Dirichlet series
| L(s) = 1 | + 256·2-s + 2.91e3·3-s + 4.09e4·4-s − 5.96e4·5-s + 7.46e5·6-s − 3.47e5·7-s + 5.24e6·8-s + 5.31e6·9-s − 1.52e7·10-s + 3.99e6·11-s + 1.19e8·12-s − 1.41e7·13-s − 8.88e7·14-s − 1.73e8·15-s + 5.87e8·16-s − 1.98e8·17-s + 1.36e9·18-s − 1.88e8·19-s − 2.44e9·20-s − 1.01e9·21-s + 1.02e9·22-s − 8.90e8·23-s + 1.52e10·24-s − 2.82e8·25-s − 3.62e9·26-s + 7.74e9·27-s − 1.42e10·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 5·4-s − 1.70·5-s + 6.53·6-s − 1.11·7-s + 7.07·8-s + 10/3·9-s − 4.82·10-s + 0.679·11-s + 11.5·12-s − 0.813·13-s − 3.15·14-s − 3.94·15-s + 35/4·16-s − 1.99·17-s + 9.42·18-s − 0.917·19-s − 8.53·20-s − 2.57·21-s + 1.92·22-s − 1.25·23-s + 16.3·24-s − 0.231·25-s − 2.30·26-s + 3.84·27-s − 5.57·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(14-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 19^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.23305\times 10^{8}\) |
| Root analytic conductor: | \(11.0563\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 19^{4} ,\ ( \ : 13/2, 13/2, 13/2, 13/2 ),\ 1 )\) |
Particular Values
| \(L(7)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{15}{2})\) | 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 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) | |
| 19 | $C_1$ | \( ( 1 + p^{6} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 59656 T + 768323377 p T^{2} + 6415276690438 p^{2} T^{3} + 10600249233153784 p^{4} T^{4} + 6415276690438 p^{15} T^{5} + 768323377 p^{27} T^{6} + 59656 p^{39} T^{7} + p^{52} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 347178 T + 308920757137 T^{2} + 70591220899537374 T^{3} + \)\(55\!\cdots\!96\)\( p T^{4} + 70591220899537374 p^{13} T^{5} + 308920757137 p^{26} T^{6} + 347178 p^{39} T^{7} + p^{52} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 3990830 T + 7378729158727 p T^{2} - 3662569947548417858 p^{2} T^{3} + \)\(26\!\cdots\!80\)\( p^{3} T^{4} - 3662569947548417858 p^{15} T^{5} + 7378729158727 p^{27} T^{6} - 3990830 p^{39} T^{7} + p^{52} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 1089622 p T + 1087918316206408 T^{2} + \)\(82\!\cdots\!02\)\( p T^{3} + \)\(46\!\cdots\!90\)\( T^{4} + \)\(82\!\cdots\!02\)\( p^{14} T^{5} + 1087918316206408 p^{26} T^{6} + 1089622 p^{40} T^{7} + p^{52} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 198884958 T + 15766830301209053 T^{2} + \)\(20\!\cdots\!30\)\( T^{3} + \)\(31\!\cdots\!00\)\( T^{4} + \)\(20\!\cdots\!30\)\( p^{13} T^{5} + 15766830301209053 p^{26} T^{6} + 198884958 p^{39} T^{7} + p^{52} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 890361502 T + 949998482092829444 T^{2} + \)\(35\!\cdots\!26\)\( T^{3} + \)\(25\!\cdots\!58\)\( T^{4} + \)\(35\!\cdots\!26\)\( p^{13} T^{5} + 949998482092829444 p^{26} T^{6} + 890361502 p^{39} T^{7} + p^{52} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 4608498908 T + 27322103385343623188 T^{2} + \)\(92\!\cdots\!12\)\( T^{3} + \)\(33\!\cdots\!62\)\( T^{4} + \)\(92\!