Dirichlet series
| L(s) = 1 | − 64·2-s + 324·3-s + 2.56e3·4-s − 2.50e3·5-s − 2.07e4·6-s − 8.79e3·7-s − 8.19e4·8-s + 6.56e4·9-s + 1.60e5·10-s + 6.37e3·11-s + 8.29e5·12-s + 1.14e5·13-s + 5.62e5·14-s − 8.10e5·15-s + 2.29e6·16-s + 4.42e5·17-s − 4.19e6·18-s − 9.83e5·19-s − 6.40e6·20-s − 2.84e6·21-s − 4.07e5·22-s − 1.80e6·23-s − 2.65e7·24-s + 3.90e6·25-s − 7.31e6·26-s + 1.06e7·27-s − 2.25e7·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 5·4-s − 1.78·5-s − 6.53·6-s − 1.38·7-s − 7.07·8-s + 10/3·9-s + 5.05·10-s + 0.131·11-s + 11.5·12-s + 1.10·13-s + 3.91·14-s − 4.13·15-s + 35/4·16-s + 1.28·17-s − 9.42·18-s − 1.73·19-s − 8.94·20-s − 3.19·21-s − 0.371·22-s − 1.34·23-s − 16.3·24-s + 2·25-s − 3.13·26-s + 3.84·27-s − 6.92·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.62782\times 10^{9}\) |
| Root analytic conductor: | \(14.1726\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
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 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| 5 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| good | 7 | $C_2 \wr S_4$ | \( 1 + 8794 T + 12065275 p T^{2} + 13122312186 p^{2} T^{3} + 17021306770140 p^{3} T^{4} + 13122312186 p^{11} T^{5} + 12065275 p^{19} T^{6} + 8794 p^{27} T^{7} + p^{36} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 6374 T + 5029984709 T^{2} - 89280934439142 T^{3} + 14818582630197161676 T^{4} - 89280934439142 p^{9} T^{5} + 5029984709 p^{18} T^{6} - 6374 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 442802 T + 475150699593 T^{2} - 147330714698712762 T^{3} + \)\(84\!\cdots\!24\)\( T^{4} - 147330714698712762 p^{9} T^{5} + 475150699593 p^{18} T^{6} - 442802 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 983116 T + 1502338415056 T^{2} + 948202698801074732 T^{3} + \)\(76\!\cdots\!66\)\( T^{4} + 948202698801074732 p^{9} T^{5} + 1502338415056 p^{18} T^{6} + 983116 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 1803906 T + 3150202072037 T^{2} + 4515833905014192674 T^{3} + \)\(47\!\cdots\!24\)\( T^{4} + 4515833905014192674 p^{9} T^{5} + 3150202072037 p^{18} T^{6} + 1803906 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 131360 T + 1130379002988 p T^{2} + 21606170983041649056 T^{3} + \)\(53\!\cdots\!86\)\( T^{4} + 21606170983041649056 p^{9} T^{5} + 1130379002988 p^{19} T^{6} + 131360 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 5766812 T + 65376644592112 T^{2} + \)\(41\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!02\)\( T^{4} + \)\(41\!\cdots\!60\)\( p^{9} T^{5} + 65376644592112 p^{18} T^{6} + 5766812 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 884206 T + 367394450534033 T^{2} + \)\(39\!\cdots\!58\)\( T^{3} + \)\(60\!\cdots\!48\)\( T^{4} + \)\(39\!\cdots\!58\)\( p^{9} T^{5} + 367394450534033 p^{18} T^{6} - 884206 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 9417982 T + 872027324556513 T^{2} - \)\(27\!\cdots\!14\)\( T^{3} + \)\(33\!\cdots\!80\)\( T^{4} - \)\(27\!\cdots\!14\)\( p^{9} T^{5} + 872027324556513 p^{18} T^{6} - 9417982 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 32848920 T + 1563797704731708 T^{2} - \)\(31\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!02\)\( T^{4} - \)\(31\!\cdots\!36\)\( p^{9} T^{5} + 1563797704731708 p^{18} T^{6} - 32848920 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 43654412 T + 4762788596416896 T^{2} - \)\(13\!\cdots\!56\)\( T^{3} + \)\(80\!\cdots\!26\)\( T^{4} - \)\(13\!\cdots\!56\)\( p^{9} T^{5} + 4762788596416896 p^{18} T^{6} - 43654412 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 139512322 T + 18061041714785225 T^{2} - \)\(13\!\cdots\!82\)\( T^{3} + \)\(96\!\cdots\!52\)\( T^{4} - \)\(13\!\cdots\!82\)\( p^{9} T^{5} + 18061041714785225 p^{18} T^{6} - 139512322 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 67052752 T + 19538317267284796 T^{2} - \)\(58\!