Dirichlet series
| L(s) = 1 | + 70·3-s + 1.02e3·5-s + 9.60e3·7-s − 3.08e4·9-s − 3.14e4·11-s + 1.14e4·13-s + 7.15e4·15-s + 3.97e5·17-s + 2.46e5·19-s + 6.72e5·21-s + 1.97e6·23-s − 2.84e6·25-s − 1.68e6·27-s + 3.71e6·29-s + 1.11e7·31-s − 2.20e6·33-s + 9.81e6·35-s + 3.02e7·37-s + 8.02e5·39-s + 2.71e7·41-s + 6.40e7·43-s − 3.15e7·45-s + 1.04e8·47-s + 5.76e7·49-s + 2.77e7·51-s + 2.05e7·53-s − 3.21e7·55-s + ⋯ |
| L(s) = 1 | + 0.498·3-s + 0.731·5-s + 1.51·7-s − 1.56·9-s − 0.648·11-s + 0.111·13-s + 0.364·15-s + 1.15·17-s + 0.434·19-s + 0.754·21-s + 1.47·23-s − 1.45·25-s − 0.610·27-s + 0.976·29-s + 2.16·31-s − 0.323·33-s + 1.10·35-s + 2.65·37-s + 0.0555·39-s + 1.50·41-s + 2.85·43-s − 1.14·45-s + 3.12·47-s + 10/7·49-s + 0.575·51-s + 0.358·53-s − 0.474·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(9834496\) = \(2^{12} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(691993.\) |
| Root analytic conductor: | \(5.37047\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 9834496,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(12.27956715\) |
| \(L(\frac12)\) | \(\approx\) | \(12.27956715\) |
| \(L(\frac{11}{2})\) | 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 | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 - 70 T + 35728 T^{2} - 110138 p^{3} T^{3} + 118701758 p^{2} T^{4} - 110138 p^{12} T^{5} + 35728 p^{18} T^{6} - 70 p^{27} T^{7} + p^{36} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 1022 T + 3885236 T^{2} - 152222546 p^{2} T^{3} + 474253832134 p^{2} T^{4} - 152222546 p^{11} T^{5} + 3885236 p^{18} T^{6} - 1022 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 31476 T + 4380397484 T^{2} + 201908733784308 T^{3} + 11686055140738167126 T^{4} + 201908733784308 p^{9} T^{5} + 4380397484 p^{18} T^{6} + 31476 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 882 p T + 18012059452 T^{2} - 10047323485206 p^{2} T^{3} + 816678926783593686 p^{2} T^{4} - 10047323485206 p^{11} T^{5} + 18012059452 p^{18} T^{6} - 882 p^{28} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 397012 T + 145574384852 T^{2} + 50074046515806740 T^{3} - \)\(18\!\cdots\!34\)\( T^{4} + 50074046515806740 p^{9} T^{5} + 145574384852 p^{18} T^{6} - 397012 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 246610 T - 549062765408 T^{2} - 20510971622336314 T^{3} + \)\(28\!\cdots\!66\)\( T^{4} - 20510971622336314 p^{9} T^{5} - 549062765408 p^{18} T^{6} - 246610 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 1976552 T + 3707594729756 T^{2} - 3281923196317765256 T^{3} + \)\(41\!\cdots\!14\)\( T^{4} - 3281923196317765256 p^{9} T^{5} + 3707594729756 p^{18} T^{6} - 1976552 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 3718940 T + 45544141177892 T^{2} - \)\(12\!\cdots\!16\)\( T^{3} + \)\(92\!\cdots\!86\)\( T^{4} - \)\(12\!\cdots\!16\)\( p^{9} T^{5} + 45544141177892 p^{18} T^{6} - 3718940 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 11118436 T + 62993200170220 T^{2} - 48454988397995435284 T^{3} - \)\(38\!\cdots\!06\)\( T^{4} - 48454988397995435284 p^{9} T^{5} + 62993200170220 p^{18} T^{6} - 11118436 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 30221340 T + 845835241436452 T^{2} - \)\(13\!\cdots\!28\)\( T^{3} + \)\(18\!\cdots\!02\)\( T^{4} - \)\(13\!\cdots\!28\)\( p^{9} T^{5} + 845835241436452 p^{18} T^{6} - 30221340 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 27162604 T + 1043629846287716 T^{2} - \)\(15\!\cdots\!44\)\( T^{3} + \)\(39\!\cdots\!50\)\( T^{4} - \)\(15\!\cdots\!44\)\( p^{9} T^{5} + 1043629846287716 p^{18} T^{6} - 27162604 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 64096964 T + 2439218914892044 T^{2} - \)\(60\!\cdots\!48\)\( T^{3} + \)\(14\!\cdots\!86\)\( T^{4} - \)\(60\!\cdots\!48\)\( p^{9} T^{5} + 2439218914892044 p^{18} T^{6} - 64096964 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 104378204 T + 6687499386104684 T^{2} - \)\(32\!\cdots\!28\)\( T^{3} + \)\(12\!\cdots\!14\)\( T^{4} - \)\(32\!\cdots\!28\)\( p^{9} T^{5} + 6687499386104684 p^{18} T^{6} - 104378204 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 20587304 T + 8130860046889292 T^{2} - \)\(93\!