Dirichlet series
| L(s) = 1 | − 1.15e4·2-s − 2.47e6·3-s − 2.24e7·4-s + 4.88e9·5-s + 2.85e10·6-s − 2.15e11·7-s + 1.35e12·8-s + 7.04e11·9-s − 5.63e13·10-s − 1.07e14·11-s + 5.54e13·12-s − 2.27e15·13-s + 2.48e15·14-s − 1.20e16·15-s − 4.29e15·16-s + 2.75e16·17-s − 8.13e15·18-s − 6.05e17·19-s − 1.09e17·20-s + 5.31e17·21-s + 1.24e18·22-s − 5.94e18·23-s − 3.34e18·24-s + 1.49e19·25-s + 2.63e19·26-s + 9.04e18·27-s + 4.82e18·28-s + ⋯ |
| L(s) = 1 | − 0.996·2-s − 0.895·3-s − 0.167·4-s + 1.78·5-s + 0.893·6-s − 0.838·7-s + 0.868·8-s + 0.0923·9-s − 1.78·10-s − 0.938·11-s + 0.149·12-s − 2.08·13-s + 0.836·14-s − 1.60·15-s − 0.238·16-s + 0.674·17-s − 0.0921·18-s − 3.30·19-s − 0.298·20-s + 0.751·21-s + 0.935·22-s − 2.46·23-s − 0.777·24-s + 2·25-s + 2.07·26-s + 0.429·27-s + 0.140·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+27/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(625\) = \(5^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(284383.\) |
| Root analytic conductor: | \(4.80549\) |
| Motivic weight: | \(27\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 625,\ (\ :27/2, 27/2, 27/2, 27/2),\ 1)\) |
Particular Values
| \(L(14)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{29}{2})\) | 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 | 5 | $C_1$ | \( ( 1 - p^{13} T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + 5775 p T + 608675 p^{8} T^{2} + 10807775 p^{16} T^{3} + 5676341913 p^{16} T^{4} + 10807775 p^{43} T^{5} + 608675 p^{62} T^{6} + 5775 p^{82} T^{7} + p^{108} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + 824600 p T + 22284607700 p^{5} T^{2} + 44210792331400 p^{10} T^{3} + 16607589661332651506 p^{13} T^{4} + 44210792331400 p^{37} T^{5} + 22284607700 p^{59} T^{6} + 824600 p^{82} T^{7} + p^{108} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 + 4388065000 p^{2} T + \)\(21\!\cdots\!00\)\( p^{2} T^{2} - \)\(17\!\cdots\!00\)\( p^{3} T^{3} + \)\(33\!\cdots\!02\)\( p^{6} T^{4} - \)\(17\!\cdots\!00\)\( p^{30} T^{5} + \)\(21\!\cdots\!00\)\( p^{56} T^{6} + 4388065000 p^{83} T^{7} + p^{108} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 107427307660512 T + \)\(40\!\cdots\!08\)\( p T^{2} + \)\(30\!\cdots\!84\)\( p^{2} T^{3} + \)\(57\!\cdots\!70\)\( p^{4} T^{4} + \)\(30\!\cdots\!84\)\( p^{29} T^{5} + \)\(40\!\cdots\!08\)\( p^{55} T^{6} + 107427307660512 p^{81} T^{7} + p^{108} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 175213927748200 p T + \)\(18\!\cdots\!00\)\( p^{2} T^{2} + \)\(65\!\cdots\!00\)\( p^{3} T^{3} + \)\(29\!\cdots\!98\)\( p^{4} T^{4} + \)\(65\!\cdots\!00\)\( p^{30} T^{5} + \)\(18\!\cdots\!00\)\( p^{56} T^{6} + 175213927748200 p^{82} T^{7} + p^{108} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 27536367778002600 T + \)\(60\!\cdots\!00\)\( T^{2} - \)\(70\!\cdots\!00\)\( p T^{3} + \)\(50\!\cdots\!22\)\( p^{2} T^{4} - \)\(70\!\cdots\!00\)\( p^{28} T^{5} + \)\(60\!\cdots\!00\)\( p^{54} T^{6} - 27536367778002600 p^{81} T^{7} + p^{108} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 605613718108524400 T + \)\(21\!\cdots\!56\)\( T^{2} + \)\(28\!\cdots\!00\)\( p T^{3} + \)\(30\!\cdots\!66\)\( p^{2} T^{4} + \)\(28\!\cdots\!00\)\( p^{28} T^{5} + \)\(21\!\cdots\!56\)\( p^{54} T^{6} + 605613718108524400 p^{81} T^{7} + p^{108} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 258575084590717800 p T + \)\(45\!