Dirichlet series
| L(s) = 1 | − 84·3-s − 1.39e3·5-s − 1.23e4·7-s − 2.75e4·9-s + 1.04e5·11-s + 1.20e5·13-s + 1.17e5·15-s − 4.12e5·17-s − 5.21e5·19-s + 1.03e6·21-s − 3.01e6·23-s + 1.94e6·25-s − 1.06e6·27-s + 6.15e6·29-s − 1.27e7·31-s − 8.75e6·33-s + 1.71e7·35-s + 2.05e7·37-s − 1.01e7·39-s + 1.16e7·41-s − 7.69e6·43-s + 3.84e7·45-s + 3.15e7·47-s + 4.50e6·49-s + 3.46e7·51-s + 7.25e7·53-s − 1.45e8·55-s + ⋯ |
| L(s) = 1 | − 0.598·3-s − 0.998·5-s − 1.93·7-s − 1.40·9-s + 2.14·11-s + 1.17·13-s + 0.597·15-s − 1.19·17-s − 0.917·19-s + 1.15·21-s − 2.24·23-s + 0.996·25-s − 0.385·27-s + 1.61·29-s − 2.48·31-s − 1.28·33-s + 1.93·35-s + 1.79·37-s − 0.700·39-s + 0.642·41-s − 0.343·43-s + 1.39·45-s + 0.944·47-s + 0.111·49-s + 0.716·51-s + 1.26·53-s − 2.14·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{16} \cdot 19^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.00958\times 10^{8}\) |
| Root analytic conductor: | \(12.5128\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 2^{16} \cdot 19^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(0.6378952377\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6378952377\) |
| \(L(\frac{11}{2})\) | 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 | \( 1 \) | |
| 19 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 28 p T + 34625 T^{2} + 2096378 p T^{3} + 72317420 p^{2} T^{4} + 2096378 p^{10} T^{5} + 34625 p^{18} T^{6} + 28 p^{28} T^{7} + p^{36} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 279 p T - 206 p T^{2} + 65059389 p^{2} T^{3} + 62045170554 p^{3} T^{4} + 65059389 p^{11} T^{5} - 206 p^{19} T^{6} + 279 p^{28} T^{7} + p^{36} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 + 12307 T + 146953147 T^{2} + 177878821772 p T^{3} + 182915235883424 p^{2} T^{4} + 177878821772 p^{10} T^{5} + 146953147 p^{18} T^{6} + 12307 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 104249 T + 12830631428 T^{2} - 70344030474447 p T^{3} + 49503846191918961558 T^{4} - 70344030474447 p^{10} T^{5} + 12830631428 p^{18} T^{6} - 104249 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 120486 T + 11611428199 T^{2} - 1537985136135464 T^{3} + \)\(26\!\cdots\!96\)\( T^{4} - 1537985136135464 p^{9} T^{5} + 11611428199 p^{18} T^{6} - 120486 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 412139 T + 99978579665 T^{2} + 47122140457475886 T^{3} + \)\(22\!\cdots\!54\)\( T^{4} + 47122140457475886 p^{9} T^{5} + 99978579665 p^{18} T^{6} + 412139 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 3010300 T + 8067019202447 T^{2} + 26728667320812612 p^{2} T^{3} + \)\(22\!\cdots\!92\)\( T^{4} + 26728667320812612 p^{11} T^{5} + 8067019202447 p^{18} T^{6} + 3010300 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 6153240 T + 66652695791987 T^{2} - \)\(26\!\cdots\!04\)\( T^{3} + \)\(15\!\cdots\!08\)\( T^{4} - \)\(26\!\cdots\!04\)\( p^{9} T^{5} + 66652695791987 p^{18} T^{6} - 6153240 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 12774024 T + 157641860418028 T^{2} + \)\(10\!\cdots\!12\)\( T^{3} + \)\(69\!\cdots\!98\)\( T^{4} + \)\(10\!\cdots\!12\)\( p^{9} T^{5} + 157641860418028 p^{18} T^{6} + 12774024 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 20506048 T + 451640199111340 T^{2} - \)\(57\!\cdots\!32\)\( T^{3} + \)\(77\!\cdots\!30\)\( T^{4} - \)\(57\!\cdots\!32\)\( p^{9} T^{5} + 451640199111340 p^{18} T^{6} - 20506048 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 11620300 T + 777621524436344 T^{2} - \)\(46\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!26\)\( T^{4} - \)\(46\!\cdots\!00\)\( p^{9} T^{5} + 777621524436344 p^{18} T^{6} - 11620300 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 7698327 T + 1116374999600644 T^{2} + \)\(20\!\cdots\!79\)\( T^{3} + \)\(59\!\cdots\!74\)\( T^{4} + \)\(20\!\cdots\!79\)\( p^{9} T^{5} + 1116374999600644 p^{18} T^{6} + 7698327 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 31581083 T + 59729788569388 p T^{2} - \)\(86\!\cdots\!23\)\( T^{3} + \)\(43\!\cdots\!46\)\( T^{4} - \)\(86\!\cdots\!23\)\( p^{9} T^{5} + 59729788569388 p^{19} T^{6} - 31581083 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 72549422 T + 9181124351782847 T^{2} - \)\(40\!