Dirichlet series
| L(s) = 1 | + 512·2-s + 1.63e5·4-s − 5.52e4·5-s − 3.29e6·7-s + 4.19e7·8-s − 2.82e7·10-s − 4.24e7·11-s + 2.25e8·13-s − 1.68e9·14-s + 9.39e9·16-s − 1.06e9·17-s + 1.49e9·19-s − 9.04e9·20-s − 2.17e10·22-s + 1.76e10·23-s − 7.15e10·25-s + 1.15e11·26-s − 5.39e11·28-s + 4.86e10·29-s − 6.69e10·31-s + 1.92e12·32-s − 5.46e11·34-s + 1.81e11·35-s − 1.65e11·37-s + 7.62e11·38-s − 2.31e12·40-s − 7.59e11·41-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s − 0.316·5-s − 1.51·7-s + 7.07·8-s − 0.893·10-s − 0.656·11-s + 0.996·13-s − 4.27·14-s + 35/4·16-s − 0.631·17-s + 0.382·19-s − 1.58·20-s − 1.85·22-s + 1.08·23-s − 2.34·25-s + 2.81·26-s − 7.55·28-s + 0.523·29-s − 0.437·31-s + 9.89·32-s − 1.78·34-s + 0.477·35-s − 0.285·37-s + 1.08·38-s − 2.23·40-s − 0.609·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(16-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+15/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{8} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.04495\times 10^{9}\) |
| Root analytic conductor: | \(13.4087\) |
| Motivic weight: | \(15\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 15/2, 15/2, 15/2, 15/2 ),\ 1 )\) |
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 | \( 1 \) | ||
| 7 | $C_1$ | \( ( 1 + p^{7} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 55216 T + 14928989764 p T^{2} + 113554168098448 p^{2} T^{3} + 24496758575590229918 p^{3} T^{4} + 113554168098448 p^{17} T^{5} + 14928989764 p^{31} T^{6} + 55216 p^{45} T^{7} + p^{60} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 3855472 p T + 1081594634834980 p T^{2} + \)\(29\!\cdots\!16\)\( p^{2} T^{3} + \)\(47\!\cdots\!18\)\( p^{3} T^{4} + \)\(29\!\cdots\!16\)\( p^{17} T^{5} + 1081594634834980 p^{31} T^{6} + 3855472 p^{46} T^{7} + p^{60} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 225416856 T + 5531186220502780 p T^{2} - \)\(56\!\cdots\!60\)\( p^{2} T^{3} + \)\(78\!\cdots\!82\)\( p^{3} T^{4} - \)\(56\!\cdots\!60\)\( p^{17} T^{5} + 5531186220502780 p^{31} T^{6} - 225416856 p^{45} T^{7} + p^{60} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 1067868368 T + 7405514702544767780 T^{2} + \)\(68\!\cdots\!88\)\( T^{3} + \)\(30\!\cdots\!82\)\( T^{4} + \)\(68\!\cdots\!88\)\( p^{15} T^{5} + 7405514702544767780 p^{30} T^{6} + 1067868368 p^{45} T^{7} + p^{60} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 1490154848 T + 12654090096320545228 T^{2} + \)\(87\!\cdots\!00\)\( T^{3} - \)\(10\!\cdots\!74\)\( T^{4} + \)\(87\!\cdots\!00\)\( p^{15} T^{5} + 12654090096320545228 p^{30} T^{6} - 1490154848 p^{45} T^{7} + p^{60} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 17651202640 T + \)\(36\!\cdots\!56\)\( T^{2} - \)\(16\!\cdots\!44\)\( T^{3} + \)\(37\!\cdots\!86\)\( T^{4} - \)\(16\!\cdots\!44\)\( p^{15} T^{5} + \)\(36\!\cdots\!56\)\( p^{30} T^{6} - 17651202640 p^{45} T^{7} + p^{60} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 48603833216 T + \)\(22\!\cdots\!40\)\( T^{2} - \)\(32\!\cdots\!72\)\( p T^{3} + \)\(26\!\cdots\!26\)\( T^{4} - \)\(32\!