Dirichlet series
| L(s) = 1 | − 6·3-s + 41·9-s + 14·11-s − 82·17-s + 94·19-s − 17·25-s − 190·27-s − 84·33-s + 120·41-s − 40·43-s − 102·49-s + 492·51-s − 564·57-s − 62·59-s + 178·67-s + 54·73-s + 102·75-s + 898·81-s + 392·83-s − 26·89-s − 184·97-s + 574·99-s − 26·107-s + 248·113-s + 533·121-s − 720·123-s + 127-s + ⋯ |
| L(s) = 1 | − 2·3-s + 41/9·9-s + 1.27·11-s − 4.82·17-s + 4.94·19-s − 0.679·25-s − 7.03·27-s − 2.54·33-s + 2.92·41-s − 0.930·43-s − 2.08·49-s + 9.64·51-s − 9.89·57-s − 1.05·59-s + 2.65·67-s + 0.739·73-s + 1.35·75-s + 11.0·81-s + 4.72·83-s − 0.292·89-s − 1.89·97-s + 5.79·99-s − 0.242·107-s + 2.19·113-s + 4.40·121-s − 5.85·123-s + 0.00787·127-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{60} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.67298\times 10^{9}\) |
| Root analytic conductor: | \(2.47053\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{60} \cdot 7^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(1.953095513\) |
| \(L(\frac12)\) | \(\approx\) | \(1.953095513\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 102 T^{2} + 1209 p T^{4} + 1244 p^{3} T^{6} + 1209 p^{5} T^{8} + 102 p^{8} T^{10} + p^{12} T^{12} \) | |
| good | 3 | \( ( 1 + p T - 7 T^{2} - 22 T^{3} + 7 T^{4} - 61 T^{5} - 350 T^{6} - 61 p^{2} T^{7} + 7 p^{4} T^{8} - 22 p^{6} T^{9} - 7 p^{8} T^{10} + p^{11} T^{11} + p^{12} T^{12} )^{2} \) |
| 5 | \( 1 + 17 T^{2} - 437 T^{4} - 28164 T^{6} - 10839 p^{2} T^{8} + 5916803 T^{10} + 456152134 T^{12} + 5916803 p^{4} T^{14} - 10839 p^{10} T^{16} - 28164 p^{12} T^{18} - 437 p^{16} T^{20} + 17 p^{20} T^{22} + p^{24} T^{24} \) | |
| 11 | \( ( 1 - 7 T - 193 T^{2} + 2268 T^{3} + 15415 T^{4} - 1421 p^{2} T^{5} - 336170 T^{6} - 1421 p^{4} T^{7} + 15415 p^{4} T^{8} + 2268 p^{6} T^{9} - 193 p^{8} T^{10} - 7 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 13 | \( ( 1 - 950 T^{2} + 386271 T^{4} - 85941828 T^{6} + 386271 p^{4} T^{8} - 950 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 17 | \( ( 1 + 41 T + 471 T^{2} + 5456 T^{3} + 12309 p T^{4} + 1462951 T^{5} - 27854634 T^{6} + 1462951 p^{2} T^{7} + 12309 p^{5} T^{8} + 5456 p^{6} T^{9} + 471 p^{8} T^{10} + 41 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 19 | \( ( 1 - 47 T + 487 T^{2} - 8532 T^{3} + 642195 T^{4} - 10028861 T^{5} + 46257550 T^{6} - 10028861 p^{2} T^{7} + 642195 p^{4} T^{8} - 8532 p^{6} T^{9} + 487 p^{8} T^{10} - 47 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 23 | \( 1 + 2381 T^{2} + 2947123 T^{4} + 2827993812 T^{6} + 99941036331 p T^{8} + 1531108455942719 T^{10} + 862820921151363886 T^{12} + 1531108455942719 p^{4} T^{14} + 99941036331 p^{9} T^{16} + 2827993812 p^{12} T^{18} + 2947123 p^{16} T^{20} + 2381 p^{20} T^{22} + p^{24} T^{24} \) | |
| 29 | \( ( 1 - 1838 T^{2} + 2723567 T^{4} - 2423469812 T^{6} + 2723567 p^{4} T^{8} - 1838 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 31 | \( 1 + 3329 T^{2} + 4631887 T^{4} + 6475839496 T^{6} + 10686875408317 T^{8} + 11546553377441207 T^{10} + 9928800654054771830 T^{12} + 11546553377441207 p^{4} T^{14} + 10686875408317 p^{8} T^{16} + 6475839496 p^{12} T^{18} + 4631887 p^{16} T^{20} + 3329 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( 1 + 173 p T^{2} + 22091395 T^{4} + 55331146644 T^{6} + 111395987301489 T^{8} + 188098942339055675 T^{10} + \)\(27\!\cdots\!06\)\( T^{12} + 188098942339055675 p^{4} T^{14} + 111395987301489 p^{8} T^{16} + 55331146644 p^{12} T^{18} + 22091395 p^{16} T^{20} + 173 p^{21} T^{22} + p^{24} T^{24} \) | |
| 41 | \( ( 1 - 30 T + 4755 T^{2} - 100496 T^{3} + 4755 p^{2} T^{4} - 30 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 43 | \( ( 1 + 10 T + 3855 T^{2} + 53388 T^{3} + 3855 p^{2} T^{4} + 10 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 47 | \( 1 + 8313 T^{2} + 34702703 T^{4} + 101154992824 T^{6} + 247422958854189 T^{8} + 586359622432093519 T^{10} + \)\(13\!\cdots\!26\)\( T^{12} + 586359622432093519 p^{4} T^{14} + 247422958854189 p^{8} T^{16} + 101154992824 p^{12} T^{18} + 34702703 p^{16} T^{20} + 8313 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( 1 + 9273 T^{2} + 34839939 T^{4} + 130830291652 T^{6} + 620089921398753 T^{8} + 1876428047292173523 T^{10} + \)\(44\!\cdots\!94\)\( T^{12} + 1876428047292173523 p^{4} T^{14} + 620089921398753 p^{8} T^{16} + 130830291652 p^{12} T^{18} + 34839939 p^{16} T^{20} + 9273 p^{20} T^{22} + p^{24} T^{24} \) | |
