Dirichlet series
| L(s) = 1 | + 2.62e4·3-s − 4.48e5·5-s − 2.30e7·7-s + 4.30e8·9-s + 1.18e9·11-s − 2.25e9·13-s − 1.17e10·15-s − 3.50e10·17-s + 1.19e9·19-s − 6.05e11·21-s − 1.38e11·23-s − 1.43e12·25-s + 5.64e12·27-s + 4.75e12·29-s + 3.54e12·31-s + 3.10e13·33-s + 1.03e13·35-s + 5.56e12·37-s − 5.91e13·39-s − 2.83e13·41-s − 5.02e13·43-s − 1.93e14·45-s − 4.22e14·47-s + 3.32e14·49-s − 9.19e14·51-s − 6.13e14·53-s − 5.30e14·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.513·5-s − 1.51·7-s + 10/3·9-s + 1.66·11-s − 0.766·13-s − 1.18·15-s − 1.21·17-s + 0.0161·19-s − 3.49·21-s − 0.367·23-s − 1.87·25-s + 3.84·27-s + 1.76·29-s + 0.747·31-s + 3.83·33-s + 0.776·35-s + 0.260·37-s − 1.77·39-s − 0.554·41-s − 0.656·43-s − 1.71·45-s − 2.58·47-s + 10/7·49-s − 2.81·51-s − 1.35·53-s − 0.853·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+17/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(49787136\) = \(2^{8} \cdot 3^{4} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.61084\times 10^{8}\) |
| Root analytic conductor: | \(12.4059\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 49787136,\ (\ :17/2, 17/2, 17/2, 17/2),\ 1)\) |
Particular Values
| \(L(9)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{19}{2})\) | 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 | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 - p^{8} T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 + p^{8} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 448644 T + 326868783492 p T^{2} + 1112798574194028 p^{4} T^{3} + \)\(49\!\cdots\!58\)\( p^{5} T^{4} + 1112798574194028 p^{21} T^{5} + 326868783492 p^{35} T^{6} + 448644 p^{51} T^{7} + p^{68} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 1181407500 T + 178083558240066324 p T^{2} - \)\(12\!\cdots\!40\)\( p^{2} T^{3} + \)\(10\!\cdots\!30\)\( p^{3} T^{4} - \)\(12\!\cdots\!40\)\( p^{19} T^{5} + 178083558240066324 p^{35} T^{6} - 1181407500 p^{51} T^{7} + p^{68} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 173434632 p T + 22227327802714452364 T^{2} + \)\(32\!\cdots\!56\)\( p T^{3} + \)\(14\!\cdots\!62\)\( p^{2} T^{4} + \)\(32\!\cdots\!56\)\( p^{18} T^{5} + 22227327802714452364 p^{34} T^{6} + 173434632 p^{52} T^{7} + p^{68} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 35032850604 T + 68664540002353545924 p T^{2} - \)\(18\!\cdots\!92\)\( T^{3} - \)\(23\!\cdots\!42\)\( T^{4} - \)\(18\!\cdots\!92\)\( p^{17} T^{5} + 68664540002353545924 p^{35} T^{6} + 35032850604 p^{51} T^{7} + p^{68} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 63073496 p T + \)\(89\!\cdots\!00\)\( T^{2} - \)\(26\!\cdots\!92\)\( T^{3} + \)\(60\!\cdots\!06\)\( T^{4} - \)\(26\!\cdots\!92\)\( p^{17} T^{5} + \)\(89\!\cdots\!00\)\( p^{34} T^{6} - 63073496 p^{52} T^{7} + p^{68} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 138142670436 T + \)\(36\!\cdots\!04\)\( T^{2} + \)\(55\!\cdots\!28\)\( T^{3} + \)\(67\!\cdots\!98\)\( T^{4} + \)\(55\!\cdots\!28\)\( p^{17} T^{5} + \)\(36\!\cdots\!04\)\( p^{34} T^{6} + 138142670436 p^{51} T^{7} + p^{68} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 4751074860096 T + \)\(28\!\cdots\!88\)\( T^{2} - \)\(94\!