Dirichlet series
| L(s) = 1 | + 2.04e3·2-s + 7.87e4·3-s + 2.62e6·4-s − 5.28e6·5-s + 1.61e8·6-s − 1.33e8·7-s + 2.68e9·8-s + 3.87e9·9-s − 1.08e10·10-s − 6.61e9·11-s + 2.06e11·12-s + 4.24e10·13-s − 2.73e11·14-s − 4.16e11·15-s + 2.40e12·16-s − 2.81e11·17-s + 7.93e12·18-s − 2.82e12·19-s − 1.38e13·20-s − 1.05e13·21-s − 1.35e13·22-s + 7.08e12·23-s + 2.11e14·24-s − 2.79e13·25-s + 8.68e13·26-s + 1.52e14·27-s − 3.50e14·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 5·4-s − 1.21·5-s + 6.53·6-s − 1.25·7-s + 7.07·8-s + 10/3·9-s − 3.42·10-s − 0.846·11-s + 11.5·12-s + 1.10·13-s − 3.53·14-s − 2.79·15-s + 35/4·16-s − 0.575·17-s + 9.42·18-s − 2.00·19-s − 6.05·20-s − 2.89·21-s − 2.39·22-s + 0.819·23-s + 16.3·24-s − 1.46·25-s + 3.13·26-s + 3.84·27-s − 6.25·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+19/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(37015056\) = \(2^{4} \cdot 3^{4} \cdot 13^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.01468\times 10^{9}\) |
| Root analytic conductor: | \(13.3595\) |
| Motivic weight: | \(19\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 37015056,\ (\ :19/2, 19/2, 19/2, 19/2),\ 1)\) |
Particular Values
| \(L(10)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{21}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{9} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - p^{9} T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 - p^{9} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 1057126 p T + 2233847204208 p^{2} T^{2} + 1764982085322375234 p^{3} T^{3} + \)\(21\!\cdots\!26\)\( p^{4} T^{4} + 1764982085322375234 p^{22} T^{5} + 2233847204208 p^{40} T^{6} + 1057126 p^{58} T^{7} + p^{76} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 133621986 T + 4484131175622500 p T^{2} + \)\(38\!\cdots\!02\)\( p^{2} T^{3} + \)\(14\!\cdots\!78\)\( p^{4} T^{4} + \)\(38\!\cdots\!02\)\( p^{21} T^{5} + 4484131175622500 p^{39} T^{6} + 133621986 p^{57} T^{7} + p^{76} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 6617703860 T + \)\(14\!\cdots\!76\)\( T^{2} + \)\(13\!\cdots\!48\)\( p T^{3} + \)\(77\!\cdots\!90\)\( p^{2} T^{4} + \)\(13\!\cdots\!48\)\( p^{20} T^{5} + \)\(14\!\cdots\!76\)\( p^{38} T^{6} + 6617703860 p^{57} T^{7} + p^{76} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 281304945440 T + \)\(16\!\cdots\!60\)\( p T^{2} + \)\(46\!\cdots\!44\)\( p^{2} T^{3} + \)\(14\!\cdots\!94\)\( p^{3} T^{4} + \)\(46\!\cdots\!44\)\( p^{21} T^{5} + \)\(16\!\cdots\!60\)\( p^{39} T^{6} + 281304945440 p^{57} T^{7} + p^{76} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 148509483594 p T + \)\(71\!\cdots\!28\)\( T^{2} + \)\(59\!\cdots\!70\)\( p T^{3} + \)\(17\!\cdots\!62\)\( T^{4} + \)\(59\!\cdots\!70\)\( p^{20} T^{5} + \)\(71\!\cdots\!28\)\( p^{38} T^{6} + 148509483594 p^{58} T^{7} + p^{76} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 7080855130008 T + \)\(97\!\cdots\!72\)\( T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + \)\(32\!\cdots\!06\)\( T^{4} - \)\(22\!\cdots\!20\)\( p^{19} T^{5} + \)\(97\!\cdots\!72\)\( p^{38} T^{6} - 7080855130008 p^{57} T^{7} + p^{76} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 230151366838620 T + \)\(31\!