\cdots\!12\)\( p^{13} T^{5} + 27322103385343623188 p^{26} T^{6} + 4608498908 p^{39} T^{7} + p^{52} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 2889582466 T + 90762629199372970840 T^{2} - \)\(19\!\cdots\!62\)\( T^{3} + \)\(32\!\cdots\!14\)\( T^{4} - \)\(19\!\cdots\!62\)\( p^{13} T^{5} + 90762629199372970840 p^{26} T^{6} - 2889582466 p^{39} T^{7} + p^{52} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 14283620366 T + \)\(75\!\cdots\!32\)\( T^{2} + \)\(89\!\cdots\!22\)\( T^{3} + \)\(26\!\cdots\!58\)\( T^{4} + \)\(89\!\cdots\!22\)\( p^{13} T^{5} + \)\(75\!\cdots\!32\)\( p^{26} T^{6} + 14283620366 p^{39} T^{7} + p^{52} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 11437523984 T + \)\(29\!\cdots\!88\)\( T^{2} + \)\(28\!\cdots\!92\)\( T^{3} + \)\(37\!\cdots\!90\)\( T^{4} + \)\(28\!\cdots\!92\)\( p^{13} T^{5} + \)\(29\!\cdots\!88\)\( p^{26} T^{6} + 11437523984 p^{39} T^{7} + p^{52} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 92776634126 T + \)\(80\!\cdots\!93\)\( T^{2} + \)\(38\!\cdots\!38\)\( T^{3} + \)\(19\!\cdots\!28\)\( T^{4} + \)\(38\!\cdots\!38\)\( p^{13} T^{5} + \)\(80\!\cdots\!93\)\( p^{26} T^{6} + 92776634126 p^{39} T^{7} + p^{52} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 133438366420 T + \)\(15\!\cdots\!97\)\( T^{2} + \)\(11\!\cdots\!50\)\( T^{3} + \)\(91\!\cdots\!08\)\( T^{4} + \)\(11\!\cdots\!50\)\( p^{13} T^{5} + \)\(15\!\cdots\!97\)\( p^{26} T^{6} + 133438366420 p^{39} T^{7} + p^{52} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 103641883528 T + \)\(20\!\cdots\!40\)\( T^{2} + \)\(43\!\cdots\!88\)\( T^{3} + \)\(53\!\cdots\!74\)\( T^{4} + \)\(43\!\cdots\!88\)\( p^{13} T^{5} + \)\(20\!\cdots\!40\)\( p^{26} T^{6} + 103641883528 p^{39} T^{7} + p^{52} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 348482171580 T + \)\(36\!\cdots\!92\)\( T^{2} + \)\(91\!\cdots\!08\)\( T^{3} + \)\(53\!\cdots\!94\)\( T^{4} + \)\(91\!\cdots\!08\)\( p^{13} T^{5} + \)\(36\!\cdots\!92\)\( p^{26} T^{6} + 348482171580 p^{39} T^{7} + p^{52} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 1308096081982 T + \)\(10\!\cdots\!33\)\( T^{2} + \)\(57\!\cdots\!74\)\( T^{3} + \)\(25\!\cdots\!20\)\( T^{4} + \)\(57\!\cdots\!74\)\( p^{13} T^{5} + \)\(10\!\cdots\!33\)\( p^{26} T^{6} + 1308096081982 p^{39} T^{7} + p^{52} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 38242208400 T + \)\(73\!\cdots\!12\)\( T^{2} + \)\(42\!\cdots\!92\)\( T^{3} + \)\(35\!\cdots\!94\)\( T^{4} + \)\(42\!\cdots\!92\)\( p^{13} T^{5} + \)\(73\!\cdots\!12\)\( p^{26} T^{6} - 38242208400 p^{39} T^{7} + p^{52} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 449953124 p T + \)\(30\!\cdots\!64\)\( T^{2} + \)\(47\!\cdots\!88\)\( T^{3} + \)\(49\!\cdots\!46\)\( T^{4} + \)\(47\!\cdots\!88\)\( p^{13} T^{5} + \)\(30\!\cdots\!64\)\( p^{26} T^{6} + 449953124 p^{40} T^{7} + p^{52} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 1922685504006 T + \)\(46\!\cdots\!61\)\( T^{2} + \)\(86\!\cdots\!54\)\( T^{3} + \)\(10\!\cdots\!52\)\( T^{4} + \)\(86\!\cdots\!54\)\( p^{13} T^{5} + \)\(46\!