Dirichlet series
| L(s) = 1 | − 4·7-s + 24·19-s − 82·25-s + 84·31-s − 68·37-s − 80·43-s − 84·49-s + 216·61-s − 56·67-s + 156·73-s − 28·79-s + 408·103-s − 244·109-s + 302·121-s + 127-s + 131-s − 96·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.17e3·169-s + 173-s + 328·175-s + ⋯ |
| L(s) = 1 | − 4/7·7-s + 1.26·19-s − 3.27·25-s + 2.70·31-s − 1.83·37-s − 1.86·43-s − 1.71·49-s + 3.54·61-s − 0.835·67-s + 2.13·73-s − 0.354·79-s + 3.96·103-s − 2.23·109-s + 2.49·121-s + 0.00787·127-s + 0.00763·131-s − 0.721·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 6.93·169-s + 0.00578·173-s + 1.87·175-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.04888\times 10^{23}\) |
| Root analytic conductor: | \(5.24080\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(73.89313258\) |
| \(L(\frac12)\) | \(\approx\) | \(73.89313258\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 + 2 T + 48 T^{2} + 628 T^{3} + 215 p T^{4} + 628 p^{2} T^{5} + 48 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| good | 5 | \( 1 + 82 T^{2} + 3739 T^{4} + 22294 p T^{6} + 2031449 T^{8} + 8796436 T^{10} - 172156922 p T^{12} - 1553872984 p^{2} T^{14} - 1128416066174 T^{16} - 1553872984 p^{6} T^{18} - 172156922 p^{9} T^{20} + 8796436 p^{12} T^{22} + 2031449 p^{16} T^{24} + 22294 p^{21} T^{26} + 3739 p^{24} T^{28} + 82 p^{28} T^{30} + p^{32} T^{32} \) |
| 11 | \( 1 - 302 T^{2} + 24283 T^{4} + 1726094 T^{6} - 362836855 T^{8} + 28865177524 T^{10} - 1821143178610 T^{12} - 556112338083928 T^{14} + 142719685713224002 T^{16} - 556112338083928 p^{4} T^{18} - 1821143178610 p^{8} T^{20} + 28865177524 p^{12} T^{22} - 362836855 p^{16} T^{24} + 1726094 p^{20} T^{26} + 24283 p^{24} T^{28} - 302 p^{28} T^{30} + p^{32} T^{32} \) | |
| 13 | \( ( 1 - 586 T^{2} + 188625 T^{4} - 46908482 T^{6} + 9169031972 T^{8} - 46908482 p^{4} T^{10} + 188625 p^{8} T^{12} - 586 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 17 | \( 1 + 1392 T^{2} + 916596 T^{4} + 440632992 T^{6} + 193241832490 T^{8} + 77329454247888 T^{10} + 27388275550379856 T^{12} + 8869385639621454096 T^{14} + \)\(26\!\cdots\!39\)\( T^{16} + 8869385639621454096 p^{4} T^{18} + 27388275550379856 p^{8} T^{20} + 77329454247888 p^{12} T^{22} + 193241832490 p^{16} T^{24} + 440632992 p^{20} T^{26} + 916596 p^{24} T^{28} + 1392 p^{28} T^{30} + p^{32} T^{32} \) | |
| 19 | \( ( 1 - 12 T + 71 p T^{2} - 15612 T^{3} + 1057809 T^{4} - 10682160 T^{5} + 572544622 T^{6} - 5131216248 T^{7} + 236195825318 T^{8} - 5131216248 p^{2} T^{9} + 572544622 p^{4} T^{10} - 10682160 p^{6} T^{11} + 1057809 p^{8} T^{12} - 15612 p^{10} T^{13} + 71 p^{13} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 - 1080 T^{2} + 300212 T^{4} + 182751984 T^{6} - 183823831190 T^{8} + 72593627119992 T^{10} - 6770670169575472 T^{12} - 10525257868033494216 T^{14} + \)\(78\!\cdots\!47\)\( T^{16} - 10525257868033494216 p^{4} T^{18} - 6770670169575472 p^{8} T^{20} + 72593627119992 p^{12} T^{22} - 183823831190 p^{16} T^{24} + 182751984 p^{20} T^{26} + 300212 p^{24} T^{28} - 1080 p^{28} T^{30} + p^{32} T^{32} \) | |
