Dirichlet series
| L(s) = 1 | + 64·2-s + 2.56e3·4-s − 879·5-s + 9.60e3·7-s + 8.19e4·8-s − 5.62e4·10-s + 3.39e4·11-s − 1.12e5·13-s + 6.14e5·14-s + 2.29e6·16-s − 4.37e5·17-s − 6.44e5·19-s − 2.25e6·20-s + 2.17e6·22-s − 1.11e6·23-s − 2.26e6·25-s − 7.20e6·26-s + 2.45e7·28-s − 6.91e6·29-s − 8.09e6·31-s + 5.87e7·32-s − 2.80e7·34-s − 8.44e6·35-s − 2.22e7·37-s − 4.12e7·38-s − 7.20e7·40-s + 5.23e6·41-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s − 0.628·5-s + 1.51·7-s + 7.07·8-s − 1.77·10-s + 0.699·11-s − 1.09·13-s + 4.27·14-s + 35/4·16-s − 1.27·17-s − 1.13·19-s − 3.14·20-s + 1.97·22-s − 0.833·23-s − 1.16·25-s − 3.09·26-s + 7.55·28-s − 1.81·29-s − 1.57·31-s + 9.89·32-s − 3.59·34-s − 0.950·35-s − 1.95·37-s − 3.20·38-s − 4.44·40-s + 0.289·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{12} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.43653\times 10^{9}\) |
| Root analytic conductor: | \(13.9529\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | 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 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) |
| 3 | \( 1 \) | ||
| 7 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 879 T + 3038639 T^{2} + 5388443226 T^{3} + 1351938639936 p T^{4} + 5388443226 p^{9} T^{5} + 3038639 p^{18} T^{6} + 879 p^{27} T^{7} + p^{36} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 33954 T + 5258415812 T^{2} - 95060688219810 T^{3} + 14863908842376801126 T^{4} - 95060688219810 p^{9} T^{5} + 5258415812 p^{18} T^{6} - 33954 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 112609 T + 28051598080 T^{2} + 2871871832304331 T^{3} + \)\(39\!\cdots\!82\)\( T^{4} + 2871871832304331 p^{9} T^{5} + 28051598080 p^{18} T^{6} + 112609 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 437952 T + 505706485010 T^{2} + 149406267677248512 T^{3} + \)\(91\!\cdots\!07\)\( T^{4} + 149406267677248512 p^{9} T^{5} + 505706485010 p^{18} T^{6} + 437952 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 644464 T + 1209373413544 T^{2} + 535606765593101584 T^{3} + \)\(56\!\cdots\!46\)\( T^{4} + 535606765593101584 p^{9} T^{5} + 1209373413544 p^{18} T^{6} + 644464 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 1119183 T + 1346807335514 T^{2} + 3176177800727094975 T^{3} + \)\(68\!\cdots\!02\)\( T^{4} + 3176177800727094975 p^{9} T^{5} + 1346807335514 p^{18} T^{6} + 1119183 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 6915879 T + 74186000344778 T^{2} + \)\(31\!\cdots\!09\)\( T^{3} + \)\(17\!\cdots\!94\)\( T^{4} + \)\(31\!\cdots\!09\)\( p^{9} T^{5} + 74186000344778 p^{18} T^{6} + 6915879 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 8097031 T + 105464241151984 T^{2} + \)\(57\!\cdots\!71\)\( T^{3} + \)\(42\!\cdots\!66\)\( T^{4} + \)\(57\!\cdots\!71\)\( p^{9} T^{5} + 105464241151984 p^{18} T^{6} + 8097031 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 22282003 T + 5929298491753 p T^{2} - \)\(52\!\cdots\!42\)\( T^{3} - \)\(20\!\cdots\!46\)\( T^{4} - \)\(52\!\cdots\!42\)\( p^{9} T^{5} + 5929298491753 p^{19} T^{6} + 22282003 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 5233221 T + 922266190210793 T^{2} - \)\(41\!\cdots\!14\)\( T^{3} + \)\(42\!\cdots\!62\)\( T^{4} - \)\(41\!\cdots\!14\)\( p^{9} T^{5} + 922266190210793 p^{18} T^{6} - 5233221 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 13659382 T + 1239067654271524 T^{2} + \)\(15\!\cdots\!20\)\( T^{3} + \)\(89\!\cdots\!77\)\( T^{4} + \)\(15\!\cdots\!20\)\( p^{9} T^{5} + 1239067654271524 p^{18} T^{6} + 13659382 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 16787355 T + 283601054640389 T^{2} + \)\(86\!\cdots\!96\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} + \)\(86\!\cdots\!96\)\( p^{9} T^{5} + 283601054640389 p^{18} T^{6} + 16787355 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 42033297 T + 6934465036949774 T^{2} + \)\(37\!\cdots\!55\)\( T^{3} + \)\(29\!\cdots\!62\)\( T^{4} + \)\(37\!