\cdots\!96\)\( T^{3} + \)\(36\!\cdots\!60\)\( T^{4} - \)\(40\!\cdots\!96\)\( p^{9} T^{5} + 9181124351782847 p^{18} T^{6} - 72549422 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 149234120 T + 38463872258171009 T^{2} - \)\(37\!\cdots\!70\)\( T^{3} + \)\(51\!\cdots\!56\)\( T^{4} - \)\(37\!\cdots\!70\)\( p^{9} T^{5} + 38463872258171009 p^{18} T^{6} - 149234120 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 129004373 T + 36134043355765018 T^{2} - \)\(36\!\cdots\!19\)\( T^{3} + \)\(61\!\cdots\!14\)\( T^{4} - \)\(36\!\cdots\!19\)\( p^{9} T^{5} + 36134043355765018 p^{18} T^{6} - 129004373 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 132595266 T + 50476271812135045 T^{2} + \)\(24\!\cdots\!06\)\( T^{3} + \)\(10\!\cdots\!12\)\( T^{4} + \)\(24\!\cdots\!06\)\( p^{9} T^{5} + 50476271812135045 p^{18} T^{6} + 132595266 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 47138482 T + 109274701906688168 T^{2} + \)\(43\!\cdots\!18\)\( T^{3} + \)\(54\!\cdots\!62\)\( T^{4} + \)\(43\!\cdots\!18\)\( p^{9} T^{5} + 109274701906688168 p^{18} T^{6} - 47138482 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 39332795 T + 147831632963514661 T^{2} - \)\(30\!\cdots\!82\)\( T^{3} + \)\(10\!\cdots\!82\)\( T^{4} - \)\(30\!\cdots\!82\)\( p^{9} T^{5} + 147831632963514661 p^{18} T^{6} + 39332795 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 307010840 T + 250097675403470728 T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(34\!\cdots\!42\)\( T^{4} - \)\(12\!\cdots\!20\)\( p^{9} T^{5} + 250097675403470728 p^{18} T^{6} - 307010840 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 746568232 T + 900563380389680840 T^{2} - \)\(42\!\cdots\!92\)\( T^{3} + \)\(26\!\cdots\!54\)\( T^{4} - \)\(42\!\cdots\!92\)\( p^{9} T^{5} + 900563380389680840 p^{18} T^{6} - 746568232 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 286943482 T + 949493363525023412 T^{2} - \)\(20\!\cdots\!34\)\( T^{3} + \)\(42\!\cdots\!38\)\( T^{4} - \)\(20\!\cdots\!34\)\( p^{9} T^{5} + 949493363525023412 p^{18} T^{6} - 286943482 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 793519958 T - 696016064440921328 T^{2} - \)\(23\!\cdots\!78\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} - \)\(23\!\cdots\!78\)\( p^{9} T^{5} - 696016064440921328 p^{18} T^{6} - 793519958 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−6.80715859736183390726353552076, −6.49307186886305444648294246166, −6.38666209660008428193485764707, −6.23960725519416946564721502182, −6.13153618781262389701001343879, −5.76412481469894656736823697907, −5.35433799012612295962919156778, −5.28740640121683661068689015258, −4.92919379019416666210630894597, −4.11463649793330364659045660893, −4.04842801061585416095675725414, −4.02213298955425410027426213333, −4.00776089930757694121618414001, −3.66244354619710327523136304973, −3.11990027886109754409993634304, −2.87866474983833328862427545482, −2.87546862138939864438561856409, −2.03769425949122112064498712382, −2.00436938204342424161334707989, −1.97693317435961445335773872256, −1.17364406135648021649684064042, −0.974012881045426772962236627111, −0.52599049246472262833301564235, −0.47021738668907560771795035954, −0.14697719436405866887815610436, 0.14697719436405866887815610436, 0.47021738668907560771795035954, 0.52599049246472262833301564235, 0.974012881045426772962236627111, 1.17364406135648021649684064042, 1.97693317435961445335773872256, 2.00436938204342424161334707989, 2.03769425949122112064498712382, 2.87546862138939864438561856409, 2.87866474983833328862427545482, 3.11990027886109754409993634304, 3.66244354619710327523136304973, 4.00776089930757694121618414001, 4.02213298955425410027426213333, 4.04842801061585416095675725414, 4.11463649793330364659045660893, 4.92919379019416666210630894597, 5.28740640121683661068689015258, 5.35433799012612295962919156778, 5.76412481469894656736823697907, 6.13153618781262389701001343879, 6.23960725519416946564721502182, 6.38666209660008428193485764707, 6.49307186886305444648294246166, 6.80715859736183390726353552076