\cdots\!76\)\( T^{2} + \)\(30\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!66\)\( T^{4} + \)\(30\!\cdots\!40\)\( p^{19} T^{5} + \)\(31\!\cdots\!76\)\( p^{38} T^{6} + 230151366838620 p^{57} T^{7} + p^{76} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 78872110802554 T + \)\(52\!\cdots\!16\)\( T^{2} + \)\(13\!\cdots\!38\)\( T^{3} + \)\(73\!\cdots\!22\)\( T^{4} + \)\(13\!\cdots\!38\)\( p^{19} T^{5} + \)\(52\!\cdots\!16\)\( p^{38} T^{6} + 78872110802554 p^{57} T^{7} + p^{76} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 56371510188684 T + \)\(10\!\cdots\!40\)\( T^{2} + \)\(24\!\cdots\!72\)\( T^{3} + \)\(90\!\cdots\!98\)\( T^{4} + \)\(24\!\cdots\!72\)\( p^{19} T^{5} + \)\(10\!\cdots\!40\)\( p^{38} T^{6} + 56371510188684 p^{57} T^{7} + p^{76} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 1999953686269094 T + \)\(97\!\cdots\!12\)\( T^{2} + \)\(18\!\cdots\!82\)\( T^{3} + \)\(63\!\cdots\!22\)\( T^{4} + \)\(18\!\cdots\!82\)\( p^{19} T^{5} + \)\(97\!\cdots\!12\)\( p^{38} T^{6} + 1999953686269094 p^{57} T^{7} + p^{76} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 53743147710480 p T + \)\(29\!\cdots\!24\)\( T^{2} - \)\(48\!\cdots\!60\)\( T^{3} + \)\(43\!\cdots\!38\)\( T^{4} - \)\(48\!\cdots\!60\)\( p^{19} T^{5} + \)\(29\!\cdots\!24\)\( p^{38} T^{6} - 53743147710480 p^{58} T^{7} + p^{76} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 593305688836832 T + \)\(92\!\cdots\!00\)\( T^{2} - \)\(46\!\cdots\!68\)\( T^{3} + \)\(47\!\cdots\!82\)\( T^{4} - \)\(46\!\cdots\!68\)\( p^{19} T^{5} + \)\(92\!\cdots\!00\)\( p^{38} T^{6} + 593305688836832 p^{57} T^{7} + p^{76} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 83356324797428312 T + \)\(21\!\cdots\!72\)\( T^{2} - \)\(82\!\cdots\!80\)\( T^{3} - \)\(12\!\cdots\!74\)\( T^{4} - \)\(82\!\cdots\!80\)\( p^{19} T^{5} + \)\(21\!\cdots\!72\)\( p^{38} T^{6} + 83356324797428312 p^{57} T^{7} + p^{76} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 53299266282870928 T + \)\(13\!\cdots\!92\)\( T^{2} + \)\(58\!\cdots\!36\)\( T^{3} + \)\(84\!\cdots\!34\)\( T^{4} + \)\(58\!\cdots\!36\)\( p^{19} T^{5} + \)\(13\!\cdots\!92\)\( p^{38} T^{6} + 53299266282870928 p^{57} T^{7} + p^{76} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 192521673929867700 T + \)\(32\!\cdots\!76\)\( T^{2} + \)\(33\!\cdots\!28\)\( T^{3} + \)\(35\!\cdots\!10\)\( T^{4} + \)\(33\!\cdots\!28\)\( p^{19} T^{5} + \)\(32\!\cdots\!76\)\( p^{38} T^{6} + 192521673929867700 p^{57} T^{7} + p^{76} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 677923304686247746 T + \)\(32\!\cdots\!68\)\( T^{2} + \)\(15\!\cdots\!30\)\( p T^{3} + \)\(27\!\cdots\!66\)\( T^{4} + \)\(15\!\cdots\!30\)\( p^{20} T^{5} + \)\(32\!\cdots\!68\)\( p^{38} T^{6} + 677923304686247746 p^{57} T^{7} + p^{76} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 751979312823206012 T + \)\(58\!\cdots\!24\)\( T^{2} - \)\(25\!\cdots\!16\)\( T^{3} + \)\(12\!\cdots\!66\)\( T^{4} - \)\(25\!\cdots\!16\)\( p^{19} T^{5} + \)\(58\!\cdots\!24\)\( p^{38} T^{6} - 751979312823206012 p^{57} T^{7} + p^{76} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 1250463885947279332 T + \)\(13\!