| 59 | \( ( 1 + 31 T - 7567 T^{2} - 176030 T^{3} + 36214647 T^{4} + 412965247 T^{5} - 131958052142 T^{6} + 412965247 p^{2} T^{7} + 36214647 p^{4} T^{8} - 176030 p^{6} T^{9} - 7567 p^{8} T^{10} + 31 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 61 | \( 1 - 503 T^{2} - 13520125 T^{4} - 91919186268 T^{6} + 20373908662561 T^{8} + 680714329682975843 T^{10} + \)\(44\!\cdots\!06\)\( T^{12} + 680714329682975843 p^{4} T^{14} + 20373908662561 p^{8} T^{16} - 91919186268 p^{12} T^{18} - 13520125 p^{16} T^{20} - 503 p^{20} T^{22} + p^{24} T^{24} \) | |
| 67 | \( ( 1 - 89 T - 19 T^{2} + 251662 T^{3} - 15244389 T^{4} + 746405371 T^{5} - 32107721582 T^{6} + 746405371 p^{2} T^{7} - 15244389 p^{4} T^{8} + 251662 p^{6} T^{9} - 19 p^{8} T^{10} - 89 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 71 | \( ( 1 - 7830 T^{2} + 72414895 T^{4} - 385201987444 T^{6} + 72414895 p^{4} T^{8} - 7830 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 73 | \( ( 1 - 27 T - 12877 T^{2} + 216188 T^{3} + 104738937 T^{4} - 821298961 T^{5} - 619422939130 T^{6} - 821298961 p^{2} T^{7} + 104738937 p^{4} T^{8} + 216188 p^{6} T^{9} - 12877 p^{8} T^{10} - 27 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 79 | \( 1 + 27485 T^{2} + 389982499 T^{4} + 4285373244996 T^{6} + 40173777270526125 T^{8} + \)\(31\!\cdots\!47\)\( T^{10} + \)\(20\!\cdots\!50\)\( T^{12} + \)\(31\!\cdots\!47\)\( p^{4} T^{14} + 40173777270526125 p^{8} T^{16} + 4285373244996 p^{12} T^{18} + 389982499 p^{16} T^{20} + 27485 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( ( 1 - 98 T + 12183 T^{2} - 526716 T^{3} + 12183 p^{2} T^{4} - 98 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 89 | \( ( 1 + 13 T - 19181 T^{2} - 141748 T^{3} + 218594713 T^{4} + 635752183 T^{5} - 1943770650106 T^{6} + 635752183 p^{2} T^{7} + 218594713 p^{4} T^{8} - 141748 p^{6} T^{9} - 19181 p^{8} T^{10} + 13 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 97 | \( ( 1 + 46 T + 18423 T^{2} + 458352 T^{3} + 18423 p^{2} T^{4} + 46 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.97256433637790924193682420110, −3.89934490577362362251534907628, −3.85997961165616840996302178521, −3.65029997422377795657035263138, −3.59449405064223387114099222430, −3.32956178759877097052544287630, −3.31022766617557348395717155816, −3.27281725096903306202975248616, −3.21912957369588634093440978432, −2.69348968367828781808740602645, −2.67638798329395471783090056871, −2.53177490704446529448379826994, −2.37915131339159751132135411339, −2.33694330103633170611671393991, −1.98131575852895218766637724338, −1.98028145966925020246656176206, −1.81532487332953749779403490281, −1.54017641309873042957105057455, −1.48292988462864581631385363814, −1.25924873854389714736093802889, −1.13124862105895924576446937378, −0.75185866039655683022941711910, −0.67272534294187073252386857235, −0.62960669443335950223162762254, −0.14416254364885373163806064266, 0.14416254364885373163806064266, 0.62960669443335950223162762254, 0.67272534294187073252386857235, 0.75185866039655683022941711910, 1.13124862105895924576446937378, 1.25924873854389714736093802889, 1.48292988462864581631385363814, 1.54017641309873042957105057455, 1.81532487332953749779403490281, 1.98028145966925020246656176206, 1.98131575852895218766637724338, 2.33694330103633170611671393991, 2.37915131339159751132135411339, 2.53177490704446529448379826994, 2.67638798329395471783090056871, 2.69348968367828781808740602645, 3.21912957369588634093440978432, 3.27281725096903306202975248616, 3.31022766617557348395717155816, 3.32956178759877097052544287630, 3.59449405064223387114099222430, 3.65029997422377795657035263138, 3.85997961165616840996302178521, 3.89934490577362362251534907628, 3.97256433637790924193682420110
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.