Dirichlet series
| L(s) = 1 | − 256·2-s − 2.91e3·3-s + 4.09e4·4-s + 7.46e5·6-s − 1.39e5·7-s − 5.24e6·8-s + 5.31e6·9-s + 5.84e6·11-s − 1.19e8·12-s − 1.35e7·13-s + 3.58e7·14-s + 5.87e8·16-s − 8.61e7·17-s − 1.36e9·18-s + 2.65e8·19-s + 4.07e8·21-s − 1.49e9·22-s − 7.66e7·23-s + 1.52e10·24-s + 3.47e9·26-s − 7.74e9·27-s − 5.72e9·28-s + 5.04e9·29-s + 3.91e9·31-s − 6.01e10·32-s − 1.70e10·33-s + 2.20e10·34-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s + 6.53·6-s − 0.449·7-s − 7.07·8-s + 10/3·9-s + 0.995·11-s − 11.5·12-s − 0.780·13-s + 1.27·14-s + 35/4·16-s − 0.866·17-s − 9.42·18-s + 1.29·19-s + 1.03·21-s − 2.81·22-s − 0.107·23-s + 16.3·24-s + 2.20·26-s − 3.84·27-s − 2.24·28-s + 1.57·29-s + 0.791·31-s − 9.89·32-s − 2.29·33-s + 2.44·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(14-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.69337\times 10^{8}\) |
| Root analytic conductor: | \(12.6825\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 13/2, 13/2, 13/2, 13/2 ),\ 1 )\) |
Particular Values
| \(L(7)\) | \(\approx\) | \(0.2462005754\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2462005754\) |
| \(L(\frac{15}{2})\) | 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 | $C_1$ | \( ( 1 + p^{6} T )^{4} \) | |
| 5 | \( 1 \) | ||
| good | 7 | $C_2 \wr S_4$ | \( 1 + 139878 T + 86680013072 T^{2} - 4218062767835670 p T^{3} + 28529687515373479614 p^{2} T^{4} - 4218062767835670 p^{14} T^{5} + 86680013072 p^{26} T^{6} + 139878 p^{39} T^{7} + p^{52} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 5847318 T + 7704071953728 p T^{2} - 316149345059738346 p^{3} T^{3} + \)\(31\!\cdots\!70\)\( p^{3} T^{4} - 316149345059738346 p^{16} T^{5} + 7704071953728 p^{27} T^{6} - 5847318 p^{39} T^{7} + p^{52} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 1045098 p T + 640910685318928 T^{2} + \)\(10\!\cdots\!10\)\( p T^{3} + \)\(20\!\cdots\!46\)\( T^{4} + \)\(10\!\cdots\!10\)\( p^{14} T^{5} + 640910685318928 p^{26} T^{6} + 1045098 p^{40} T^{7} + p^{52} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 86186938 T + 16149577807552952 T^{2} + \)\(86\!\cdots\!10\)\( T^{3} + \)\(98\!\cdots\!66\)\( T^{4} + \)\(86\!\cdots\!10\)\( p^{13} T^{5} + 16149577807552952 p^{26} T^{6} + 86186938 p^{39} T^{7} + p^{52} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 265575360 T + 171710457984697036 T^{2} - \)\(30\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!86\)\( T^{4} - \)\(30\!\cdots\!20\)\( p^{13} T^{5} + 171710457984697036 p^{26} T^{6} - 265575360 p^{39} T^{7} + p^{52} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 76634464 T + 448845846248321468 T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!46\)\( T^{4} + \)\(17\!\cdots\!20\)\( p^{13} T^{5} + 448845846248321468 p^{26} T^{6} + 76634464 p^{39} T^{7} + p^{52} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 5044195890 T + 32732315055522991556 T^{2} - \)\(12\!\cdots\!30\)\( T^{3} + \)\(51\!\cdots\!26\)\( T^{4} - \)\(12\!\cdots\!30\)\( p^{13} T^{5} + 32732315055522991556 p^{26} T^{6} - 5044195890 p^{39} T^{7} + p^{52} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 3912055828 T + 37555833572363880508 T^{2} - \)\(57\!\cdots\!16\)\( T^{3} + \)\(44\!\cdots\!70\)\( T^{4} - \)\(57\!\cdots\!16\)\( p^{13} T^{5} + 37555833572363880508 p^{26} T^{6} - 3912055828 p^{39} T^{7} + p^{52} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 13026005442 T + \)\(20\!\cdots\!12\)\( T^{2} - \)\(54\!\cdots\!90\)\( T^{3} + \)\(11\!\cdots\!26\)\( T^{4} - \)\(54\!\cdots\!90\)\( p^{13} T^{5} + \)\(20\!\cdots\!12\)\( p^{26} T^{6} - 13026005442 p^{39} T^{7} + p^{52} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 434985228 p T + \)\(30\!\cdots\!48\)\( T^{2} - \)\(46\!\cdots\!36\)\( T^{3} + \)\(39\!\cdots\!70\)\( T^{4} - \)\(46\!\cdots\!36\)\( p^{13} T^{5} + \)\(30\!\cdots\!48\)\( p^{26} T^{6} - 434985228 p^{40} T^{7} + p^{52} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 19967004864 T + \)\(33\!\cdots\!08\)\( T^{2} + \)\(13\!\cdots\!40\)\( T^{3} + \)\(64\!\cdots\!26\)\( T^{4} + \)\(13\!\cdots\!40\)\( p^{13} T^{5} + \)\(33\!\cdots\!08\)\( p^{26} T^{6} + 19967004864 p^{39} T^{7} + p^{52} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 14674295108 T + \)\(11\!\cdots\!32\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(57\!\cdots\!86\)\( T^{4} - \)\(28\!\cdots\!20\)\( p^{13} T^{5} + \)\(11\!\cdots\!32\)\( p^{26} T^{6} + 14674295108 p^{39} T^{7} + p^{52} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 148695426 T + \)\(44\!\cdots\!08\)\( T^{2} + \)\(23\!