Dirichlet series
| L(s) = 1 | + 6.77e5·5-s + 9.63e6·7-s − 6.03e8·11-s − 2.47e9·13-s + 7.42e9·17-s − 1.95e10·19-s + 4.58e11·23-s − 7.88e11·25-s − 8.44e11·29-s + 1.24e12·31-s + 6.52e12·35-s + 1.76e13·37-s − 3.31e13·41-s + 2.41e13·43-s − 1.19e13·47-s − 4.17e14·49-s + 2.56e14·53-s − 4.09e14·55-s + 9.05e14·59-s + 2.51e14·61-s − 1.67e15·65-s + 2.67e15·67-s + 4.18e15·71-s − 1.41e15·73-s − 5.81e15·77-s + 7.57e15·79-s + 1.48e16·83-s + ⋯ |
| L(s) = 1 | + 0.776·5-s + 0.631·7-s − 0.849·11-s − 0.841·13-s + 0.258·17-s − 0.264·19-s + 1.22·23-s − 1.03·25-s − 0.313·29-s + 0.262·31-s + 0.490·35-s + 0.826·37-s − 0.647·41-s + 0.314·43-s − 0.0732·47-s − 1.79·49-s + 0.566·53-s − 0.658·55-s + 0.802·59-s + 0.167·61-s − 0.653·65-s + 0.803·67-s + 0.768·71-s − 0.204·73-s − 0.536·77-s + 0.561·79-s + 0.724·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+17/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(26873856\) = \(2^{12} \cdot 3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.02859\times 10^{8}\) |
| Root analytic conductor: | \(11.4856\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 26873856,\ (\ :17/2, 17/2, 17/2, 17/2),\ 1)\) |
Particular Values
| \(L(9)\) | \(\approx\) | \(0.0006271267144\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0006271267144\) |
| \(L(\frac{19}{2})\) | 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 | \( 1 \) | ||
| good | 5 | $C_2 \wr S_4$ | \( 1 - 677824 T + 249566118148 p T^{2} - 2377297831999552 p^{4} T^{3} + \)\(36\!\cdots\!22\)\( p^{5} T^{4} - 2377297831999552 p^{21} T^{5} + 249566118148 p^{35} T^{6} - 677824 p^{51} T^{7} + p^{68} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 9632880 T + 72930471741604 p T^{2} - \)\(13\!\cdots\!60\)\( p^{2} T^{3} + \)\(61\!\cdots\!58\)\( p^{4} T^{4} - \)\(13\!\cdots\!60\)\( p^{19} T^{5} + 72930471741604 p^{35} T^{6} - 9632880 p^{51} T^{7} + p^{68} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 603687680 T + 1093698267630025004 T^{2} + \)\(52\!\cdots\!40\)\( p T^{3} + \)\(60\!\cdots\!50\)\( p^{2} T^{4} + \)\(52\!\cdots\!40\)\( p^{18} T^{5} + 1093698267630025004 p^{34} T^{6} + 603687680 p^{51} T^{7} + p^{68} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 2476119800 T + 1553297207541801340 p T^{2} + \)\(23\!\cdots\!00\)\( p^{2} T^{3} + \)\(10\!\cdots\!74\)\( p^{3} T^{4} + \)\(23\!\cdots\!00\)\( p^{19} T^{5} + 1553297207541801340 p^{35} T^{6} + 2476119800 p^{51} T^{7} + p^{68} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 7423708544 T + \)\(16\!\cdots\!28\)\( p T^{2} - \)\(19\!\cdots\!12\)\( T^{3} + \)\(33\!\cdots\!10\)\( T^{4} - \)\(19\!\cdots\!12\)\( p^{17} T^{5} + \)\(16\!\cdots\!28\)\( p^{35} T^{6} - 7423708544 p^{51} T^{7} + p^{68} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 19551988960 T + \)\(19\!\cdots\!04\)\( T^{2} + \)\(26\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!46\)\( T^{4} + \)\(26\!\cdots\!60\)\( p^{17} T^{5} + \)\(19\!\cdots\!04\)\( p^{34} T^{6} + 19551988960 p^{51} T^{7} + p^{68} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 458239426048 T + \)\(19\!\cdots\!04\)\( T^{2} - \)\(82\!\cdots\!16\)\( T^{3} + \)\(47\!\cdots\!70\)\( T^{4} - \)\(82\!\cdots\!16\)\( p^{17} T^{5} + \)\(19\!