Properties

Label 40-71e20-1.1-c1e20-0-0
Degree $40$
Conductor $1.060\times 10^{37}$
Sign $1$
Analytic cond. $1.17681\times 10^{-5}$
Root an. cond. $0.752952$
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  − 6·2-s − 3-s + 18·4-s − 5-s + 6·6-s − 7-s − 39·8-s + 3·9-s + 6·10-s − 11-s − 18·12-s + 3·13-s + 6·14-s + 15-s + 68·16-s − 2·17-s − 18·18-s + 3·19-s − 18·20-s + 21-s + 6·22-s − 22·23-s + 39·24-s + 20·25-s − 18·26-s − 27-s − 18·28-s + ⋯
L(s)  = 1  − 4.24·2-s − 0.577·3-s + 9·4-s − 0.447·5-s + 2.44·6-s − 0.377·7-s − 13.7·8-s + 9-s + 1.89·10-s − 0.301·11-s − 5.19·12-s + 0.832·13-s + 1.60·14-s + 0.258·15-s + 17·16-s − 0.485·17-s − 4.24·18-s + 0.688·19-s − 4.02·20-s + 0.218·21-s + 1.27·22-s − 4.58·23-s + 7.96·24-s + 4·25-s − 3.53·26-s − 0.192·27-s − 3.40·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(71^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(71^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(71^{20}\)
Sign: $1$
Analytic conductor: \(1.17681\times 10^{-5}\)
Root analytic conductor: \(0.752952\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{71} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 71^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.01291654541\)
\(L(\frac12)\) \(\approx\) \(0.01291654541\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 - 45 T + 1175 T^{2} - 21650 T^{3} + 304245 T^{4} - 3348579 T^{5} + 28434485 T^{6} - 171181750 T^{7} + 456111725 T^{8} + 3978774115 T^{9} - 60430064769 T^{10} + 3978774115 p T^{11} + 456111725 p^{2} T^{12} - 171181750 p^{3} T^{13} + 28434485 p^{4} T^{14} - 3348579 p^{5} T^{15} + 304245 p^{6} T^{16} - 21650 p^{7} T^{17} + 1175 p^{8} T^{18} - 45 p^{9} T^{19} + p^{10} T^{20} \)
good2 \( 1 + 3 p T + 9 p T^{2} + 39 T^{3} + 19 p^{2} T^{4} + 71 p T^{5} + 125 p T^{6} + 13 p^{5} T^{7} + 165 p^{2} T^{8} + 31 p^{5} T^{9} + 89 p^{4} T^{10} + 987 p T^{11} + 2641 T^{12} + 3493 T^{13} + 4765 T^{14} + 3317 p T^{15} + 4541 p T^{16} + 12441 T^{17} + 4385 p^{2} T^{18} + 785 p^{5} T^{19} + 35721 T^{20} + 785 p^{6} T^{21} + 4385 p^{4} T^{22} + 12441 p^{3} T^{23} + 4541 p^{5} T^{24} + 3317 p^{6} T^{25} + 4765 p^{6} T^{26} + 3493 p^{7} T^{27} + 2641 p^{8} T^{28} + 987 p^{10} T^{29} + 89 p^{14} T^{30} + 31 p^{16} T^{31} + 165 p^{14} T^{32} + 13 p^{18} T^{33} + 125 p^{15} T^{34} + 71 p^{16} T^{35} + 19 p^{18} T^{36} + 39 p^{17} T^{37} + 9 p^{19} T^{38} + 3 p^{20} T^{39} + p^{20} T^{40} \)
3 \( 1 + T - 2 T^{2} - 4 T^{3} + p^{2} T^{4} + 4 p^{2} T^{5} + 23 T^{6} + p T^{7} + 28 T^{8} + 166 T^{9} + 299 T^{10} + 536 T^{11} + 190 p^{2} T^{12} + 1028 p T^{13} + 1016 p T^{14} + 8 p^{3} T^{15} + 155 p^{4} T^{16} + 38836 T^{17} + 6055 p^{2} T^{18} + 61736 T^{19} + 48817 T^{20} + 61736 p T^{21} + 6055 p^{4} T^{22} + 38836 p^{3} T^{23} + 155 p^{8} T^{24} + 8 p^{8} T^{25} + 1016 p^{7} T^{26} + 1028 p^{8} T^{27} + 190 p^{10} T^{28} + 536 p^{9} T^{29} + 299 p^{10} T^{30} + 166 p^{11} T^{31} + 28 p^{12} T^{32} + p^{14} T^{33} + 23 p^{14} T^{34} + 4 p^{17} T^{35} + p^{18} T^{36} - 4 p^{17} T^{37} - 2 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \)