\cdots\!70\)\( T^{3} + \)\(12\!\cdots\!06\)\( T^{4} + \)\(23\!\cdots\!70\)\( p^{13} T^{5} + \)\(44\!\cdots\!08\)\( p^{26} T^{6} - 148695426 p^{39} T^{7} + p^{52} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 8794032210 p T + \)\(50\!\cdots\!16\)\( T^{2} - \)\(16\!\cdots\!30\)\( T^{3} + \)\(84\!\cdots\!46\)\( T^{4} - \)\(16\!\cdots\!30\)\( p^{13} T^{5} + \)\(50\!\cdots\!16\)\( p^{26} T^{6} - 8794032210 p^{40} T^{7} + p^{52} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 251259255832 T + \)\(36\!\cdots\!08\)\( T^{2} + \)\(97\!\cdots\!24\)\( T^{3} + \)\(72\!\cdots\!70\)\( T^{4} + \)\(97\!\cdots\!24\)\( p^{13} T^{5} + \)\(36\!\cdots\!08\)\( p^{26} T^{6} + 251259255832 p^{39} T^{7} + p^{52} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 2514654315828 T + \)\(44\!\cdots\!92\)\( T^{2} + \)\(49\!\cdots\!80\)\( T^{3} + \)\(43\!\cdots\!46\)\( T^{4} + \)\(49\!\cdots\!80\)\( p^{13} T^{5} + \)\(44\!\cdots\!92\)\( p^{26} T^{6} + 2514654315828 p^{39} T^{7} + p^{52} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 949540288212 T + \)\(26\!\cdots\!48\)\( T^{2} + \)\(21\!\cdots\!04\)\( T^{3} + \)\(46\!\cdots\!70\)\( T^{4} + \)\(21\!\cdots\!04\)\( p^{13} T^{5} + \)\(26\!\cdots\!48\)\( p^{26} T^{6} + 949540288212 p^{39} T^{7} + p^{52} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 4011104951964 T + \)\(11\!\cdots\!68\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(34\!\cdots\!46\)\( T^{4} + \)\(22\!\cdots\!20\)\( p^{13} T^{5} + \)\(11\!\cdots\!68\)\( p^{26} T^{6} + 4011104951964 p^{39} T^{7} + p^{52} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 1584840509460 T + \)\(72\!\cdots\!56\)\( T^{2} + \)\(51\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!26\)\( T^{4} + \)\(51\!\cdots\!20\)\( p^{13} T^{5} + \)\(72\!\cdots\!56\)\( p^{26} T^{6} + 1584840509460 p^{39} T^{7} + p^{52} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 8074338864064 T + \)\(45\!\cdots\!88\)\( T^{2} + \)\(17\!\cdots\!80\)\( T^{3} + \)\(58\!\cdots\!86\)\( T^{4} + \)\(17\!\cdots\!80\)\( p^{13} T^{5} + \)\(45\!\cdots\!88\)\( p^{26} T^{6} + 8074338864064 p^{39} T^{7} + p^{52} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 8116316700840 T + \)\(75\!\cdots\!76\)\( T^{2} + \)\(35\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!66\)\( T^{4} + \)\(35\!\cdots\!80\)\( p^{13} T^{5} + \)\(75\!\cdots\!76\)\( p^{26} T^{6} + 8116316700840 p^{39} T^{7} + p^{52} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 4725419480208 T + \)\(17\!\cdots\!32\)\( T^{2} + \)\(85\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!86\)\( T^{4} + \)\(85\!\cdots\!80\)\( p^{13} T^{5} + \)\(17\!\cdots\!32\)\( p^{26} T^{6} + 4725419480208 p^{39} T^{7} + p^{52} T^{8} \) | |
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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
−7.13599482533882690918175579129, −6.85050700715774202477429796258, −6.62331982585796931875427728581, −6.52803352154884366965840903392, −6.34968676396073525364711549885, −5.88569918757722791770375997073, −5.55571534226450177129389983713, −5.53263159343816306507701583669, −5.26342387628576344280743389920, −4.56476174175430456253485828935, −4.33523700912225410709266860158, −4.23555711913719494463676881140, −4.01408000070111840315177414336, −3.08996201673159177706644385841, −2.85484001761556742722361506997, −2.79111360134014451363184297235, −2.73595370936268261371459824074, −1.68669210272087441458109327254, −1.65616724802373087435430565365, −1.49677422282887308133149022603, −1.39265823404593501449375423852, −0.75945657273886399487394428647, −0.65602923483267854874326064819, −0.41312058860770029314536945764, −0.18249808669071027298042502519, 0.18249808669071027298042502519, 0.41312058860770029314536945764, 0.65602923483267854874326064819, 0.75945657273886399487394428647, 1.39265823404593501449375423852, 1.49677422282887308133149022603, 1.65616724802373087435430565365, 1.68669210272087441458109327254, 2.73595370936268261371459824074, 2.79111360134014451363184297235, 2.85484001761556742722361506997, 3.08996201673159177706644385841, 4.01408000070111840315177414336, 4.23555711913719494463676881140, 4.33523700912225410709266860158, 4.56476174175430456253485828935, 5.26342387628576344280743389920, 5.53263159343816306507701583669, 5.55571534226450177129389983713, 5.88569918757722791770375997073, 6.34968676396073525364711549885, 6.52803352154884366965840903392, 6.62331982585796931875427728581, 6.85050700715774202477429796258, 7.13599482533882690918175579129