\cdots\!04\)\( p^{34} T^{6} - 458239426048 p^{51} T^{7} + p^{68} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 844849583040 T + \)\(14\!\cdots\!40\)\( T^{2} + \)\(76\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!58\)\( T^{4} + \)\(76\!\cdots\!60\)\( p^{17} T^{5} + \)\(14\!\cdots\!40\)\( p^{34} T^{6} + 844849583040 p^{51} T^{7} + p^{68} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 1246685956208 T + \)\(25\!\cdots\!40\)\( T^{2} - \)\(42\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!38\)\( T^{4} - \)\(42\!\cdots\!84\)\( p^{17} T^{5} + \)\(25\!\cdots\!40\)\( p^{34} T^{6} - 1246685956208 p^{51} T^{7} + p^{68} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 17661470383320 T + \)\(60\!\cdots\!32\)\( T^{2} - \)\(50\!\cdots\!60\)\( T^{3} + \)\(49\!\cdots\!10\)\( T^{4} - \)\(50\!\cdots\!60\)\( p^{17} T^{5} + \)\(60\!\cdots\!32\)\( p^{34} T^{6} - 17661470383320 p^{51} T^{7} + p^{68} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 33100198696320 T + \)\(77\!\cdots\!00\)\( T^{2} + \)\(20\!\cdots\!40\)\( T^{3} + \)\(28\!\cdots\!78\)\( T^{4} + \)\(20\!\cdots\!40\)\( p^{17} T^{5} + \)\(77\!\cdots\!00\)\( p^{34} T^{6} + 33100198696320 p^{51} T^{7} + p^{68} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 24133870623520 T + \)\(99\!\cdots\!68\)\( T^{2} + \)\(15\!\cdots\!60\)\( T^{3} + \)\(57\!\cdots\!50\)\( T^{4} + \)\(15\!\cdots\!60\)\( p^{17} T^{5} + \)\(99\!\cdots\!68\)\( p^{34} T^{6} - 24133870623520 p^{51} T^{7} + p^{68} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 11964304009728 T + \)\(86\!\cdots\!92\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(31\!\cdots\!46\)\( T^{4} + \)\(13\!\cdots\!80\)\( p^{17} T^{5} + \)\(86\!\cdots\!92\)\( p^{34} T^{6} + 11964304009728 p^{51} T^{7} + p^{68} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 256855748281408 T + \)\(42\!\cdots\!04\)\( T^{2} - \)\(65\!\cdots\!56\)\( T^{3} + \)\(85\!\cdots\!90\)\( T^{4} - \)\(65\!\cdots\!56\)\( p^{17} T^{5} + \)\(42\!\cdots\!04\)\( p^{34} T^{6} - 256855748281408 p^{51} T^{7} + p^{68} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 905361180505600 T + \)\(34\!\cdots\!76\)\( T^{2} - \)\(32\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!70\)\( T^{4} - \)\(32\!\cdots\!00\)\( p^{17} T^{5} + \)\(34\!\cdots\!76\)\( p^{34} T^{6} - 905361180505600 p^{51} T^{7} + p^{68} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 251141035416376 T + \)\(76\!\cdots\!12\)\( T^{2} - \)\(11\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!86\)\( T^{4} - \)\(11\!\cdots\!80\)\( p^{17} T^{5} + \)\(76\!\cdots\!12\)\( p^{34} T^{6} - 251141035416376 p^{51} T^{7} + p^{68} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 2671944825444800 T + \)\(35\!\cdots\!88\)\( T^{2} - \)\(91\!\cdots\!00\)\( T^{3} + \)\(79\!\cdots\!94\)\( p T^{4} - \)\(91\!\cdots\!00\)\( p^{17} T^{5} + \)\(35\!\cdots\!88\)\( p^{34} T^{6} - 2671944825444800 p^{51} T^{7} + p^{68} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 4181227521935360 T + \)\(40\!\cdots\!56\)\( T^{2} - \)\(18\!\cdots\!40\)\( T^{3} + \)\(54\!\cdots\!46\)\( T^{4} - \)\(18\!\cdots\!40\)\( p^{17} T^{5} + \)\(40\!\cdots\!56\)\( p^{34} T^{6} - 4181227521935360 p^{51} T^{7} + p^{68} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 1410879602368040 T + \)\(10\!