5 \( 1 + T - 19 T^{2} - 16 T^{3} + 171 T^{4} - 8 T^{5} - 1127 T^{6} + 1322 T^{7} + 6107 T^{8} - 10116 T^{9} - 20898 T^{10} + 42716 T^{11} + 49356 T^{12} - 88932 T^{13} - 339216 T^{14} - 440187 T^{15} + 3176796 T^{16} + 144623 p^{2} T^{17} - 4766294 p T^{18} - 288043 p^{2} T^{19} + 5657731 p^{2} T^{20} - 288043 p^{3} T^{21} - 4766294 p^{3} T^{22} + 144623 p^{5} T^{23} + 3176796 p^{4} T^{24} - 440187 p^{5} T^{25} - 339216 p^{6} T^{26} - 88932 p^{7} T^{27} + 49356 p^{8} T^{28} + 42716 p^{9} T^{29} - 20898 p^{10} T^{30} - 10116 p^{11} T^{31} + 6107 p^{12} T^{32} + 1322 p^{13} T^{33} - 1127 p^{14} T^{34} - 8 p^{15} T^{35} + 171 p^{16} T^{36} - 16 p^{17} T^{37} - 19 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \)
7 \( 1 + T - 4 p T^{2} - 53 T^{3} + 377 T^{4} + 935 T^{5} - 2949 T^{6} - 8474 T^{7} + 16005 T^{8} + 36082 T^{9} - 104553 T^{10} + 18856 T^{11} + 1093648 T^{12} - 687788 T^{13} - 9317943 T^{14} - 415417 p T^{15} + 43972720 T^{16} + 66765241 T^{17} - 23961284 T^{18} - 36507767 p T^{19} - 836653299 T^{20} - 36507767 p^{2} T^{21} - 23961284 p^{2} T^{22} + 66765241 p^{3} T^{23} + 43972720 p^{4} T^{24} - 415417 p^{6} T^{25} - 9317943 p^{6} T^{26} - 687788 p^{7} T^{27} + 1093648 p^{8} T^{28} + 18856 p^{9} T^{29} - 104553 p^{10} T^{30} + 36082 p^{11} T^{31} + 16005 p^{12} T^{32} - 8474 p^{13} T^{33} - 2949 p^{14} T^{34} + 935 p^{15} T^{35} + 377 p^{16} T^{36} - 53 p^{17} T^{37} - 4 p^{19} T^{38} + p^{19} T^{39} + p^{20} T^{40} \)
11 \( 1 + T - 26 T^{2} - 38 T^{3} + 397 T^{4} - 1240 T^{5} - 6249 T^{6} + 34830 T^{7} + 104299 T^{8} - 528081 T^{9} + 210449 T^{10} + 656605 p T^{11} - 14115674 T^{12} - 100030262 T^{13} + 235316516 T^{14} + 458885923 T^{15} - 3147002943 T^{16} - 3290497879 T^{17} + 42332287130 T^{18} + 8554370331 T^{19} - 325781118819 T^{20} + 8554370331 p T^{21} + 42332287130 p^{2} T^{22} - 3290497879 p^{3} T^{23} - 3147002943 p^{4} T^{24} + 458885923 p^{5} T^{25} + 235316516 p^{6} T^{26} - 100030262 p^{7} T^{27} - 14115674 p^{8} T^{28} + 656605 p^{10} T^{29} + 210449 p^{10} T^{30} - 528081 p^{11} T^{31} + 104299 p^{12} T^{32} + 34830 p^{13} T^{33} - 6249 p^{14} T^{34} - 1240 p^{15} T^{35} + 397 p^{16} T^{36} - 38 p^{17} T^{37} - 26 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \)