\cdots\!12\)\( T^{3} + \)\(30\!\cdots\!10\)\( T^{4} - \)\(94\!\cdots\!12\)\( p^{17} T^{5} + \)\(28\!\cdots\!88\)\( p^{34} T^{6} - 4751074860096 p^{51} T^{7} + p^{68} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 3549041900856 T + \)\(44\!\cdots\!12\)\( T^{2} - \)\(20\!\cdots\!12\)\( T^{3} + \)\(13\!\cdots\!58\)\( T^{4} - \)\(20\!\cdots\!12\)\( p^{17} T^{5} + \)\(44\!\cdots\!12\)\( p^{34} T^{6} - 3549041900856 p^{51} T^{7} + p^{68} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 5568436246472 T + \)\(15\!\cdots\!96\)\( T^{2} - \)\(19\!\cdots\!08\)\( p T^{3} + \)\(10\!\cdots\!86\)\( T^{4} - \)\(19\!\cdots\!08\)\( p^{18} T^{5} + \)\(15\!\cdots\!96\)\( p^{34} T^{6} - 5568436246472 p^{51} T^{7} + p^{68} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 28341130943076 T + \)\(36\!\cdots\!92\)\( T^{2} + \)\(95\!\cdots\!88\)\( T^{3} + \)\(14\!\cdots\!82\)\( T^{4} + \)\(95\!\cdots\!88\)\( p^{17} T^{5} + \)\(36\!\cdots\!92\)\( p^{34} T^{6} + 28341130943076 p^{51} T^{7} + p^{68} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 50291277184288 T + \)\(15\!\cdots\!28\)\( T^{2} + \)\(45\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!98\)\( T^{4} + \)\(45\!\cdots\!84\)\( p^{17} T^{5} + \)\(15\!\cdots\!28\)\( p^{34} T^{6} + 50291277184288 p^{51} T^{7} + p^{68} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 422196808118280 T + \)\(83\!\cdots\!60\)\( T^{2} + \)\(79\!\cdots\!92\)\( T^{3} + \)\(73\!\cdots\!22\)\( T^{4} + \)\(79\!\cdots\!92\)\( p^{17} T^{5} + \)\(83\!\cdots\!60\)\( p^{34} T^{6} + 422196808118280 p^{51} T^{7} + p^{68} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 613391348268456 T + \)\(66\!\cdots\!48\)\( T^{2} + \)\(28\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!90\)\( T^{4} + \)\(28\!\cdots\!80\)\( p^{17} T^{5} + \)\(66\!\cdots\!48\)\( p^{34} T^{6} + 613391348268456 p^{51} T^{7} + p^{68} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 1327961664592920 T + \)\(21\!\cdots\!08\)\( T^{2} + \)\(94\!\cdots\!68\)\( T^{3} + \)\(18\!\cdots\!22\)\( T^{4} + \)\(94\!\cdots\!68\)\( p^{17} T^{5} + \)\(21\!\cdots\!08\)\( p^{34} T^{6} + 1327961664592920 p^{51} T^{7} + p^{68} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 638673818939760 T + \)\(28\!\cdots\!72\)\( T^{2} - \)\(10\!\cdots\!96\)\( T^{3} + \)\(24\!\cdots\!02\)\( T^{4} - \)\(10\!\cdots\!96\)\( p^{17} T^{5} + \)\(28\!\cdots\!72\)\( p^{34} T^{6} + 638673818939760 p^{51} T^{7} + p^{68} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 1978637675022152 T + \)\(87\!\cdots\!36\)\( T^{2} - \)\(47\!\cdots\!56\)\( T^{3} + \)\(11\!\cdots\!78\)\( T^{4} - \)\(47\!\cdots\!56\)\( p^{17} T^{5} + \)\(87\!\cdots\!36\)\( p^{34} T^{6} + 1978637675022152 p^{51} T^{7} + p^{68} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 10940103622140252 T + \)\(13\!\cdots\!56\)\( T^{2} + \)\(89\!\cdots\!16\)\( T^{3} + \)\(62\!\cdots\!30\)\( T^{4} + \)\(89\!\cdots\!16\)\( p^{17} T^{5} + \)\(13\!\cdots\!56\)\( p^{34} T^{6} + 10940103622140252 p^{51} T^{7} + p^{68} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 7197922436085584 T + \)\(63\!\cdots\!16\)\( T^{2} + \)\(42\!