\cdots\!40\)\( T^{2} + \)\(88\!\cdots\!20\)\( T^{3} + \)\(60\!\cdots\!18\)\( T^{4} + \)\(88\!\cdots\!20\)\( p^{17} T^{5} + \)\(10\!\cdots\!40\)\( p^{34} T^{6} + 1410879602368040 p^{51} T^{7} + p^{68} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 7574345714454640 T + \)\(49\!\cdots\!08\)\( T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!78\)\( T^{4} - \)\(13\!\cdots\!20\)\( p^{17} T^{5} + \)\(49\!\cdots\!08\)\( p^{34} T^{6} - 7574345714454640 p^{51} T^{7} + p^{68} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 14868236752232192 T + \)\(12\!\cdots\!24\)\( T^{2} - \)\(11\!\cdots\!84\)\( T^{3} + \)\(69\!\cdots\!10\)\( T^{4} - \)\(11\!\cdots\!84\)\( p^{17} T^{5} + \)\(12\!\cdots\!24\)\( p^{34} T^{6} - 14868236752232192 p^{51} T^{7} + p^{68} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 31667797088981760 T + \)\(54\!\cdots\!52\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!34\)\( T^{4} - \)\(12\!\cdots\!00\)\( p^{17} T^{5} + \)\(54\!\cdots\!52\)\( p^{34} T^{6} - 31667797088981760 p^{51} T^{7} + p^{68} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 17206349902780360 T + \)\(17\!\cdots\!88\)\( T^{2} + \)\(25\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!98\)\( T^{4} + \)\(25\!\cdots\!60\)\( p^{17} T^{5} + \)\(17\!\cdots\!88\)\( p^{34} T^{6} + 17206349902780360 p^{51} T^{7} + p^{68} T^{8} \) | |
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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
−7.84299080345332210885934956360, −7.04437448552984461155420266916, −6.99828147415231160913006791304, −6.70913337947464768057643571944, −6.51127063730699555495746365069, −5.94293798611760825911908211751, −5.62043670362182508078096618537, −5.43104061908610520528024723830, −5.40878118945960184889756026303, −4.90573804619097706447441374173, −4.49361951974435938824129600387, −4.30696106489029772530207083322, −4.29598528338025468141111727881, −3.30336268130311990699258367671, −3.27274070737630396360717222982, −3.24699492777819507384740191490, −2.67214774428279301862804494608, −2.18104169695929780874858720149, −2.06131684638569357729204252837, −1.83558044414904979038879852634, −1.73763349293803007177621148440, −0.815206933874207551358617193397, −0.805212605200805553625626525195, −0.78013964063741653424442486401, −0.00216523353714934902619855260, 0.00216523353714934902619855260, 0.78013964063741653424442486401, 0.805212605200805553625626525195, 0.815206933874207551358617193397, 1.73763349293803007177621148440, 1.83558044414904979038879852634, 2.06131684638569357729204252837, 2.18104169695929780874858720149, 2.67214774428279301862804494608, 3.24699492777819507384740191490, 3.27274070737630396360717222982, 3.30336268130311990699258367671, 4.29598528338025468141111727881, 4.30696106489029772530207083322, 4.49361951974435938824129600387, 4.90573804619097706447441374173, 5.40878118945960184889756026303, 5.43104061908610520528024723830, 5.62043670362182508078096618537, 5.94293798611760825911908211751, 6.51127063730699555495746365069, 6.70913337947464768057643571944, 6.99828147415231160913006791304, 7.04437448552984461155420266916, 7.84299080345332210885934956360