\cdots\!72\)\( p^{16} T^{5} + \)\(22\!\cdots\!40\)\( p^{30} T^{6} - 48603833216 p^{45} T^{7} + p^{60} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 66948471072 T + \)\(51\!\cdots\!92\)\( T^{2} - \)\(45\!\cdots\!64\)\( T^{3} + \)\(11\!\cdots\!30\)\( T^{4} - \)\(45\!\cdots\!64\)\( p^{15} T^{5} + \)\(51\!\cdots\!92\)\( p^{30} T^{6} + 66948471072 p^{45} T^{7} + p^{60} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 165011511304 T + \)\(90\!\cdots\!40\)\( T^{2} + \)\(99\!\cdots\!76\)\( T^{3} + \)\(40\!\cdots\!42\)\( T^{4} + \)\(99\!\cdots\!76\)\( p^{15} T^{5} + \)\(90\!\cdots\!40\)\( p^{30} T^{6} + 165011511304 p^{45} T^{7} + p^{60} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 759932074608 T + \)\(19\!\cdots\!40\)\( T^{2} + \)\(33\!\cdots\!56\)\( T^{3} + \)\(32\!\cdots\!06\)\( T^{4} + \)\(33\!\cdots\!56\)\( p^{15} T^{5} + \)\(19\!\cdots\!40\)\( p^{30} T^{6} + 759932074608 p^{45} T^{7} + p^{60} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 939420947792 T - \)\(65\!\cdots\!12\)\( T^{2} - \)\(21\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(21\!\cdots\!80\)\( p^{15} T^{5} - \)\(65\!\cdots\!12\)\( p^{30} T^{6} - 939420947792 p^{45} T^{7} + p^{60} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 3606135631392 T + \)\(43\!\cdots\!44\)\( T^{2} + \)\(12\!\cdots\!44\)\( T^{3} + \)\(76\!\cdots\!70\)\( T^{4} + \)\(12\!\cdots\!44\)\( p^{15} T^{5} + \)\(43\!\cdots\!44\)\( p^{30} T^{6} + 3606135631392 p^{45} T^{7} + p^{60} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 15571497448800 T + \)\(35\!\cdots\!00\)\( T^{2} + \)\(34\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!98\)\( T^{4} + \)\(34\!\cdots\!00\)\( p^{15} T^{5} + \)\(35\!\cdots\!00\)\( p^{30} T^{6} + 15571497448800 p^{45} T^{7} + p^{60} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 36433798220512 T + \)\(15\!\cdots\!08\)\( T^{2} + \)\(33\!\cdots\!20\)\( T^{3} + \)\(81\!\cdots\!26\)\( T^{4} + \)\(33\!\cdots\!20\)\( p^{15} T^{5} + \)\(15\!\cdots\!08\)\( p^{30} T^{6} + 36433798220512 p^{45} T^{7} + p^{60} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 31107553629432 T + \)\(25\!\cdots\!52\)\( T^{2} + \)\(53\!\cdots\!36\)\( T^{3} + \)\(23\!\cdots\!30\)\( T^{4} + \)\(53\!\cdots\!36\)\( p^{15} T^{5} + \)\(25\!\cdots\!52\)\( p^{30} T^{6} + 31107553629432 p^{45} T^{7} + p^{60} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 66339747458960 T + \)\(43\!\cdots\!16\)\( T^{2} + \)\(68\!\cdots\!40\)\( T^{3} + \)\(45\!\cdots\!78\)\( T^{4} + \)\(68\!\cdots\!40\)\( p^{15} T^{5} + \)\(43\!\cdots\!16\)\( p^{30} T^{6} + 66339747458960 p^{45} T^{7} + p^{60} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 35524074310928 T + \)\(14\!\cdots\!64\)\( T^{2} + \)\(26\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!86\)\( T^{4} + \)\(26\!\cdots\!80\)\( p^{15} T^{5} + \)\(14\!\cdots\!64\)\( p^{30} T^{6} + 35524074310928 p^{45} T^{7} + p^{60} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 214521476058344 T + \)\(25\!\cdots\!20\)\( T^{2} + \)\(30\!\cdots\!04\)\( T^{3} + \)\(34\!\cdots\!90\)\( T^{4} + \)\(30\!