\cdots\!00\)\( p^{2} T^{2} + \)\(64\!\cdots\!00\)\( p^{3} T^{3} + \)\(75\!\cdots\!98\)\( p^{4} T^{4} + \)\(64\!\cdots\!00\)\( p^{30} T^{5} + \)\(45\!\cdots\!00\)\( p^{56} T^{6} + 258575084590717800 p^{82} T^{7} + p^{108} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 71787455948677266600 T + \)\(10\!\cdots\!36\)\( T^{2} + \)\(58\!\cdots\!00\)\( T^{3} + \)\(48\!\cdots\!86\)\( T^{4} + \)\(58\!\cdots\!00\)\( p^{27} T^{5} + \)\(10\!\cdots\!36\)\( p^{54} T^{6} + 71787455948677266600 p^{81} T^{7} + p^{108} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + \)\(16\!\cdots\!52\)\( T + \)\(34\!\cdots\!08\)\( T^{2} - \)\(48\!\cdots\!96\)\( T^{3} + \)\(11\!\cdots\!70\)\( T^{4} - \)\(48\!\cdots\!96\)\( p^{27} T^{5} + \)\(34\!\cdots\!08\)\( p^{54} T^{6} + \)\(16\!\cdots\!52\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - \)\(77\!\cdots\!00\)\( T + \)\(84\!\cdots\!00\)\( T^{2} - \)\(48\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!78\)\( T^{4} - \)\(48\!\cdots\!00\)\( p^{27} T^{5} + \)\(84\!\cdots\!00\)\( p^{54} T^{6} - \)\(77\!\cdots\!00\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - \)\(13\!\cdots\!28\)\( T + \)\(18\!\cdots\!68\)\( T^{2} - \)\(14\!\cdots\!76\)\( T^{3} + \)\(10\!\cdots\!70\)\( T^{4} - \)\(14\!\cdots\!76\)\( p^{27} T^{5} + \)\(18\!\cdots\!68\)\( p^{54} T^{6} - \)\(13\!\cdots\!28\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + \)\(13\!\cdots\!00\)\( T + \)\(44\!\cdots\!00\)\( T^{2} + \)\(42\!\cdots\!00\)\( T^{3} + \)\(82\!\cdots\!98\)\( T^{4} + \)\(42\!\cdots\!00\)\( p^{27} T^{5} + \)\(44\!\cdots\!00\)\( p^{54} T^{6} + \)\(13\!\cdots\!00\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + \)\(14\!\cdots\!00\)\( T + \)\(11\!\cdots\!00\)\( T^{2} + \)\(63\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!38\)\( T^{4} + \)\(63\!\cdots\!00\)\( p^{27} T^{5} + \)\(11\!\cdots\!00\)\( p^{54} T^{6} + \)\(14\!\cdots\!00\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + \)\(23\!\cdots\!00\)\( T + \)\(43\!\cdots\!00\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!38\)\( T^{4} + \)\(10\!\cdots\!00\)\( p^{27} T^{5} + \)\(43\!\cdots\!00\)\( p^{54} T^{6} + \)\(23\!\cdots\!00\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - \)\(16\!\cdots\!00\)\( T + \)\(21\!\cdots\!76\)\( T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!66\)\( T^{4} - \)\(29\!\cdots\!00\)\( p^{27} T^{5} + \)\(21\!\cdots\!76\)\( p^{54} T^{6} - \)\(16\!\cdots\!00\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - \)\(28\!\cdots\!88\)\( T + \)\(80\!\cdots\!88\)\( T^{2} - \)\(13\!\cdots\!36\)\( T^{3} + \)\(20\!\cdots\!70\)\( T^{4} - \)\(13\!\cdots\!36\)\( p^{27} T^{5} + \)\(80\!\cdots\!88\)\( p^{54} T^{6} - \)\(28\!\cdots\!88\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + \)\(10\!\cdots\!00\)\( T + \)\(87\!\cdots\!00\)\( T^{2} + \)\(52\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!58\)\( T^{4} + \)\(52\!\cdots\!00\)\( p^{27} T^{5} + \)\(87\!\cdots\!00\)\( p^{54} T^{6} + \)\(10\!\cdots\!00\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - \)\(16\!\cdots\!68\)\( T + \)\(34\!\cdots\!48\)\( T^{2} - \)\(40\!\cdots\!16\)\( T^{3} + \)\(45\!\cdots\!70\)\( T^{4} - \)\(40\!\cdots\!16\)\( p^{27} T^{5} + \)\(34\!