13 \( 1 - 3 T - 41 T^{2} + 123 T^{3} + 516 T^{4} - 205 T^{5} - 1165 T^{6} - 58255 T^{7} + 570 p T^{8} + 904087 T^{9} + 74115 p T^{10} - 343513 T^{11} - 56667437 T^{12} - 92151476 T^{13} + 724830330 T^{14} + 1872692940 T^{15} + 2052639795 T^{16} - 41892252655 T^{17} - 112679541886 T^{18} + 332971218467 T^{19} + 1420108454333 T^{20} + 332971218467 p T^{21} - 112679541886 p^{2} T^{22} - 41892252655 p^{3} T^{23} + 2052639795 p^{4} T^{24} + 1872692940 p^{5} T^{25} + 724830330 p^{6} T^{26} - 92151476 p^{7} T^{27} - 56667437 p^{8} T^{28} - 343513 p^{9} T^{29} + 74115 p^{11} T^{30} + 904087 p^{11} T^{31} + 570 p^{13} T^{32} - 58255 p^{13} T^{33} - 1165 p^{14} T^{34} - 205 p^{15} T^{35} + 516 p^{16} T^{36} + 123 p^{17} T^{37} - 41 p^{18} T^{38} - 3 p^{19} T^{39} + p^{20} T^{40} \)
17 \( 1 + 2 T + 12 T^{2} + 88 T^{3} + 469 T^{4} + 5812 T^{5} + 15667 T^{6} + 286 p^{2} T^{7} + 29034 p T^{8} + 2187112 T^{9} + 17771796 T^{10} + 53796158 T^{11} + 254934950 T^{12} + 75228059 p T^{13} + 5414798262 T^{14} + 35367674388 T^{15} + 107891855645 T^{16} + 478378380062 T^{17} + 2113650316740 T^{18} + 8470423685163 T^{19} + 49406910004077 T^{20} + 8470423685163 p T^{21} + 2113650316740 p^{2} T^{22} + 478378380062 p^{3} T^{23} + 107891855645 p^{4} T^{24} + 35367674388 p^{5} T^{25} + 5414798262 p^{6} T^{26} + 75228059 p^{8} T^{27} + 254934950 p^{8} T^{28} + 53796158 p^{9} T^{29} + 17771796 p^{10} T^{30} + 2187112 p^{11} T^{31} + 29034 p^{13} T^{32} + 286 p^{15} T^{33} + 15667 p^{14} T^{34} + 5812 p^{15} T^{35} + 469 p^{16} T^{36} + 88 p^{17} T^{37} + 12 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \)
19 \( 1 - 3 T + 12 T^{2} - 67 T^{3} + 1134 T^{4} - 5602 T^{5} + 15908 T^{6} - 77209 T^{7} + 614290 T^{8} - 3057001 T^{9} + 5755245 T^{10} - 1362844 p T^{11} + 140709427 T^{12} - 256706170 T^{13} - 2478435293 T^{14} + 10785887248 T^{15} - 32765171524 T^{16} + 433518549952 T^{17} - 161016130711 p T^{18} + 11119903834432 T^{19} - 29541453300337 T^{20} + 11119903834432 p T^{21} - 161016130711 p^{3} T^{22} + 433518549952 p^{3} T^{23} - 32765171524 p^{4} T^{24} + 10785887248 p^{5} T^{25} - 2478435293 p^{6} T^{26} - 256706170 p^{7} T^{27} + 140709427 p^{8} T^{28} - 1362844 p^{10} T^{29} + 5755245 p^{10} T^{30} - 3057001 p^{11} T^{31} + 614290 p^{12} T^{32} - 77209 p^{13} T^{33} + 15908 p^{14} T^{34} - 5602 p^{15} T^{35} + 1134 p^{16} T^{36} - 67 p^{17} T^{37} + 12 p^{18} T^{38} - 3 p^{19} T^{39} + p^{20} T^{40} \)
23 \( ( 1 + 11 T + 153 T^{2} + 1216 T^{3} + 10822 T^{4} + 73003 T^{5} + 510320 T^{6} + 2980216 T^{7} + 17486120 T^{8} + 89829009 T^{9} + 458008117 T^{10} + 89829009 p T^{11} + 17486120 p^{2} T^{12} + 2980216 p^{3} T^{13} + 510320 p^{4} T^{14} + 73003 p^{5} T^{15} + 10822 p^{6} T^{16} + 1216 p^{7} T^{17} + 153 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