\cdots\!24\)\( T^{2} + \)\(91\!\cdots\!20\)\( T^{3} + \)\(55\!\cdots\!86\)\( T^{4} + \)\(91\!\cdots\!20\)\( p^{19} T^{5} + \)\(13\!\cdots\!24\)\( p^{38} T^{6} + 1250463885947279332 p^{57} T^{7} + p^{76} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 1518877215887415832 T + \)\(32\!\cdots\!92\)\( T^{2} + \)\(15\!\cdots\!96\)\( T^{3} + \)\(37\!\cdots\!94\)\( T^{4} + \)\(15\!\cdots\!96\)\( p^{19} T^{5} + \)\(32\!\cdots\!92\)\( p^{38} T^{6} + 1518877215887415832 p^{57} T^{7} + p^{76} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 4984740214465560408 T + \)\(17\!\cdots\!76\)\( T^{2} + \)\(43\!\cdots\!36\)\( T^{3} + \)\(85\!\cdots\!38\)\( T^{4} + \)\(43\!\cdots\!36\)\( p^{19} T^{5} + \)\(17\!\cdots\!76\)\( p^{38} T^{6} + 4984740214465560408 p^{57} T^{7} + p^{76} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 10978502476734915526 T + \)\(70\!\cdots\!20\)\( T^{2} - \)\(30\!\cdots\!66\)\( T^{3} + \)\(12\!\cdots\!02\)\( p T^{4} - \)\(30\!\cdots\!66\)\( p^{19} T^{5} + \)\(70\!\cdots\!20\)\( p^{38} T^{6} - 10978502476734915526 p^{57} T^{7} + p^{76} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 7738572853018656108 T + \)\(21\!\cdots\!32\)\( p T^{2} - \)\(13\!\cdots\!16\)\( T^{3} + \)\(16\!\cdots\!70\)\( T^{4} - \)\(13\!\cdots\!16\)\( p^{19} T^{5} + \)\(21\!\cdots\!32\)\( p^{39} T^{6} - 7738572853018656108 p^{57} T^{7} + p^{76} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.86037318823018204236307435125, −7.29704572159823731616287078401, −7.19552570069776783207597323532, −6.92822844339524332674356197202, −6.57756694977957942055355644180, −6.15640773681150131106476613704, −5.98852858484406885132141921791, −5.91000373204266926663179972297, −5.54100822588673974724625967283, −4.82376540228602495076330060337, −4.62524123613979501614840315330, −4.46112248634109932331754474478, −4.44333709357121644297113641053, −3.66002398679094580177753449107, −3.64931200712865803645056798255, −3.63205177880286237054790263068, −3.39160982558588762674183754151, −3.11858219227889540504242455847, −2.71178849945468755520627894388, −2.43272601203532045198433702583, −2.37985198774273667691818599630, −1.75764341389742593885400416470, −1.66937374223182717785345669018, −1.45155684484364737540042546373, −1.25274373742216990853754555091, 0, 0, 0, 0, 1.25274373742216990853754555091, 1.45155684484364737540042546373, 1.66937374223182717785345669018, 1.75764341389742593885400416470, 2.37985198774273667691818599630, 2.43272601203532045198433702583, 2.71178849945468755520627894388, 3.11858219227889540504242455847, 3.39160982558588762674183754151, 3.63205177880286237054790263068, 3.64931200712865803645056798255, 3.66002398679094580177753449107, 4.44333709357121644297113641053, 4.46112248634109932331754474478, 4.62524123613979501614840315330, 4.82376540228602495076330060337, 5.54100822588673974724625967283, 5.91000373204266926663179972297, 5.98852858484406885132141921791, 6.15640773681150131106476613704, 6.57756694977957942055355644180, 6.92822844339524332674356197202, 7.19552570069776783207597323532, 7.29704572159823731616287078401, 7.86037318823018204236307435125