\cdots\!48\)\( p^{54} T^{6} - \)\(16\!\cdots\!68\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + \)\(18\!\cdots\!00\)\( T + \)\(87\!\cdots\!00\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(37\!\cdots\!66\)\( p T^{4} + \)\(10\!\cdots\!00\)\( p^{27} T^{5} + \)\(87\!\cdots\!00\)\( p^{54} T^{6} + \)\(18\!\cdots\!00\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + \)\(38\!\cdots\!00\)\( T + \)\(65\!\cdots\!36\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!86\)\( T^{4} + \)\(17\!\cdots\!00\)\( p^{27} T^{5} + \)\(65\!\cdots\!36\)\( p^{54} T^{6} + \)\(38\!\cdots\!00\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + \)\(35\!\cdots\!00\)\( T + \)\(60\!\cdots\!00\)\( T^{2} + \)\(67\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!58\)\( T^{4} + \)\(67\!\cdots\!00\)\( p^{27} T^{5} + \)\(60\!\cdots\!00\)\( p^{54} T^{6} + \)\(35\!\cdots\!00\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - \)\(51\!\cdots\!00\)\( T + \)\(25\!\cdots\!16\)\( T^{2} - \)\(70\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!46\)\( T^{4} - \)\(70\!\cdots\!00\)\( p^{27} T^{5} + \)\(25\!\cdots\!16\)\( p^{54} T^{6} - \)\(51\!\cdots\!00\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + \)\(90\!\cdots\!00\)\( T + \)\(16\!\cdots\!00\)\( T^{2} + \)\(90\!\cdots\!00\)\( T^{3} + \)\(97\!\cdots\!38\)\( T^{4} + \)\(90\!\cdots\!00\)\( p^{27} T^{5} + \)\(16\!\cdots\!00\)\( p^{54} T^{6} + \)\(90\!\cdots\!00\)\( p^{81} T^{7} + p^{108} T^{8} \) | |
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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
−12.69278402890317700240313179630, −11.96458774774300468427219438953, −11.25507282812652510678807777242, −10.93234987010626124115522750856, −10.23666517936782384566256991631, −10.16928272418296365723116610496, −9.898611133528876851345967820080, −9.388265511444021117592947741263, −9.266651108792605807378440153570, −8.264403747745638834962669940015, −8.225593358336636671553241578207, −7.62982015484408550703464721746, −6.93801418135339779160637816163, −6.55498132323741145539046070528, −6.00472494448965829369289926727, −5.82352782919270826490030915429, −5.29942206780169066722718645992, −4.87575117703384413375490745794, −4.32136601327995384498125927218, −3.87007075371049794182385065458, −2.90292323254855494762462876129, −2.55735092025425359870604158433, −2.02748686087092218438579694522, −1.71380610193653052328344350711, −1.37609273198182441622915377216, 0, 0, 0, 0, 1.37609273198182441622915377216, 1.71380610193653052328344350711, 2.02748686087092218438579694522, 2.55735092025425359870604158433, 2.90292323254855494762462876129, 3.87007075371049794182385065458, 4.32136601327995384498125927218, 4.87575117703384413375490745794, 5.29942206780169066722718645992, 5.82352782919270826490030915429, 6.00472494448965829369289926727, 6.55498132323741145539046070528, 6.93801418135339779160637816163, 7.62982015484408550703464721746, 8.225593358336636671553241578207, 8.264403747745638834962669940015, 9.266651108792605807378440153570, 9.388265511444021117592947741263, 9.898611133528876851345967820080, 10.16928272418296365723116610496, 10.23666517936782384566256991631, 10.93234987010626124115522750856, 11.25507282812652510678807777242, 11.96458774774300468427219438953, 12.69278402890317700240313179630