Dirichlet series
| L(s) = 1 | − 512·2-s + 8.74e3·3-s + 1.63e5·4-s − 1.40e5·5-s − 4.47e6·6-s − 4.54e5·7-s − 4.19e7·8-s + 4.78e7·9-s + 7.17e7·10-s − 5.32e7·11-s + 1.43e9·12-s − 2.50e8·13-s + 2.32e8·14-s − 1.22e9·15-s + 9.39e9·16-s + 1.64e9·17-s − 2.44e10·18-s + 4.42e9·19-s − 2.29e10·20-s − 3.97e9·21-s + 2.72e10·22-s + 2.14e10·23-s − 3.66e11·24-s − 3.54e10·25-s + 1.28e11·26-s + 2.09e11·27-s − 7.44e10·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 5·4-s − 0.802·5-s − 6.53·6-s − 0.208·7-s − 7.07·8-s + 10/3·9-s + 2.26·10-s − 0.824·11-s + 11.5·12-s − 1.10·13-s + 0.589·14-s − 1.85·15-s + 35/4·16-s + 0.969·17-s − 9.42·18-s + 1.13·19-s − 4.01·20-s − 0.481·21-s + 2.33·22-s + 1.31·23-s − 16.3·24-s − 1.16·25-s + 3.13·26-s + 3.84·27-s − 1.04·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+15/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(37015056\) = \(2^{4} \cdot 3^{4} \cdot 13^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.53460\times 10^{8}\) |
| Root analytic conductor: | \(10.5499\) |
| Motivic weight: | \(15\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 37015056,\ (\ :15/2, 15/2, 15/2, 15/2),\ 1)\) |
Particular Values
| \(L(8)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{17}{2})\) | 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 | $C_1$ | \( ( 1 + p^{7} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - p^{7} T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 + p^{7} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 140164 T + 11015187628 p T^{2} + 406251448925404 p^{2} T^{3} + 19036965043877042558 p^{3} T^{4} + 406251448925404 p^{17} T^{5} + 11015187628 p^{31} T^{6} + 140164 p^{45} T^{7} + p^{60} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 454364 T + 744864720988 p T^{2} + 225869897849026604 p^{2} T^{3} + \)\(13\!\cdots\!46\)\( p^{3} T^{4} + 225869897849026604 p^{17} T^{5} + 744864720988 p^{31} T^{6} + 454364 p^{45} T^{7} + p^{60} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 53279408 T + 16123123710605396 T^{2} + \)\(58\!\cdots\!48\)\( p T^{3} + \)\(82\!\cdots\!54\)\( p^{2} T^{4} + \)\(58\!\cdots\!48\)\( p^{16} T^{5} + 16123123710605396 p^{30} T^{6} + 53279408 p^{45} T^{7} + p^{60} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 1640673024 T + 3679140154626270284 T^{2} - \)\(33\!\cdots\!28\)\( T^{3} + \)\(12\!\cdots\!06\)\( T^{4} - \)\(33\!\cdots\!28\)\( p^{15} T^{5} + 3679140154626270284 p^{30} T^{6} - 1640673024 p^{45} T^{7} + p^{60} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 4423126956 T + 66618608249954644564 T^{2} - \)\(20\!\cdots\!96\)\( T^{3} + \)\(15\!\cdots\!06\)\( T^{4} - \)\(20\!\cdots\!96\)\( p^{15} T^{5} + 66618608249954644564 p^{30} T^{6} - 4423126956 p^{45} T^{7} + p^{60} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 21422358656 T + \)\(87\!\cdots\!92\)\( T^{2} - \)\(10\!\cdots\!52\)\( T^{3} + \)\(28\!\cdots\!18\)\( T^{4} - \)\(10\!\cdots\!52\)\( p^{15} T^{5} + \)\(87\!\cdots\!92\)\( p^{30} T^{6} - 21422358656 p^{45} T^{7} + p^{60} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 73688032744 T + \)\(20\!