| 29 | \( ( 1 + 2174 T^{2} + 2516985 T^{4} + 2194531654 T^{6} + 1859538564788 T^{8} + 2194531654 p^{4} T^{10} + 2516985 p^{8} T^{12} + 2174 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 42 T + 2300 T^{2} - 71904 T^{3} + 2032911 T^{4} - 14635488 T^{5} - 1293212528 T^{6} + 80862191466 T^{7} - 3110801971768 T^{8} + 80862191466 p^{2} T^{9} - 1293212528 p^{4} T^{10} - 14635488 p^{6} T^{11} + 2032911 p^{8} T^{12} - 71904 p^{10} T^{13} + 2300 p^{12} T^{14} - 42 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 34 T - 2145 T^{2} - 30850 T^{3} + 3634805 T^{4} + 5704116 T^{5} - 1780269014 T^{6} - 3524539760 T^{7} - 1686604222566 T^{8} - 3524539760 p^{2} T^{9} - 1780269014 p^{4} T^{10} + 5704116 p^{6} T^{11} + 3634805 p^{8} T^{12} - 30850 p^{10} T^{13} - 2145 p^{12} T^{14} + 34 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 2928 T^{2} + 8748060 T^{4} - 17665300368 T^{6} + 34276513378886 T^{8} - 17665300368 p^{4} T^{10} + 8748060 p^{8} T^{12} - 2928 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 43 | \( ( 1 + 20 T + 1495 T^{2} - 12412 T^{3} + 2043148 T^{4} - 12412 p^{2} T^{5} + 1495 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 47 | \( 1 + 4696 T^{2} - 843020 T^{4} - 14658249520 T^{6} + 68447569420394 T^{8} + 110538060039885928 T^{10} - \)\(44\!\cdots\!96\)\( T^{12} - \)\(22\!\cdots\!12\)\( T^{14} + \)\(23\!\cdots\!87\)\( T^{16} - \)\(22\!\cdots\!12\)\( p^{4} T^{18} - \)\(44\!\cdots\!96\)\( p^{8} T^{20} + 110538060039885928 p^{12} T^{22} + 68447569420394 p^{16} T^{24} - 14658249520 p^{20} T^{26} - 843020 p^{24} T^{28} + 4696 p^{28} T^{30} + p^{32} T^{32} \) | |
| 53 | \( 1 - 286 p T^{2} + 115836523 T^{4} - 636002551354 T^{6} + 2894917765070345 T^{8} - 11373981523270195724 T^{10} + \)\(39\!\cdots\!30\)\( T^{12} - \)\(12\!\cdots\!12\)\( T^{14} + \)\(36\!\cdots\!22\)\( T^{16} - \)\(12\!\cdots\!12\)\( p^{4} T^{18} + \)\(39\!\cdots\!30\)\( p^{8} T^{20} - 11373981523270195724 p^{12} T^{22} + 2894917765070345 p^{16} T^{24} - 636002551354 p^{20} T^{26} + 115836523 p^{24} T^{28} - 286 p^{29} T^{30} + p^{32} T^{32} \) | |
| 59 | \( 1 + 17810 T^{2} + 156138171 T^{4} + 16624955882 p T^{6} + 5254571132379353 T^{8} + 25367067778945476180 T^{10} + \)\(11\!\cdots\!22\)\( T^{12} + \)\(42\!\cdots\!40\)\( T^{14} + \)\(15\!\cdots\!46\)\( T^{16} + \)\(42\!\cdots\!40\)\( p^{4} T^{18} + \)\(11\!\cdots\!22\)\( p^{8} T^{20} + 25367067778945476180 p^{12} T^{22} + 5254571132379353 p^{16} T^{24} + 16624955882 p^{21} T^{26} + 156138171 p^{24} T^{28} + 17810 p^{28} T^{30} + p^{32} T^{32} \) | |
| 61 | \( ( 1 - 108 T + 17824 T^{2} - 1505088 T^{3} + 156521554 T^{4} - 10470661020 T^{5} + 866515376608 T^{6} - 50055017743404 T^{7} + 3585736190320435 T^{8} - 50055017743404 p^{2} T^{9} + 866515376608 p^{4} T^{10} - 10470661020 p^{6} T^{11} + 156521554 p^{8} T^{12} - 1505088 p^{10} T^{13} + 17824 p^{12} T^{14} - 108 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 28 T - 11639 T^{2} - 589540 T^{3} + 70030853 T^{4} + 3940623904 T^{5} - 222792035734 T^{6} - 9372718997992 T^{7} + 705843999464446 T^{8} - 9372718997992 p^{2} T^{9} - 222792035734 p^{4} T^{10} + 3940623904 p^{6} T^{11} + 70030853 p^{8} T^{12} - 589540 p^{10} T^{13} - 11639 p^{12} T^{14} + 28 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 71 | \( ( 1 + 33784 T^{2} + 525329932 T^{4} + 4912356704008 T^{6} + 30191341632221542 T^{8} + 4912356704008 p^{4} T^{10} + 525329932 p^{8} T^{12} + 33784 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 78 T + 11475 T^{2} - 736866 T^{3} + 60095337 T^{4} - 2492814828 T^{5} - 74210744370 T^{6} + 5556108007320 T^{7} - 