\cdots\!56\)\( T^{3} + \)\(35\!\cdots\!94\)\( T^{4} - \)\(93\!\cdots\!56\)\( p^{9} T^{5} + 8130860046889292 p^{18} T^{6} - 20587304 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 130065474 T + 37663300523348912 T^{2} - \)\(32\!\cdots\!02\)\( T^{3} + \)\(84\!\cdots\!70\)\( p T^{4} - \)\(32\!\cdots\!02\)\( p^{9} T^{5} + 37663300523348912 p^{18} T^{6} - 130065474 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 2204762 p T + 21870514355282404 T^{2} + \)\(40\!\cdots\!74\)\( T^{3} + \)\(85\!\cdots\!70\)\( T^{4} + \)\(40\!\cdots\!74\)\( p^{9} T^{5} + 21870514355282404 p^{18} T^{6} + 2204762 p^{28} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 104031144 T + 69277515948625420 T^{2} - \)\(46\!\cdots\!36\)\( T^{3} + \)\(22\!\cdots\!02\)\( T^{4} - \)\(46\!\cdots\!36\)\( p^{9} T^{5} + 69277515948625420 p^{18} T^{6} - 104031144 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 71827896 T + 146322967233650204 T^{2} + \)\(81\!\cdots\!28\)\( T^{3} + \)\(94\!\cdots\!26\)\( T^{4} + \)\(81\!\cdots\!28\)\( p^{9} T^{5} + 146322967233650204 p^{18} T^{6} + 71827896 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 586388432 T + 287367725317903660 T^{2} + \)\(86\!\cdots\!92\)\( T^{3} + \)\(24\!\cdots\!62\)\( T^{4} + \)\(86\!\cdots\!92\)\( p^{9} T^{5} + 287367725317903660 p^{18} T^{6} + 586388432 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 164282440 T + 374476269578262652 T^{2} - \)\(34\!\cdots\!56\)\( T^{3} + \)\(60\!\cdots\!06\)\( T^{4} - \)\(34\!\cdots\!56\)\( p^{9} T^{5} + 374476269578262652 p^{18} T^{6} - 164282440 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 130538394 T + 377102872233050528 T^{2} - \)\(44\!\cdots\!50\)\( T^{3} + \)\(70\!\cdots\!06\)\( T^{4} - \)\(44\!\cdots\!50\)\( p^{9} T^{5} + 377102872233050528 p^{18} T^{6} - 130538394 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 2036960408 T + 2826473714232778556 T^{2} + \)\(25\!\cdots\!36\)\( T^{3} + \)\(17\!\cdots\!46\)\( T^{4} + \)\(25\!\cdots\!36\)\( p^{9} T^{5} + 2826473714232778556 p^{18} T^{6} + 2036960408 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 2049365612 T + 3192890854147630036 T^{2} + \)\(37\!\cdots\!32\)\( T^{3} + \)\(37\!\cdots\!94\)\( T^{4} + \)\(37\!\cdots\!32\)\( p^{9} T^{5} + 3192890854147630036 p^{18} T^{6} + 2049365612 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−9.403613159711551110145161074814, −8.870861653097992729055624387131, −8.765955014795698025127764386505, −8.298380929398718635956759408793, −8.174024261304149969154381203850, −7.85341896366426524742960930714, −7.42423089851295923849544395524, −7.31198041867354981657355971770, −6.85823193485481271837084724330, −5.91589514745588508270350897637, −5.82212738134335334978610819453, −5.81928383822772694006924475595, −5.61992348588834658560229673231, −4.75900480317033727744582683808, −4.67853383643659627214215438362, −4.08156061254797022888569173525, −3.98816829751607567870208128470, −2.95104569017411972192904587669, −2.72395516419427395735665804738, −2.53294778499738433501055480949, −2.47505681115703576202395916157, −1.52239516414818728958455560929, −1.02437846945909118016048495455, −0.920556206311076365743083171587, −0.50262602583751291862891939303, 0.50262602583751291862891939303, 0.920556206311076365743083171587, 1.02437846945909118016048495455, 1.52239516414818728958455560929, 2.47505681115703576202395916157, 2.53294778499738433501055480949, 2.72395516419427395735665804738, 2.95104569017411972192904587669, 3.98816829751607567870208128470, 4.08156061254797022888569173525, 4.67853383643659627214215438362, 4.75900480317033727744582683808, 5.61992348588834658560229673231, 5.81928383822772694006924475595, 5.82212738134335334978610819453, 5.91589514745588508270350897637, 6.85823193485481271837084724330, 7.31198041867354981657355971770, 7.42423089851295923849544395524, 7.85341896366426524742960930714, 8.174024261304149969154381203850, 8.298380929398718635956759408793, 8.765955014795698025127764386505, 8.870861653097992729055624387131, 9.403613159711551110145161074814