\cdots\!92\)\( T^{3} + \)\(18\!\cdots\!90\)\( T^{4} - \)\(58\!\cdots\!92\)\( p^{9} T^{5} + 19538317267284796 p^{18} T^{6} - 67052752 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 114920146 T + 34915558596436321 T^{2} + \)\(31\!\cdots\!86\)\( T^{3} + \)\(59\!\cdots\!40\)\( T^{4} + \)\(31\!\cdots\!86\)\( p^{9} T^{5} + 34915558596436321 p^{18} T^{6} + 114920146 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 220285232 T + 60882890717761788 T^{2} + \)\(11\!\cdots\!12\)\( T^{3} + \)\(22\!\cdots\!54\)\( T^{4} + \)\(11\!\cdots\!12\)\( p^{9} T^{5} + 60882890717761788 p^{18} T^{6} + 220285232 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 380309122 T + 143547998061528053 T^{2} + \)\(36\!\cdots\!34\)\( T^{3} + \)\(99\!\cdots\!80\)\( T^{4} + \)\(36\!\cdots\!34\)\( p^{9} T^{5} + 143547998061528053 p^{18} T^{6} + 380309122 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 149634936 T + 93426299132259404 T^{2} - \)\(22\!\cdots\!84\)\( T^{3} + \)\(29\!\cdots\!54\)\( T^{4} - \)\(22\!\cdots\!84\)\( p^{9} T^{5} + 93426299132259404 p^{18} T^{6} - 149634936 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 41054514 T + 153468926340733133 T^{2} - \)\(87\!\cdots\!62\)\( T^{3} - \)\(43\!\cdots\!80\)\( T^{4} - \)\(87\!\cdots\!62\)\( p^{9} T^{5} + 153468926340733133 p^{18} T^{6} + 41054514 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 139364640 T + 159513060425338076 T^{2} + \)\(39\!\cdots\!32\)\( T^{3} + \)\(17\!\cdots\!50\)\( T^{4} + \)\(39\!\cdots\!32\)\( p^{9} T^{5} + 159513060425338076 p^{18} T^{6} - 139364640 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 504813726 T + 1414320412703932937 T^{2} + \)\(52\!\cdots\!18\)\( T^{3} + \)\(74\!\cdots\!60\)\( T^{4} + \)\(52\!\cdots\!18\)\( p^{9} T^{5} + 1414320412703932937 p^{18} T^{6} + 504813726 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 3087579830 T + 5300116773867116049 T^{2} + \)\(66\!\cdots\!46\)\( T^{3} + \)\(65\!\cdots\!20\)\( T^{4} + \)\(66\!\cdots\!46\)\( p^{9} T^{5} + 5300116773867116049 p^{18} T^{6} + 3087579830 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−7.52126184709893929189073692089, −6.89062130825199446173849311038, −6.84220760461515139186632974506, −6.82597434438784488223037404396, −6.64745876427381812529875344728, −5.98157684848845235771074634442, −5.77258698832416188988346563628, −5.64773005529113281715690480064, −5.54894302526255248982068714879, −4.43246612014080427247510226609, −4.30229883915915725733944132822, −4.29101084681683456804222716370, −4.00902764992251830230302636863, −3.48742551944716354032743815253, −3.38622043016768102169913616711, −3.25713365262980177246045773153, −3.10262485251307405855068406434, −2.53815500576917342491144797292, −2.29489444740784924030763643860, −2.19286884668829066530993901690, −2.09069552329273074487601253034, −1.26835401615716236032526243291, −1.21183274278562668934480676281, −1.05385212730021988198811958411, −0.963218631458537494118416118886, 0, 0, 0, 0, 0.963218631458537494118416118886, 1.05385212730021988198811958411, 1.21183274278562668934480676281, 1.26835401615716236032526243291, 2.09069552329273074487601253034, 2.19286884668829066530993901690, 2.29489444740784924030763643860, 2.53815500576917342491144797292, 3.10262485251307405855068406434, 3.25713365262980177246045773153, 3.38622043016768102169913616711, 3.48742551944716354032743815253, 4.00902764992251830230302636863, 4.29101084681683456804222716370, 4.30229883915915725733944132822, 4.43246612014080427247510226609, 5.54894302526255248982068714879, 5.64773005529113281715690480064, 5.77258698832416188988346563628, 5.98157684848845235771074634442, 6.64745876427381812529875344728, 6.82597434438784488223037404396, 6.84220760461515139186632974506, 6.89062130825199446173849311038, 7.52126184709893929189073692089