29 \( 1 + T - 56 T^{2} - 325 T^{3} + 2064 T^{4} + 25728 T^{5} - 21574 T^{6} - 1213743 T^{7} - 3832471 T^{8} + 35636956 T^{9} + 262761196 T^{10} - 326245892 T^{11} - 9941797675 T^{12} - 27444562508 T^{13} + 224447033558 T^{14} + 1655421747763 T^{15} - 3364970160 T^{16} - 48959719035835 T^{17} - 202808359483695 T^{18} + 615006425245159 T^{19} + 8213049094264605 T^{20} + 615006425245159 p T^{21} - 202808359483695 p^{2} T^{22} - 48959719035835 p^{3} T^{23} - 3364970160 p^{4} T^{24} + 1655421747763 p^{5} T^{25} + 224447033558 p^{6} T^{26} - 27444562508 p^{7} T^{27} - 9941797675 p^{8} T^{28} - 326245892 p^{9} T^{29} + 262761196 p^{10} T^{30} + 35636956 p^{11} T^{31} - 3832471 p^{12} T^{32} - 1213743 p^{13} T^{33} - 21574 p^{14} T^{34} + 25728 p^{15} T^{35} + 2064 p^{16} T^{36} - 325 p^{17} T^{37} - 56 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \)
31 \( 1 - 6 T - 119 T^{2} + 891 T^{3} + 5695 T^{4} - 71277 T^{5} - 35683 T^{6} + 3563443 T^{7} - 11488598 T^{8} - 104844585 T^{9} + 830218713 T^{10} + 517463115 T^{11} - 32464376880 T^{12} + 120894419844 T^{13} + 675583788839 T^{14} - 7064490760638 T^{15} + 5438563583890 T^{16} + 215856008751226 T^{17} - 1057174171801940 T^{18} - 2788038940090673 T^{19} + 44613582852956405 T^{20} - 2788038940090673 p T^{21} - 1057174171801940 p^{2} T^{22} + 215856008751226 p^{3} T^{23} + 5438563583890 p^{4} T^{24} - 7064490760638 p^{5} T^{25} + 675583788839 p^{6} T^{26} + 120894419844 p^{7} T^{27} - 32464376880 p^{8} T^{28} + 517463115 p^{9} T^{29} + 830218713 p^{10} T^{30} - 104844585 p^{11} T^{31} - 11488598 p^{12} T^{32} + 3563443 p^{13} T^{33} - 35683 p^{14} T^{34} - 71277 p^{15} T^{35} + 5695 p^{16} T^{36} + 891 p^{17} T^{37} - 119 p^{18} T^{38} - 6 p^{19} T^{39} + p^{20} T^{40} \)
37 \( ( 1 + 3 T + 188 T^{2} + 321 T^{3} + 17836 T^{4} + 16807 T^{5} + 1168998 T^{6} + 561835 T^{7} + 58645484 T^{8} + 13072191 T^{9} + 2381920755 T^{10} + 13072191 p T^{11} + 58645484 p^{2} T^{12} + 561835 p^{3} T^{13} + 1168998 p^{4} T^{14} + 16807 p^{5} T^{15} + 17836 p^{6} T^{16} + 321 p^{7} T^{17} + 188 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
41 \( ( 1 + 30 T + 749 T^{2} + 12695 T^{3} + 187908 T^{4} + 2262813 T^{5} + 24430067 T^{6} + 227137601 T^{7} + 1919190616 T^{8} + 14280949101 T^{9} + 2369271569 p T^{10} + 14280949101 p T^{11} + 1919190616 p^{2} T^{12} + 227137601 p^{3} T^{13} + 24430067 p^{4} T^{14} + 2262813 p^{5} T^{15} + 187908 p^{6} T^{16} + 12695 p^{7} T^{17} + 749 p^{8} T^{18} + 30 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
43 \( 1 - 23 T + 103 T^{2} + 1098 T^{3} - 3721 T^{4} - 116276 T^{5} + 14360 p T^{6} + 4445639 T^{7} - 34894478 T^{8} - 143240743 T^{9} + 1868150775 T^{10} + 662222257 T^{11} - 63757417777 T^{12} + 136697411343 T^{13} + 35132248490 p T^{14} - 17414601261124 T^{15} + 86210744638745 T^{16} + 451693622250983 T^{17} - 5821647836642019 T^{18} - 16633519927232258 T^{19} + 393057039389131601 T^{20} - 16633519927232258 p T^{21} - 5821647836642019 p^{2} T^{22} + 451693622250983 p^{3} T^{23} + 86210744638745 p^{4} T^{24} - 17414601261124 p^{5} T^{25} + 35132248490 p^{7} T^{26} + 136697411343 p^{7} T^{27} - 63757417777 p^{8} T^{28} + 662222257 p^{9} T^{29} + 1868150775 p^{10} T^{30} - 143240743 p^{11} T^{31} - 34894478 p^{12} T^{32} + 4445639 p^{13} T^{33} + 14360 p^{15} T^{34} - 116276 p^{15} T^{35} - 3721 p^{16} T^{36} + 1098 p^{17} T^{37} + 103 p^{18} T^{38} - 23 p^{19} T^{39} + p^{20} T^{40} \)