\cdots\!04\)\( p^{15} T^{5} + \)\(25\!\cdots\!20\)\( p^{30} T^{6} + 214521476058344 p^{45} T^{7} + p^{60} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 231821535874560 T + \)\(52\!\cdots\!40\)\( T^{2} + \)\(68\!\cdots\!48\)\( T^{3} + \)\(11\!\cdots\!10\)\( T^{4} + \)\(68\!\cdots\!48\)\( p^{15} T^{5} + \)\(52\!\cdots\!40\)\( p^{30} T^{6} + 231821535874560 p^{45} T^{7} + p^{60} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 359126111005312 T + \)\(49\!\cdots\!44\)\( T^{2} + \)\(18\!\cdots\!68\)\( T^{3} - \)\(91\!\cdots\!82\)\( T^{4} + \)\(18\!\cdots\!68\)\( p^{15} T^{5} + \)\(49\!\cdots\!44\)\( p^{30} T^{6} - 359126111005312 p^{45} T^{7} + p^{60} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 188429028383568 T + \)\(25\!\cdots\!16\)\( T^{2} - \)\(31\!\cdots\!48\)\( T^{3} + \)\(62\!\cdots\!50\)\( T^{4} - \)\(31\!\cdots\!48\)\( p^{15} T^{5} + \)\(25\!\cdots\!16\)\( p^{30} T^{6} - 188429028383568 p^{45} T^{7} + p^{60} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 1107546032911976 T + \)\(21\!\cdots\!88\)\( p T^{2} + \)\(16\!\cdots\!48\)\( T^{3} + \)\(19\!\cdots\!78\)\( T^{4} + \)\(16\!\cdots\!48\)\( p^{15} T^{5} + \)\(21\!\cdots\!88\)\( p^{31} T^{6} + 1107546032911976 p^{45} T^{7} + p^{60} T^{8} \) | |
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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
−7.67319860870259056436584665539, −6.90896175265079624738237442011, −6.89530896798471687327340575382, −6.74390297981812866454653427589, −6.47960144478503690426638131490, −5.94923377848807339261765242910, −5.93105267404337869993219696606, −5.75140942358404552278761292202, −5.59473790313626660152240431189, −4.93505773493003881440518145167, −4.80999325290647316653463984336, −4.61984430470710323595149039418, −4.41659606821310619128407566079, −3.82780582038470009401611119284, −3.60070920346351172655512021121, −3.58288996753616179316005476446, −3.50105716175697991100837358657, −2.82064410537578147751626034883, −2.64592570946543735598235119068, −2.61075112176779816528200034902, −2.40022274585459660060227791233, −1.59649984833867528220257453112, −1.43321098062408019840511674150, −1.29687933839997756636607192920, −1.21099340908143310690734819256, 0, 0, 0, 0, 1.21099340908143310690734819256, 1.29687933839997756636607192920, 1.43321098062408019840511674150, 1.59649984833867528220257453112, 2.40022274585459660060227791233, 2.61075112176779816528200034902, 2.64592570946543735598235119068, 2.82064410537578147751626034883, 3.50105716175697991100837358657, 3.58288996753616179316005476446, 3.60070920346351172655512021121, 3.82780582038470009401611119284, 4.41659606821310619128407566079, 4.61984430470710323595149039418, 4.80999325290647316653463984336, 4.93505773493003881440518145167, 5.59473790313626660152240431189, 5.75140942358404552278761292202, 5.93105267404337869993219696606, 5.94923377848807339261765242910, 6.47960144478503690426638131490, 6.74390297981812866454653427589, 6.89530896798471687327340575382, 6.90896175265079624738237442011, 7.67319860870259056436584665539