Dirichlet series
| L(s) = 1 | + 512·2-s + 1.70e3·3-s + 1.63e5·4-s − 3.12e5·5-s + 8.72e5·6-s + 3.29e6·7-s + 4.19e7·8-s − 2.02e7·9-s − 1.60e8·10-s − 1.07e7·11-s + 2.79e8·12-s − 2.61e8·13-s + 1.68e9·14-s − 5.32e8·15-s + 9.39e9·16-s + 1.99e9·17-s − 1.03e10·18-s + 9.91e9·19-s − 5.12e10·20-s + 5.61e9·21-s − 5.49e9·22-s − 7.41e9·23-s + 7.15e10·24-s + 6.10e10·25-s − 1.33e11·26-s − 7.46e10·27-s + 5.39e11·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 0.450·3-s + 5·4-s − 1.78·5-s + 1.27·6-s + 1.51·7-s + 7.07·8-s − 1.41·9-s − 5.05·10-s − 0.166·11-s + 2.25·12-s − 1.15·13-s + 4.27·14-s − 0.805·15-s + 35/4·16-s + 1.17·17-s − 3.98·18-s + 2.54·19-s − 8.94·20-s + 0.680·21-s − 0.469·22-s − 0.454·23-s + 3.18·24-s + 2·25-s − 3.26·26-s − 1.37·27-s + 7.55·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+15/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(24010000\) = \(2^{4} \cdot 5^{4} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.95426\times 10^{7}\) |
| Root analytic conductor: | \(9.99427\) |
| Motivic weight: | \(15\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 24010000,\ (\ :15/2, 15/2, 15/2, 15/2),\ 1)\) |
Particular Values
| \(L(8)\) | \(\approx\) | \(95.05159693\) |
| \(L(\frac12)\) | \(\approx\) | \(95.05159693\) |
| \(L(\frac{17}{2})\) | 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 | $C_1$ | \( ( 1 - p^{7} T )^{4} \) |
| 5 | $C_1$ | \( ( 1 + p^{7} T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 - p^{7} T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 - 1705 T + 7714565 p T^{2} + 8886940 p^{4} T^{3} + 105135054704 p^{7} T^{4} + 8886940 p^{19} T^{5} + 7714565 p^{31} T^{6} - 1705 p^{45} T^{7} + p^{60} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 10735455 T + 199270642757281 p T^{2} - \)\(17\!\cdots\!80\)\( p^{3} T^{3} - \)\(12\!\cdots\!64\)\( p^{3} T^{4} - \)\(17\!\cdots\!80\)\( p^{18} T^{5} + 199270642757281 p^{31} T^{6} + 10735455 p^{45} T^{7} + p^{60} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 261103651 T + 6281710766565217 p T^{2} + \)\(65\!\cdots\!82\)\( p^{2} T^{3} + \)\(13\!\cdots\!70\)\( p^{3} T^{4} + \)\(65\!\cdots\!82\)\( p^{17} T^{5} + 6281710766565217 p^{31} T^{6} + 261103651 p^{45} T^{7} + p^{60} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 1993764411 T + 10518802983975492833 T^{2} - \)\(16\!\cdots\!10\)\( T^{3} + \)\(15\!\cdots\!34\)\( p^{2} T^{4} - \)\(16\!\cdots\!10\)\( p^{15} T^{5} + 10518802983975492833 p^{30} T^{6} - 1993764411 p^{45} T^{7} + p^{60} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 9918835754 T + 77337840478621026520 T^{2} - \)\(40\!\cdots\!78\)\( T^{3} + \)\(17\!\cdots\!58\)\( T^{4} - \)\(40\!\cdots\!78\)\( p^{15} T^{5} + 77337840478621026520 p^{30} T^{6} - 9918835754 p^{45} T^{7} + p^{60} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 7414203534 T + \)\(11\!\cdots\!32\)\( T^{2} - \)\(36\!\cdots\!06\)\( T^{3} - \)\(49\!\cdots\!50\)\( T^{4} - \)\(36\!\cdots\!06\)\( p^{15} T^{5} + \)\(11\!\cdots\!32\)\( p^{30} T^{6} + 7414203534 p^{45} T^{7} + p^{60} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 69577990773 T + \)\(28\!