47 \( 1 - 29 T + 237 T^{2} + 1866 T^{3} - 50531 T^{4} + 304962 T^{5} + 1492445 T^{6} - 33852077 T^{7} + 164513102 T^{8} + 703400781 T^{9} - 12481301060 T^{10} + 38295418779 T^{11} + 330997157353 T^{12} - 2743411387271 T^{13} - 6627122830865 T^{14} + 157856856446062 T^{15} - 138909706454230 T^{16} - 9096142046795951 T^{17} + 54188850432206649 T^{18} + 213900327859169614 T^{19} - 3896870937402661529 T^{20} + 213900327859169614 p T^{21} + 54188850432206649 p^{2} T^{22} - 9096142046795951 p^{3} T^{23} - 138909706454230 p^{4} T^{24} + 157856856446062 p^{5} T^{25} - 6627122830865 p^{6} T^{26} - 2743411387271 p^{7} T^{27} + 330997157353 p^{8} T^{28} + 38295418779 p^{9} T^{29} - 12481301060 p^{10} T^{30} + 703400781 p^{11} T^{31} + 164513102 p^{12} T^{32} - 33852077 p^{13} T^{33} + 1492445 p^{14} T^{34} + 304962 p^{15} T^{35} - 50531 p^{16} T^{36} + 1866 p^{17} T^{37} + 237 p^{18} T^{38} - 29 p^{19} T^{39} + p^{20} T^{40} \)
53 \( 1 - 2 T - 102 T^{2} + 628 T^{3} + 9529 T^{4} - 71428 T^{5} - 576418 T^{6} + 6923901 T^{7} + 26643725 T^{8} - 502809108 T^{9} - 105171691 T^{10} + 30910987233 T^{11} - 74333752257 T^{12} - 1579360650541 T^{13} + 9155549890439 T^{14} + 70100783410009 T^{15} - 732427770106904 T^{16} - 2373856076256105 T^{17} + 47599076237296451 T^{18} + 40105939580952677 T^{19} - 2638830677053959477 T^{20} + 40105939580952677 p T^{21} + 47599076237296451 p^{2} T^{22} - 2373856076256105 p^{3} T^{23} - 732427770106904 p^{4} T^{24} + 70100783410009 p^{5} T^{25} + 9155549890439 p^{6} T^{26} - 1579360650541 p^{7} T^{27} - 74333752257 p^{8} T^{28} + 30910987233 p^{9} T^{29} - 105171691 p^{10} T^{30} - 502809108 p^{11} T^{31} + 26643725 p^{12} T^{32} + 6923901 p^{13} T^{33} - 576418 p^{14} T^{34} - 71428 p^{15} T^{35} + 9529 p^{16} T^{36} + 628 p^{17} T^{37} - 102 p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \)
59 \( 1 - 31 T + 390 T^{2} - 1797 T^{3} - 14569 T^{4} + 300817 T^{5} - 1901599 T^{6} - 4477760 T^{7} + 194420391 T^{8} - 1664977606 T^{9} + 2955370541 T^{10} + 80291185495 T^{11} - 913372113856 T^{12} + 3483357169184 T^{13} + 19718959941091 T^{14} - 354711357157539 T^{15} + 2172844322902089 T^{16} - 3980585818520535 T^{17} - 38527944303198448 T^{18} + 354900197626451990 T^{19} - 2149492890803213041 T^{20} + 354900197626451990 p T^{21} - 38527944303198448 p^{2} T^{22} - 3980585818520535 p^{3} T^{23} + 2172844322902089 p^{4} T^{24} - 354711357157539 p^{5} T^{25} + 19718959941091 p^{6} T^{26} + 3483357169184 p^{7} T^{27} - 913372113856 p^{8} T^{28} + 80291185495 p^{9} T^{29} + 2955370541 p^{10} T^{30} - 1664977606 p^{11} T^{31} + 194420391 p^{12} T^{32} - 4477760 p^{13} T^{33} - 1901599 p^{14} T^{34} + 300817 p^{15} T^{35} - 14569 p^{16} T^{36} - 1797 p^{17} T^{37} + 390 p^{18} T^{38} - 31 p^{19} T^{39} + p^{20} T^{40} \)
