723508082, 2.27668666330297373085927859866, 2.35573200798291593166441127509, 2.36916759424727980800318820373, 2.49904813453131525438500628654, 2.67010648588185920453983947084, 2.75750080339041911404455371521, 2.87088073098854890033205704952, 2.95372017102436677146215278842, 2.96937396874208553017894898956, 2.97811001665262521775832877720, 3.02866207562938846791269847982, 3.05453953878134623888870215124, 3.21395782472208381261761294549, 3.31731039768377144785294118965

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

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