\cdots\!61\)\( p^{26} T^{6} + 1922685504006 p^{39} T^{7} + p^{52} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 481756655050 T + \)\(74\!\cdots\!36\)\( T^{2} + \)\(92\!\cdots\!58\)\( T^{3} + \)\(44\!\cdots\!26\)\( T^{4} + \)\(92\!\cdots\!58\)\( p^{13} T^{5} + \)\(74\!\cdots\!36\)\( p^{26} T^{6} + 481756655050 p^{39} T^{7} + p^{52} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 6572852790456 T + \)\(50\!\cdots\!72\)\( T^{2} + \)\(18\!\cdots\!68\)\( T^{3} + \)\(74\!\cdots\!38\)\( T^{4} + \)\(18\!\cdots\!68\)\( p^{13} T^{5} + \)\(50\!\cdots\!72\)\( p^{26} T^{6} + 6572852790456 p^{39} T^{7} + p^{52} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 5783342737920 T + \)\(79\!\cdots\!32\)\( T^{2} + \)\(32\!\cdots\!08\)\( T^{3} + \)\(25\!\cdots\!94\)\( T^{4} + \)\(32\!\cdots\!08\)\( p^{13} T^{5} + \)\(79\!\cdots\!32\)\( p^{26} T^{6} + 5783342737920 p^{39} T^{7} + p^{52} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 3372587617112 T + \)\(65\!\cdots\!40\)\( T^{2} - \)\(79\!\cdots\!12\)\( T^{3} - \)\(25\!\cdots\!70\)\( T^{4} - \)\(79\!\cdots\!12\)\( p^{13} T^{5} + \)\(65\!\cdots\!40\)\( p^{26} T^{6} + 3372587617112 p^{39} T^{7} + p^{52} T^{8} \) | |
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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
−8.188402346383457559114813718430, −7.44733900927542437906027690884, −7.40253999169263942898686316302, −7.13437648810339166492514213805, −7.09807099487291396178282097001, −6.48913941159506443805882901741, −6.21975901907560748854008673263, −6.14307285236324178978552795312, −6.07842268959550165940369703143, −5.10801297132946494413512669315, −4.81860440240117391018820473353, −4.79052089066347852805911762494, −4.50554295648883018854721449733, −4.03111577276760424806776965930, −3.88739665637459866090210523719, −3.72278693689331114679067255089, −3.63287666085444402830616890708, −3.05923161596490066620528967967, −2.99018643735285315160591881826, −2.74782228940565497565397046440, −2.43608094504761755493298841633, −1.88351748426559606005411166519, −1.74670386437467745748611613536, −1.52001479883642303321376938392, −1.43254047896863195878089485413, 0, 0, 0, 0, 1.43254047896863195878089485413, 1.52001479883642303321376938392, 1.74670386437467745748611613536, 1.88351748426559606005411166519, 2.43608094504761755493298841633, 2.74782228940565497565397046440, 2.99018643735285315160591881826, 3.05923161596490066620528967967, 3.63287666085444402830616890708, 3.72278693689331114679067255089, 3.88739665637459866090210523719, 4.03111577276760424806776965930, 4.50554295648883018854721449733, 4.79052089066347852805911762494, 4.81860440240117391018820473353, 5.10801297132946494413512669315, 6.07842268959550165940369703143, 6.14307285236324178978552795312, 6.21975901907560748854008673263, 6.48913941159506443805882901741, 7.09807099487291396178282097001, 7.13437648810339166492514213805, 7.40253999169263942898686316302, 7.44733900927542437906027690884, 8.188402346383457559114813718430