\cdots\!77\)\( T^{2} + \)\(14\!\cdots\!26\)\( T^{3} + \)\(35\!\cdots\!94\)\( T^{4} + \)\(14\!\cdots\!26\)\( p^{15} T^{5} + \)\(28\!\cdots\!77\)\( p^{30} T^{6} + 69577990773 p^{45} T^{7} + p^{60} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 80795631716 T + \)\(80\!\cdots\!32\)\( T^{2} - \)\(50\!\cdots\!60\)\( T^{3} + \)\(87\!\cdots\!06\)\( p T^{4} - \)\(50\!\cdots\!60\)\( p^{15} T^{5} + \)\(80\!\cdots\!32\)\( p^{30} T^{6} - 80795631716 p^{45} T^{7} + p^{60} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 553801813444 T + \)\(10\!\cdots\!48\)\( T^{2} + \)\(49\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!46\)\( T^{4} + \)\(49\!\cdots\!00\)\( p^{15} T^{5} + \)\(10\!\cdots\!48\)\( p^{30} T^{6} + 553801813444 p^{45} T^{7} + p^{60} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 4954637784606 T + \)\(12\!\cdots\!12\)\( T^{2} - \)\(23\!\cdots\!90\)\( T^{3} + \)\(34\!\cdots\!66\)\( T^{4} - \)\(23\!\cdots\!90\)\( p^{15} T^{5} + \)\(12\!\cdots\!12\)\( p^{30} T^{6} - 4954637784606 p^{45} T^{7} + p^{60} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 2229812162018 T + \)\(57\!\cdots\!80\)\( T^{2} - \)\(99\!\cdots\!42\)\( T^{3} + \)\(27\!\cdots\!26\)\( T^{4} - \)\(99\!\cdots\!42\)\( p^{15} T^{5} + \)\(57\!\cdots\!80\)\( p^{30} T^{6} - 2229812162018 p^{45} T^{7} + p^{60} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 4140013195197 T + \)\(29\!\cdots\!99\)\( T^{2} - \)\(58\!\cdots\!04\)\( T^{3} + \)\(75\!\cdots\!00\)\( p T^{4} - \)\(58\!\cdots\!04\)\( p^{15} T^{5} + \)\(29\!\cdots\!99\)\( p^{30} T^{6} - 4140013195197 p^{45} T^{7} + p^{60} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 11903963901978 T + \)\(22\!\cdots\!40\)\( T^{2} - \)\(12\!\cdots\!62\)\( T^{3} + \)\(17\!\cdots\!26\)\( T^{4} - \)\(12\!\cdots\!62\)\( p^{15} T^{5} + \)\(22\!\cdots\!40\)\( p^{30} T^{6} - 11903963901978 p^{45} T^{7} + p^{60} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 13479673577376 T + \)\(12\!\cdots\!40\)\( T^{2} - \)\(12\!\cdots\!72\)\( T^{3} + \)\(65\!\cdots\!18\)\( T^{4} - \)\(12\!\cdots\!72\)\( p^{15} T^{5} + \)\(12\!\cdots\!40\)\( p^{30} T^{6} - 13479673577376 p^{45} T^{7} + p^{60} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 9243311524034 T + \)\(11\!\cdots\!48\)\( T^{2} + \)\(57\!\cdots\!58\)\( p T^{3} + \)\(71\!\cdots\!94\)\( T^{4} + \)\(57\!\cdots\!58\)\( p^{16} T^{5} + \)\(11\!\cdots\!48\)\( p^{30} T^{6} - 9243311524034 p^{45} T^{7} + p^{60} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 52057914734696 T - \)\(10\!\cdots\!72\)\( T^{2} - \)\(74\!\cdots\!80\)\( T^{3} + \)\(75\!\cdots\!86\)\( T^{4} - \)\(74\!\cdots\!80\)\( p^{15} T^{5} - \)\(10\!\cdots\!72\)\( p^{30} T^{6} - 52057914734696 p^{45} T^{7} + p^{60} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 36316927861056 T + \)\(16\!\cdots\!32\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(21\!\cdots\!00\)\( p^{15} T^{5} + \)\(16\!\cdots\!32\)\( p^{30} T^{6} - 36316927861056 p^{45} T^{7} + p^{60} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 258794712136616 T + \)\(33\!\cdots\!32\)\( T^{2} - \)\(29\!