61 \( 1 + 2 T - 280 T^{2} + 483 T^{3} + 43964 T^{4} - 191486 T^{5} - 4501443 T^{6} + 32620163 T^{7} + 328607168 T^{8} - 3730220451 T^{9} - 15996628628 T^{10} + 333821857649 T^{11} + 246727077137 T^{12} - 24396939665243 T^{13} + 885339739391 p T^{14} + 1474853616594050 T^{15} - 7962115545743255 T^{16} - 67308235092203323 T^{17} + 721759317973587484 T^{18} + 1544287751875377516 T^{19} - 49568160190367765397 T^{20} + 1544287751875377516 p T^{21} + 721759317973587484 p^{2} T^{22} - 67308235092203323 p^{3} T^{23} - 7962115545743255 p^{4} T^{24} + 1474853616594050 p^{5} T^{25} + 885339739391 p^{7} T^{26} - 24396939665243 p^{7} T^{27} + 246727077137 p^{8} T^{28} + 333821857649 p^{9} T^{29} - 15996628628 p^{10} T^{30} - 3730220451 p^{11} T^{31} + 328607168 p^{12} T^{32} + 32620163 p^{13} T^{33} - 4501443 p^{14} T^{34} - 191486 p^{15} T^{35} + 43964 p^{16} T^{36} + 483 p^{17} T^{37} - 280 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \)
67 \( 1 + 38 T + 573 T^{2} + 3444 T^{3} - 10927 T^{4} - 213486 T^{5} + 2002205 T^{6} + 59963036 T^{7} + 484915045 T^{8} + 538366353 T^{9} - 15922945339 T^{10} + 18898118472 T^{11} + 2802404689987 T^{12} + 450105098093 p T^{13} + 122157974082329 T^{14} - 111557276947519 T^{15} + 1465464372206132 T^{16} + 94885122666839294 T^{17} + 992874200035100856 T^{18} + 4176697152117679757 T^{19} + 10767821124968522405 T^{20} + 4176697152117679757 p T^{21} + 992874200035100856 p^{2} T^{22} + 94885122666839294 p^{3} T^{23} + 1465464372206132 p^{4} T^{24} - 111557276947519 p^{5} T^{25} + 122157974082329 p^{6} T^{26} + 450105098093 p^{8} T^{27} + 2802404689987 p^{8} T^{28} + 18898118472 p^{9} T^{29} - 15922945339 p^{10} T^{30} + 538366353 p^{11} T^{31} + 484915045 p^{12} T^{32} + 59963036 p^{13} T^{33} + 2002205 p^{14} T^{34} - 213486 p^{15} T^{35} - 10927 p^{16} T^{36} + 3444 p^{17} T^{37} + 573 p^{18} T^{38} + 38 p^{19} T^{39} + p^{20} T^{40} \)
73 \( 1 + 21 T + 134 T^{2} - 703 T^{3} - 15747 T^{4} - 18209 T^{5} + 1330267 T^{6} + 4590331 T^{7} - 143796504 T^{8} - 1287596294 T^{9} + 4640537001 T^{10} + 88416519801 T^{11} - 567205365336 T^{12} - 205353623990 p T^{13} - 44234395920964 T^{14} + 982341602105481 T^{15} + 7873056123489922 T^{16} - 33454750729442138 T^{17} - 542155821343950006 T^{18} + 2022261355653925802 T^{19} + 58108947670546638341 T^{20} + 2022261355653925802 p T^{21} - 542155821343950006 p^{2} T^{22} - 33454750729442138 p^{3} T^{23} + 7873056123489922 p^{4} T^{24} + 982341602105481 p^{5} T^{25} - 44234395920964 p^{6} T^{26} - 205353623990 p^{8} T^{27} - 567205365336 p^{8} T^{28} + 88416519801 p^{9} T^{29} + 4640537001 p^{10} T^{30} - 1287596294 p^{11} T^{31} - 143796504 p^{12} T^{32} + 4590331 p^{13} T^{33} + 1330267 p^{14} T^{34} - 18209 p^{15} T^{35} - 15747 p^{16} T^{36} - 703 p^{17} T^{37} + 134 p^{18} T^{38} + 21 p^{19} T^{39} + p^{20} T^{40} \)
