Dirichlet series
| L(s) = 1 | − 176·3-s − 2.50e3·5-s + 1.39e3·7-s − 2.03e4·9-s − 7.33e4·11-s − 1.12e5·13-s + 4.40e5·15-s + 1.73e5·17-s + 1.25e5·19-s − 2.44e5·21-s + 2.85e6·23-s + 3.90e6·25-s + 5.74e6·27-s − 3.19e6·29-s + 3.85e6·31-s + 1.29e7·33-s − 3.48e6·35-s − 6.21e6·37-s + 1.97e7·39-s + 7.42e6·41-s − 1.28e7·43-s + 5.08e7·45-s − 7.73e6·47-s − 3.19e7·49-s − 3.05e7·51-s + 1.01e8·53-s + 1.83e8·55-s + ⋯ |
| L(s) = 1 | − 1.25·3-s − 1.78·5-s + 0.219·7-s − 1.03·9-s − 1.51·11-s − 1.09·13-s + 2.24·15-s + 0.503·17-s + 0.221·19-s − 0.274·21-s + 2.13·23-s + 2·25-s + 2.07·27-s − 0.837·29-s + 0.750·31-s + 1.89·33-s − 0.391·35-s − 0.545·37-s + 1.36·39-s + 0.410·41-s − 0.575·43-s + 1.84·45-s − 0.231·47-s − 0.792·49-s − 0.631·51-s + 1.76·53-s + 2.70·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{24} \cdot 5^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.37818\times 10^{8}\) |
| Root analytic conductor: | \(12.8378\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{24} \cdot 5^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | 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 | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 176 T + 17108 p T^{2} + 763696 p^{2} T^{3} + 53357602 p^{3} T^{4} + 763696 p^{11} T^{5} + 17108 p^{19} T^{6} + 176 p^{27} T^{7} + p^{36} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 1392 T + 4843156 p T^{2} + 2911247440 p^{2} T^{3} + 5161834595562 p^{3} T^{4} + 2911247440 p^{11} T^{5} + 4843156 p^{19} T^{6} - 1392 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 73344 T + 7389094124 T^{2} + 361998346858112 T^{3} + 23176949445522440406 T^{4} + 361998346858112 p^{9} T^{5} + 7389094124 p^{18} T^{6} + 73344 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 112376 T + 32920281292 T^{2} + 3025884312984744 T^{3} + \)\(46\!\cdots\!74\)\( T^{4} + 3025884312984744 p^{9} T^{5} + 32920281292 p^{18} T^{6} + 112376 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 173448 T + 349761359708 T^{2} - 47292952848422968 T^{3} + \)\(56\!\cdots\!34\)\( T^{4} - 47292952848422968 p^{9} T^{5} + 349761359708 p^{18} T^{6} - 173448 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 125728 T + 643550181196 T^{2} - 101011957946402336 T^{3} + \)\(28\!\cdots\!86\)\( T^{4} - 101011957946402336 p^{9} T^{5} + 643550181196 p^{18} T^{6} - 125728 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 2859152 T + 6727477421132 T^{2} - 11738783046643035216 T^{3} + \)\(17\!\cdots\!78\)\( T^{4} - 11738783046643035216 p^{9} T^{5} + 6727477421132 p^{18} T^{6} - 2859152 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 3191320 T + 28127632061900 T^{2} + \)\(15\!\cdots\!28\)\( T^{3} + \)\(45\!\cdots\!70\)\( T^{4} + \)\(15\!\cdots\!28\)\( p^{9} T^{5} + 28127632061900 p^{18} T^{6} + 3191320 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 3857056 T + 3000269720644 p T^{2} - \)\(28\!\cdots\!28\)\( T^{3} + \)\(35\!\cdots\!06\)\( T^{4} - \)\(28\!\cdots\!28\)\( p^{9} T^{5} + 3000269720644 p^{19} T^{6} - 3857056 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 6213208 T + 348415870869868 T^{2} + \)\(23\!\cdots\!48\)\( T^{3} + \)\(61\!\cdots\!14\)\( T^{4} + \)\(23\!\cdots\!48\)\( p^{9} T^{5} + 348415870869868 p^{18} T^{6} + 6213208 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 7425800 T + 542029194422204 T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!46\)\( T^{4} + \)\(14\!\cdots\!00\)\( p^{9} T^{5} + 542029194422204 p^{18} T^{6} - 7425800 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 12899120 T + 1377498577492636 T^{2} + \)\(18\!\cdots\!48\)\( T^{3} + \)\(90\!\cdots\!70\)\( T^{4} + \)\(18\!\cdots\!48\)\( p^{9} T^{5} + 1377498577492636 p^{18} T^{6} + 12899120 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 7730896 T + 4375909771618988 T^{2} + \)\(25\!\cdots\!72\)\( T^{3} + \)\(72\!\cdots\!98\)\( T^{4} + \)\(25\!\cdots\!72\)\( p^{9} T^{5} + 4375909771618988 p^{18} T^{6} + 7730896 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 101217512 T + 5332932068245292 T^{2} - \)\(46\!