\cdots\!72\)\( T^{3} + \)\(53\!\cdots\!98\)\( T^{4} + \)\(42\!\cdots\!72\)\( p^{17} T^{5} + \)\(63\!\cdots\!16\)\( p^{34} T^{6} + 7197922436085584 p^{51} T^{7} + p^{68} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 9665052141562152 T + \)\(52\!\cdots\!88\)\( T^{2} + \)\(41\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!22\)\( T^{4} + \)\(41\!\cdots\!64\)\( p^{17} T^{5} + \)\(52\!\cdots\!88\)\( p^{34} T^{6} + 9665052141562152 p^{51} T^{7} + p^{68} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 12051793326558816 T + \)\(10\!\cdots\!00\)\( T^{2} + \)\(89\!\cdots\!68\)\( T^{3} + \)\(55\!\cdots\!38\)\( T^{4} + \)\(89\!\cdots\!68\)\( p^{17} T^{5} + \)\(10\!\cdots\!00\)\( p^{34} T^{6} + 12051793326558816 p^{51} T^{7} + p^{68} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 31625003976492228 T + \)\(17\!\cdots\!48\)\( T^{2} + \)\(56\!\cdots\!64\)\( T^{3} + \)\(44\!\cdots\!38\)\( T^{4} + \)\(56\!\cdots\!64\)\( p^{17} T^{5} + \)\(17\!\cdots\!48\)\( p^{34} T^{6} + 31625003976492228 p^{51} T^{7} + p^{68} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 104997350891433280 T + \)\(59\!\cdots\!96\)\( T^{2} + \)\(17\!\cdots\!04\)\( T^{3} - \)\(13\!\cdots\!14\)\( T^{4} + \)\(17\!\cdots\!04\)\( p^{17} T^{5} + \)\(59\!\cdots\!96\)\( p^{34} T^{6} - 104997350891433280 p^{51} T^{7} + p^{68} T^{8} \) | |
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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
−8.098497387338475450693967838600, −7.47237561446548566578499590641, −7.31240380445859115968469841069, −7.30194179281015175517243123794, −6.92584921019536654124989129503, −6.29761668062789408996834931366, −6.29326173618892566158986655088, −6.14102392610584467670104330987, −6.06683193039234878426780672998, −4.88838283004992553017402953985, −4.83157279813576990671584054355, −4.69273954246996555448334425583, −4.44743913763713317211617592450, −3.77388874839361166232544558622, −3.63179651730177697213035205944, −3.60532729592025363367801071758, −3.50605941017816498516859921295, −2.70578442903344625722112314162, −2.65613214965981281026141293174, −2.56257558701643543399585383665, −2.26954847837621947378915329069, −1.62597428800549980268576840169, −1.39028333633406248907013117034, −1.25718554643080612647447656633, −1.15525749135569515400265189306, 0, 0, 0, 0, 1.15525749135569515400265189306, 1.25718554643080612647447656633, 1.39028333633406248907013117034, 1.62597428800549980268576840169, 2.26954847837621947378915329069, 2.56257558701643543399585383665, 2.65613214965981281026141293174, 2.70578442903344625722112314162, 3.50605941017816498516859921295, 3.60532729592025363367801071758, 3.63179651730177697213035205944, 3.77388874839361166232544558622, 4.44743913763713317211617592450, 4.69273954246996555448334425583, 4.83157279813576990671584054355, 4.88838283004992553017402953985, 6.06683193039234878426780672998, 6.14102392610584467670104330987, 6.29326173618892566158986655088, 6.29761668062789408996834931366, 6.92584921019536654124989129503, 7.30194179281015175517243123794, 7.31240380445859115968469841069, 7.47237561446548566578499590641, 8.098497387338475450693967838600