79 \( 1 + 59 T + 1407 T^{2} + 16006 T^{3} + 57430 T^{4} - 645879 T^{5} - 9235153 T^{6} - 39209248 T^{7} + 532488362 T^{8} + 12638452096 T^{9} + 81662209067 T^{10} - 380390392591 T^{11} - 6415937124780 T^{12} + 9133844072130 T^{13} + 522370845754280 T^{14} + 4451920919203603 T^{15} + 25137836698265799 T^{16} - 112221815937930156 T^{17} - 3080695386352898193 T^{18} + 2723268989098632912 T^{19} + \)\(28\!\cdots\!80\)\( T^{20} + 2723268989098632912 p T^{21} - 3080695386352898193 p^{2} T^{22} - 112221815937930156 p^{3} T^{23} + 25137836698265799 p^{4} T^{24} + 4451920919203603 p^{5} T^{25} + 522370845754280 p^{6} T^{26} + 9133844072130 p^{7} T^{27} - 6415937124780 p^{8} T^{28} - 380390392591 p^{9} T^{29} + 81662209067 p^{10} T^{30} + 12638452096 p^{11} T^{31} + 532488362 p^{12} T^{32} - 39209248 p^{13} T^{33} - 9235153 p^{14} T^{34} - 645879 p^{15} T^{35} + 57430 p^{16} T^{36} + 16006 p^{17} T^{37} + 1407 p^{18} T^{38} + 59 p^{19} T^{39} + p^{20} T^{40} \)
83 \( 1 + 15 T - 31 T^{2} - 398 T^{3} + 21978 T^{4} + 1855 p T^{5} - 745940 T^{6} + 8566194 T^{7} + 284712686 T^{8} + 1151065877 T^{9} + 526608016 T^{10} + 195213765643 T^{11} + 2419064540338 T^{12} + 5942711656065 T^{13} + 69085107152697 T^{14} + 1904119991493536 T^{15} + 14408927834079101 T^{16} + 76435222233565015 T^{17} + 1067793234087929207 T^{18} + 11770413124663781464 T^{19} + \)\(10\!\cdots\!85\)\( T^{20} + 11770413124663781464 p T^{21} + 1067793234087929207 p^{2} T^{22} + 76435222233565015 p^{3} T^{23} + 14408927834079101 p^{4} T^{24} + 1904119991493536 p^{5} T^{25} + 69085107152697 p^{6} T^{26} + 5942711656065 p^{7} T^{27} + 2419064540338 p^{8} T^{28} + 195213765643 p^{9} T^{29} + 526608016 p^{10} T^{30} + 1151065877 p^{11} T^{31} + 284712686 p^{12} T^{32} + 8566194 p^{13} T^{33} - 745940 p^{14} T^{34} + 1855 p^{16} T^{35} + 21978 p^{16} T^{36} - 398 p^{17} T^{37} - 31 p^{18} T^{38} + 15 p^{19} T^{39} + p^{20} T^{40} \)
89 \( 1 - 16 T - 238 T^{2} + 5275 T^{3} + 13646 T^{4} - 761564 T^{5} + 3127600 T^{6} + 41599003 T^{7} - 718397031 T^{8} + 4017940239 T^{9} + 54448376208 T^{10} - 913648356343 T^{11} + 1336080591231 T^{12} + 53136826237643 T^{13} - 589876560818461 T^{14} + 3404893971216846 T^{15} + 34832377314907807 T^{16} - 764368452763291051 T^{17} + 2486943928159883786 T^{18} + 36196673693766756950 T^{19} - \)\(50\!