\cdots\!88\)\( T^{3} + \)\(35\!\cdots\!94\)\( T^{4} - \)\(46\!\cdots\!88\)\( p^{9} T^{5} + 5332932068245292 p^{18} T^{6} - 101217512 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 98371040 T + 16697738628491756 T^{2} + \)\(28\!\cdots\!80\)\( T^{3} + \)\(77\!\cdots\!26\)\( T^{4} + \)\(28\!\cdots\!80\)\( p^{9} T^{5} + 16697738628491756 p^{18} T^{6} + 98371040 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 42840952 T + 511537521596284 p T^{2} + \)\(10\!\cdots\!76\)\( T^{3} + \)\(49\!\cdots\!06\)\( T^{4} + \)\(10\!\cdots\!76\)\( p^{9} T^{5} + 511537521596284 p^{19} T^{6} + 42840952 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 72829968 T + 58358414462910076 T^{2} + \)\(88\!\cdots\!88\)\( T^{3} + \)\(16\!\cdots\!14\)\( T^{4} + \)\(88\!\cdots\!88\)\( p^{9} T^{5} + 58358414462910076 p^{18} T^{6} + 72829968 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 728351072 T + 272499383306293724 T^{2} - \)\(78\!\cdots\!96\)\( T^{3} + \)\(19\!\cdots\!66\)\( T^{4} - \)\(78\!\cdots\!96\)\( p^{9} T^{5} + 272499383306293724 p^{18} T^{6} - 728351072 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 135883160 T + 125372580970067452 T^{2} + \)\(25\!\cdots\!40\)\( T^{3} + \)\(92\!\cdots\!14\)\( T^{4} + \)\(25\!\cdots\!40\)\( p^{9} T^{5} + 125372580970067452 p^{18} T^{6} + 135883160 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 932964288 T + 666869545637894716 T^{2} - \)\(33\!\cdots\!16\)\( T^{3} + \)\(12\!\cdots\!86\)\( T^{4} - \)\(33\!\cdots\!16\)\( p^{9} T^{5} + 666869545637894716 p^{18} T^{6} - 932964288 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 1006214192 T + 958538310443626940 T^{2} + \)\(52\!\cdots\!52\)\( T^{3} + \)\(27\!\cdots\!02\)\( T^{4} + \)\(52\!\cdots\!52\)\( p^{9} T^{5} + 958538310443626940 p^{18} T^{6} + 1006214192 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 734523608 T + 1021794154090964156 T^{2} + \)\(72\!\cdots\!48\)\( T^{3} + \)\(47\!\cdots\!10\)\( T^{4} + \)\(72\!\cdots\!48\)\( p^{9} T^{5} + 1021794154090964156 p^{18} T^{6} + 734523608 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 72294520 T + 54140137519539868 T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(44\!\cdots\!34\)\( T^{4} - \)\(24\!\cdots\!80\)\( p^{9} T^{5} + 54140137519539868 p^{18} T^{6} + 72294520 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−7.37025075050760284956125145735, −7.22035697460422548824598071971, −6.74232448529693797879720342044, −6.60004788578208503839946662534, −6.59965784295410030271191580801, −5.95143731109798519687385216553, −5.62994045787228444026511950805, −5.59796033561032316863650854780, −5.35211450777242105206677025826, −4.98945836306350156902615281462, −4.83929336970478306136048525007, −4.77981988660644225353173613817, −4.44549620363505789141768589689, −3.95084457898050555840560716298, −3.53857970148478860191862818554, −3.46385126206826828076574250005, −3.31759672265824507620014590782, −2.64612196982329526519452138153, −2.58917386660283566033378274944, −2.56868138820628898395760824563, −2.15250160812455631995905873708, −1.41629297074355457917309723934, −1.16605346317297221336614116398, −0.989724588621832949654467871774, −0.67866655731272818597052055444, 0, 0, 0, 0, 0.67866655731272818597052055444, 0.989724588621832949654467871774, 1.16605346317297221336614116398, 1.41629297074355457917309723934, 2.15250160812455631995905873708, 2.56868138820628898395760824563, 2.58917386660283566033378274944, 2.64612196982329526519452138153, 3.31759672265824507620014590782, 3.46385126206826828076574250005, 3.53857970148478860191862818554, 3.95084457898050555840560716298, 4.44549620363505789141768589689, 4.77981988660644225353173613817, 4.83929336970478306136048525007, 4.98945836306350156902615281462, 5.35211450777242105206677025826, 5.59796033561032316863650854780, 5.62994045787228444026511950805, 5.95143731109798519687385216553, 6.59965784295410030271191580801, 6.60004788578208503839946662534, 6.74232448529693797879720342044, 7.22035697460422548824598071971, 7.37025075050760284956125145735