\cdots\!56\)\( T^{3} + \)\(24\!\cdots\!50\)\( T^{4} - \)\(29\!\cdots\!56\)\( p^{15} T^{5} + \)\(33\!\cdots\!32\)\( p^{30} T^{6} - 258794712136616 p^{45} T^{7} + p^{60} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 219038268061787 T + \)\(66\!\cdots\!87\)\( T^{2} - \)\(54\!\cdots\!44\)\( T^{3} + \)\(14\!\cdots\!24\)\( T^{4} - \)\(54\!\cdots\!44\)\( p^{15} T^{5} + \)\(66\!\cdots\!87\)\( p^{30} T^{6} - 219038268061787 p^{45} T^{7} + p^{60} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 436995858013500 T + \)\(17\!\cdots\!80\)\( T^{2} - \)\(39\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!98\)\( T^{4} - \)\(39\!\cdots\!00\)\( p^{15} T^{5} + \)\(17\!\cdots\!80\)\( p^{30} T^{6} - 436995858013500 p^{45} T^{7} + p^{60} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 713112118708230 T + \)\(72\!\cdots\!96\)\( T^{2} - \)\(33\!\cdots\!10\)\( T^{3} + \)\(18\!\cdots\!06\)\( T^{4} - \)\(33\!\cdots\!10\)\( p^{15} T^{5} + \)\(72\!\cdots\!96\)\( p^{30} T^{6} - 713112118708230 p^{45} T^{7} + p^{60} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 72508683690623 T + \)\(24\!\cdots\!93\)\( T^{2} - \)\(14\!\cdots\!22\)\( T^{3} + \)\(22\!\cdots\!50\)\( T^{4} - \)\(14\!\cdots\!22\)\( p^{15} T^{5} + \)\(24\!\cdots\!93\)\( p^{30} T^{6} - 72508683690623 p^{45} T^{7} + p^{60} T^{8} \) | |
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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
−7.85302570221084563339493760935, −7.46879432588794368684238184623, −7.35485695486681395706704406616, −6.89433770730484142425726418325, −6.84638837924126591534716444866, −5.90110521330314758278159743072, −5.82796987083878144176740931851, −5.58899299072527219983534967025, −5.45166989693622548899411679251, −4.97218459525184718464815228852, −4.87868536694126895101633547838, −4.47124363823942621494728864116, −4.20023672736741583208176608338, −3.75571073870930186627898543886, −3.58742778360932458917255667804, −3.48006415971430808755865797679, −3.04793153848157530243692706231, −2.65709953781936695772831282050, −2.37007879192695459722962643660, −2.22599986641657258599779970609, −1.96604527776718661515054338874, −1.15353955771380276607075024342, −0.845806615260493064878269343478, −0.76277263044978692731564627512, −0.49125764447035734188898555583, 0.49125764447035734188898555583, 0.76277263044978692731564627512, 0.845806615260493064878269343478, 1.15353955771380276607075024342, 1.96604527776718661515054338874, 2.22599986641657258599779970609, 2.37007879192695459722962643660, 2.65709953781936695772831282050, 3.04793153848157530243692706231, 3.48006415971430808755865797679, 3.58742778360932458917255667804, 3.75571073870930186627898543886, 4.20023672736741583208176608338, 4.47124363823942621494728864116, 4.87868536694126895101633547838, 4.97218459525184718464815228852, 5.45166989693622548899411679251, 5.58899299072527219983534967025, 5.82796987083878144176740931851, 5.90110521330314758278159743072, 6.84638837924126591534716444866, 6.89433770730484142425726418325, 7.35485695486681395706704406616, 7.46879432588794368684238184623, 7.85302570221084563339493760935