1288705067919022 T^{8} + 5556108007320 p^{2} T^{9} - 74210744370 p^{4} T^{10} - 2492814828 p^{6} T^{11} + 60095337 p^{8} T^{12} - 736866 p^{10} T^{13} + 11475 p^{12} T^{14} - 78 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 14 T - 16532 T^{2} - 780584 T^{3} + 143583551 T^{4} + 8038302944 T^{5} - 644013235360 T^{6} - 27893453057414 T^{7} + 2749119151573864 T^{8} - 27893453057414 p^{2} T^{9} - 644013235360 p^{4} T^{10} + 8038302944 p^{6} T^{11} + 143583551 p^{8} T^{12} - 780584 p^{10} T^{13} - 16532 p^{12} T^{14} + 14 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 49746 T^{2} + 1116625161 T^{4} - 14746074482778 T^{6} + 125210198757764324 T^{8} - 14746074482778 p^{4} T^{10} + 1116625161 p^{8} T^{12} - 49746 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 89 | \( 1 + 17232 T^{2} + 163209636 T^{4} - 202837896864 T^{6} - 16120643080694966 T^{8} - \)\(18\!\cdots\!60\)\( T^{10} - \)\(38\!\cdots\!64\)\( T^{12} + \)\(78\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!83\)\( T^{16} + \)\(78\!\cdots\!28\)\( p^{4} T^{18} - \)\(38\!\cdots\!64\)\( p^{8} T^{20} - \)\(18\!\cdots\!60\)\( p^{12} T^{22} - 16120643080694966 p^{16} T^{24} - 202837896864 p^{20} T^{26} + 163209636 p^{24} T^{28} + 17232 p^{28} T^{30} + p^{32} T^{32} \) | |
| 97 | \( ( 1 - 26802 T^{2} + 578235537 T^{4} - 7769542604370 T^{6} + 87302247942617060 T^{8} - 7769542604370 p^{4} T^{10} + 578235537 p^{8} T^{12} - 26802 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
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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
−2.36302390947245037552698253769, −2.32749651334357679306110382853, −2.22085810251017245515887771503, −2.14832302174295915028268151229, −1.85393059390222362480452351062, −1.83931780748362234728933104097, −1.73603694574576617747006657342, −1.69376027832511673763876565562, −1.68146899684116583924509556192, −1.67876874860731797360117610716, −1.64219285715242441688205726446, −1.51697583352488770380053080603, −1.46827874232569200613348128202, −1.43100898459146027991940917527, −1.00210001873679269540354227623, −0.944534369693899586767846435702, −0.835183259157827505538364951883, −0.77675215911246644336049543061, −0.68129669546719909059054883811, −0.62602389351515285917770327554, −0.51957240817300006388533846855, −0.40695423232611017038036296031, −0.37342075598757384776306684917, −0.33667804440151667595043663123, −0.20206891283901335221544970258, 0.20206891283901335221544970258, 0.33667804440151667595043663123, 0.37342075598757384776306684917, 0.40695423232611017038036296031, 0.51957240817300006388533846855, 0.62602389351515285917770327554, 0.68129669546719909059054883811, 0.77675215911246644336049543061, 0.835183259157827505538364951883, 0.944534369693899586767846435702, 1.00210001873679269540354227623, 1.43100898459146027991940917527, 1.46827874232569200613348128202, 1.51697583352488770380053080603, 1.64219285715242441688205726446, 1.67876874860731797360117610716, 1.68146899684116583924509556192, 1.69376027832511673763876565562, 1.73603694574576617747006657342, 1.83931780748362234728933104097, 1.85393059390222362480452351062, 2.14832302174295915028268151229, 2.22085810251017245515887771503, 2.32749651334357679306110382853, 2.36302390947245037552698253769
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.