\cdots\!55\)\( p^{9} T^{5} + 6934465036949774 p^{18} T^{6} + 42033297 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 178647918 T + 39002555100084512 T^{2} - \)\(40\!\cdots\!68\)\( T^{3} + \)\(50\!\cdots\!57\)\( T^{4} - \)\(40\!\cdots\!68\)\( p^{9} T^{5} + 39002555100084512 p^{18} T^{6} - 178647918 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 166871080 T + 47105638347922912 T^{2} + \)\(51\!\cdots\!24\)\( T^{3} + \)\(82\!\cdots\!62\)\( T^{4} + \)\(51\!\cdots\!24\)\( p^{9} T^{5} + 47105638347922912 p^{18} T^{6} + 166871080 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 70493671 T + 64260918210942712 T^{2} + \)\(39\!\cdots\!11\)\( T^{3} + \)\(25\!\cdots\!10\)\( T^{4} + \)\(39\!\cdots\!11\)\( p^{9} T^{5} + 64260918210942712 p^{18} T^{6} + 70493671 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 255452733 T + 178877800478535044 T^{2} - \)\(33\!\cdots\!09\)\( T^{3} + \)\(12\!\cdots\!06\)\( T^{4} - \)\(33\!\cdots\!09\)\( p^{9} T^{5} + 178877800478535044 p^{18} T^{6} - 255452733 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 130022632 T + 127331176890092140 T^{2} + \)\(21\!\cdots\!68\)\( T^{3} + \)\(95\!\cdots\!30\)\( T^{4} + \)\(21\!\cdots\!68\)\( p^{9} T^{5} + 127331176890092140 p^{18} T^{6} + 130022632 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 237208751 T + 170790537082633921 T^{2} - \)\(77\!\cdots\!28\)\( T^{3} + \)\(19\!\cdots\!30\)\( T^{4} - \)\(77\!\cdots\!28\)\( p^{9} T^{5} + 170790537082633921 p^{18} T^{6} - 237208751 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 347827491 T + 377821226427633491 T^{2} + \)\(18\!\cdots\!20\)\( T^{3} + \)\(90\!\cdots\!16\)\( T^{4} + \)\(18\!\cdots\!20\)\( p^{9} T^{5} + 377821226427633491 p^{18} T^{6} + 347827491 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 231572535 T + 1026621733786945214 T^{2} + \)\(25\!\cdots\!45\)\( T^{3} + \)\(49\!\cdots\!26\)\( T^{4} + \)\(25\!\cdots\!45\)\( p^{9} T^{5} + 1026621733786945214 p^{18} T^{6} + 231572535 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 1550224882 T + 2085979997723249908 T^{2} + \)\(11\!\cdots\!42\)\( T^{3} + \)\(11\!\cdots\!94\)\( T^{4} + \)\(11\!\cdots\!42\)\( p^{9} T^{5} + 2085979997723249908 p^{18} T^{6} + 1550224882 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−7.28220143793313240798478365028, −6.83170633949306518827901606104, −6.44376055291744286675096837605, −6.34356005531107108283074514233, −6.32822857434155833960916973317, −5.60185538226800018344151685290, −5.47772418924702349957926450401, −5.42861826790430862428227495250, −5.28199527280253964825137317245, −4.80287968845476885972156134027, −4.61035543693400876613998405184, −4.32452202093521643407466538828, −4.29934142370614381587402298431, −3.79609817756114568160388824897, −3.74651198029595216621670523593, −3.58533801671873592178573542484, −3.42556318995768990688786883455, −2.63931464086437355239349357996, −2.45354882008002350066094104246, −2.32071120024522082595006385068, −2.25567577541712994424330471288, −1.60663721280327129526791808686, −1.60588925777601356746930064311, −1.37084770680647277956765640175, −1.15953479800755228143035851277, 0, 0, 0, 0, 1.15953479800755228143035851277, 1.37084770680647277956765640175, 1.60588925777601356746930064311, 1.60663721280327129526791808686, 2.25567577541712994424330471288, 2.32071120024522082595006385068, 2.45354882008002350066094104246, 2.63931464086437355239349357996, 3.42556318995768990688786883455, 3.58533801671873592178573542484, 3.74651198029595216621670523593, 3.79609817756114568160388824897, 4.29934142370614381587402298431, 4.32452202093521643407466538828, 4.61035543693400876613998405184, 4.80287968845476885972156134027, 5.28199527280253964825137317245, 5.42861826790430862428227495250, 5.47772418924702349957926450401, 5.60185538226800018344151685290, 6.32822857434155833960916973317, 6.34356005531107108283074514233, 6.44376055291744286675096837605, 6.83170633949306518827901606104, 7.28220143793313240798478365028