\cdots\!60\)\( T^{2} + \)\(15\!\cdots\!32\)\( T^{3} + \)\(23\!\cdots\!26\)\( T^{4} + \)\(15\!\cdots\!32\)\( p^{15} T^{5} + \)\(20\!\cdots\!60\)\( p^{30} T^{6} + 73688032744 p^{45} T^{7} + p^{60} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 79447997852 T + \)\(55\!\cdots\!60\)\( T^{2} + \)\(62\!\cdots\!04\)\( T^{3} + \)\(15\!\cdots\!86\)\( T^{4} + \)\(62\!\cdots\!04\)\( p^{15} T^{5} + \)\(55\!\cdots\!60\)\( p^{30} T^{6} + 79447997852 p^{45} T^{7} + p^{60} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 1314293411184 T + \)\(17\!\cdots\!76\)\( T^{2} + \)\(12\!\cdots\!16\)\( T^{3} + \)\(91\!\cdots\!38\)\( T^{4} + \)\(12\!\cdots\!16\)\( p^{15} T^{5} + \)\(17\!\cdots\!76\)\( p^{30} T^{6} + 1314293411184 p^{45} T^{7} + p^{60} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 833481381924 T - \)\(38\!\cdots\!80\)\( T^{2} + \)\(81\!\cdots\!36\)\( T^{3} + \)\(29\!\cdots\!34\)\( T^{4} + \)\(81\!\cdots\!36\)\( p^{15} T^{5} - \)\(38\!\cdots\!80\)\( p^{30} T^{6} + 833481381924 p^{45} T^{7} + p^{60} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 2610365702216 T + \)\(61\!\cdots\!40\)\( T^{2} + \)\(38\!\cdots\!16\)\( T^{3} + \)\(89\!\cdots\!70\)\( T^{4} + \)\(38\!\cdots\!16\)\( p^{15} T^{5} + \)\(61\!\cdots\!40\)\( p^{30} T^{6} + 2610365702216 p^{45} T^{7} + p^{60} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 7041910962632 T + \)\(49\!\cdots\!80\)\( T^{2} + \)\(19\!\cdots\!12\)\( T^{3} + \)\(81\!\cdots\!82\)\( T^{4} + \)\(19\!\cdots\!12\)\( p^{15} T^{5} + \)\(49\!\cdots\!80\)\( p^{30} T^{6} + 7041910962632 p^{45} T^{7} + p^{60} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 10307140760328 T + \)\(20\!\cdots\!08\)\( T^{2} + \)\(23\!\cdots\!88\)\( T^{3} + \)\(19\!\cdots\!14\)\( T^{4} + \)\(23\!\cdots\!88\)\( p^{15} T^{5} + \)\(20\!\cdots\!08\)\( p^{30} T^{6} + 10307140760328 p^{45} T^{7} + p^{60} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 23477859512984 T + \)\(11\!\cdots\!20\)\( T^{2} - \)\(20\!\cdots\!64\)\( T^{3} + \)\(61\!\cdots\!58\)\( T^{4} - \)\(20\!\cdots\!64\)\( p^{15} T^{5} + \)\(11\!\cdots\!20\)\( p^{30} T^{6} - 23477859512984 p^{45} T^{7} + p^{60} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 42648737909752 T + \)\(12\!\cdots\!96\)\( T^{2} - \)\(19\!\cdots\!16\)\( p T^{3} + \)\(13\!\cdots\!90\)\( T^{4} - \)\(19\!\cdots\!16\)\( p^{16} T^{5} + \)\(12\!\cdots\!96\)\( p^{30} T^{6} - 42648737909752 p^{45} T^{7} + p^{60} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 52056697403380 T + \)\(58\!\cdots\!48\)\( T^{2} - \)\(24\!\cdots\!24\)\( T^{3} + \)\(19\!\cdots\!62\)\( T^{4} - \)\(24\!\cdots\!24\)\( p^{15} T^{5} + \)\(58\!\cdots\!48\)\( p^{30} T^{6} - 52056697403380 p^{45} T^{7} + p^{60} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 94821926434144 T + \)\(23\!\cdots\!48\)\( T^{2} - \)\(15\!\cdots\!36\)\( T^{3} + \)\(20\!\cdots\!42\)\( T^{4} - \)\(15\!\cdots\!36\)\( p^{15} T^{5} + \)\(23\!\cdots\!48\)\( p^{30} T^{6} - 94821926434144 p^{45} T^{7} + p^{60} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 32530558183664 T + \)\(17\!\cdots\!16\)\( T^{2} + \)\(81\!\cdots\!16\)\( T^{3} + \)\(17\!