\cdots\!97\)\( T^{20} + 36196673693766756950 p T^{21} + 2486943928159883786 p^{2} T^{22} - 764368452763291051 p^{3} T^{23} + 34832377314907807 p^{4} T^{24} + 3404893971216846 p^{5} T^{25} - 589876560818461 p^{6} T^{26} + 53136826237643 p^{7} T^{27} + 1336080591231 p^{8} T^{28} - 913648356343 p^{9} T^{29} + 54448376208 p^{10} T^{30} + 4017940239 p^{11} T^{31} - 718397031 p^{12} T^{32} + 41599003 p^{13} T^{33} + 3127600 p^{14} T^{34} - 761564 p^{15} T^{35} + 13646 p^{16} T^{36} + 5275 p^{17} T^{37} - 238 p^{18} T^{38} - 16 p^{19} T^{39} + p^{20} T^{40} \)
97 \( ( 1 + 29 T + 753 T^{2} + 13598 T^{3} + 237606 T^{4} + 3447077 T^{5} + 48289004 T^{6} + 592388194 T^{7} + 7044759241 T^{8} + 75193057556 T^{9} + 779454822431 T^{10} + 75193057556 p T^{11} + 7044759241 p^{2} T^{12} + 592388194 p^{3} T^{13} + 48289004 p^{4} T^{14} + 3447077 p^{5} T^{15} + 237606 p^{6} T^{16} + 13598 p^{7} T^{17} + 753 p^{8} T^{18} + 29 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
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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

−4.15823914812318433532250178085, −4.09347890885586106255255950687, −3.94190346216448895183352224993, −3.89985120630841355373482425392, −3.89445103871393254767801166115, −3.60448417907229069849671497084, −3.53513294087320603580236648297, −3.51717706177472345292274758075, −3.47225385501369763777576278199, −3.40442502883847188904352166878, −3.28390981502080037068037936971, −2.82378970811814943473148230021, −2.80829491827303825100042367207, −2.77049752772748932450342951887, −2.73955040604680737631212248876, −2.71638596058596990902652335373, −2.42762600723552739211620877981, −2.23874608400862105064442954111, −2.15743756779052185051154876231, −2.09012560064350587838274915855, −1.84391671828380018496428100952, −1.57321639499304235469686711786, −1.47718653552201638364070714439, −1.23039057956880142318659054125, −0.870737830100476689953827958645, 0.870737830100476689953827958645, 1.23039057956880142318659054125, 1.47718653552201638364070714439, 1.57321639499304235469686711786, 1.84391671828380018496428100952, 2.09012560064350587838274915855, 2.15743756779052185051154876231, 2.23874608400862105064442954111, 2.42762600723552739211620877981, 2.71638596058596990902652335373, 2.73955040604680737631212248876, 2.77049752772748932450342951887, 2.80829491827303825100042367207, 2.82378970811814943473148230021, 3.28390981502080037068037936971, 3.40442502883847188904352166878, 3.47225385501369763777576278199, 3.51717706177472345292274758075, 3.53513294087320603580236648297, 3.60448417907229069849671497084, 3.89445103871393254767801166115, 3.89985120630841355373482425392, 3.94190346216448895183352224993, 4.09347890885586106255255950687, 4.15823914812318433532250178085

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

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