\cdots\!26\)\( T^{4} + \)\(81\!\cdots\!16\)\( p^{15} T^{5} + \)\(17\!\cdots\!16\)\( p^{30} T^{6} - 32530558183664 p^{45} T^{7} + p^{60} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 213172391613776 T + \)\(11\!\cdots\!24\)\( T^{2} - \)\(17\!\cdots\!16\)\( T^{3} + \)\(50\!\cdots\!26\)\( T^{4} - \)\(17\!\cdots\!16\)\( p^{15} T^{5} + \)\(11\!\cdots\!24\)\( p^{30} T^{6} - 213172391613776 p^{45} T^{7} + p^{60} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 241368396972552 T + \)\(14\!\cdots\!64\)\( T^{2} + \)\(28\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!86\)\( T^{4} + \)\(28\!\cdots\!40\)\( p^{15} T^{5} + \)\(14\!\cdots\!64\)\( p^{30} T^{6} + 241368396972552 p^{45} T^{7} + p^{60} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 1067524687422444 T + \)\(55\!\cdots\!80\)\( T^{2} + \)\(10\!\cdots\!68\)\( T^{3} + \)\(12\!\cdots\!38\)\( T^{4} + \)\(10\!\cdots\!68\)\( p^{15} T^{5} + \)\(55\!\cdots\!80\)\( p^{30} T^{6} + 1067524687422444 p^{45} T^{7} + p^{60} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 610342177513504 T + \)\(88\!\cdots\!96\)\( T^{2} + \)\(91\!\cdots\!80\)\( T^{3} + \)\(85\!\cdots\!86\)\( T^{4} + \)\(91\!\cdots\!80\)\( p^{15} T^{5} + \)\(88\!\cdots\!96\)\( p^{30} T^{6} + 610342177513504 p^{45} T^{7} + p^{60} T^{8} \) | |
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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
−8.381407200668332539127211672833, −7.87734481419872881218722137218, −7.84931715666437521155894788760, −7.73269069879642102924879933686, −7.59608884774561065614568134825, −6.96006473713014230663257867805, −6.85793117342565494923857742641, −6.58601027168395180626704764753, −6.57645578640379655910966815082, −5.47895981519377502268073055817, −5.35901004679021225850762149096, −5.02707422202192331890065954990, −4.99835616294072546460388433420, −3.88955590163047505420916637724, −3.69331985307116473247280832920, −3.63140527188103547019739644127, −3.36643316844419139110511087784, −2.77081966255999622645422088516, −2.72968619350957962587061981099, −2.35762557243841077125767555846, −2.23760117750358497291441502071, −1.58692249693177047626471542950, −1.38439383377166534846742907785, −1.27785518214202403180412414992, −1.06784444591142689447844222028, 0, 0, 0, 0, 1.06784444591142689447844222028, 1.27785518214202403180412414992, 1.38439383377166534846742907785, 1.58692249693177047626471542950, 2.23760117750358497291441502071, 2.35762557243841077125767555846, 2.72968619350957962587061981099, 2.77081966255999622645422088516, 3.36643316844419139110511087784, 3.63140527188103547019739644127, 3.69331985307116473247280832920, 3.88955590163047505420916637724, 4.99835616294072546460388433420, 5.02707422202192331890065954990, 5.35901004679021225850762149096, 5.47895981519377502268073055817, 6.57645578640379655910966815082, 6.58601027168395180626704764753, 6.85793117342565494923857742641, 6.96006473713014230663257867805, 7.59608884774561065614568134825, 7.73269069879642102924879933686, 7.84931715666437521155894788760, 7.87734481419872881218722137218, 8.381407200668332539127211672833