Dirichlet series
| L(s) = 1 | − 128·2-s + 972·3-s + 1.02e4·4-s + 2.86e3·5-s − 1.24e5·6-s + 1.34e5·7-s − 6.55e5·8-s + 5.90e5·9-s − 3.66e5·10-s + 3.32e5·11-s + 9.95e6·12-s + 2.50e6·13-s − 1.71e7·14-s + 2.78e6·15-s + 3.67e7·16-s + 5.67e6·17-s − 7.55e7·18-s − 1.11e6·19-s + 2.93e7·20-s + 1.30e8·21-s − 4.26e7·22-s + 2.69e7·23-s − 6.37e8·24-s − 3.22e7·25-s − 3.21e8·26-s + 2.86e8·27-s + 1.37e9·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 5·4-s + 0.409·5-s − 6.53·6-s + 3.01·7-s − 7.07·8-s + 10/3·9-s − 1.15·10-s + 0.623·11-s + 11.5·12-s + 1.87·13-s − 8.54·14-s + 0.945·15-s + 35/4·16-s + 0.970·17-s − 9.42·18-s − 0.103·19-s + 2.04·20-s + 6.97·21-s − 1.76·22-s + 0.872·23-s − 16.3·24-s − 0.661·25-s − 5.30·26-s + 3.84·27-s + 15.0·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 17^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.77243\times 10^{7}\) |
| Root analytic conductor: | \(8.85273\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 17^{4} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )\) |
Particular Values
| \(L(6)\) | \(\approx\) | \(24.66692897\) |
| \(L(\frac12)\) | \(\approx\) | \(24.66692897\) |
| \(L(\frac{13}{2})\) | 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 | $C_1$ | \( ( 1 + p^{5} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - p^{5} T )^{4} \) | |
| 17 | $C_1$ | \( ( 1 - p^{5} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 2862 T + 40466801 T^{2} - 16538571198 p T^{3} + 168105257820252 p^{2} T^{4} - 16538571198 p^{12} T^{5} + 40466801 p^{22} T^{6} - 2862 p^{33} T^{7} + p^{44} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 134264 T + 12578561836 T^{2} - 119191189295720 p T^{3} + 850377461143769254 p^{2} T^{4} - 119191189295720 p^{12} T^{5} + 12578561836 p^{22} T^{6} - 134264 p^{33} T^{7} + p^{44} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 332862 T - 17934530753 p T^{2} + 91140695990472042 T^{3} - \)\(12\!\cdots\!96\)\( T^{4} + 91140695990472042 p^{11} T^{5} - 17934530753 p^{23} T^{6} - 332862 p^{33} T^{7} + p^{44} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 2509514 T + 5058296439265 T^{2} - 9183521074013432186 T^{3} + \)\(14\!\cdots\!12\)\( T^{4} - 9183521074013432186 p^{11} T^{5} + 5058296439265 p^{22} T^{6} - 2509514 p^{33} T^{7} + p^{44} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 1112482 T + 320996210384389 T^{2} + \)\(65\!\cdots\!98\)\( T^{3} + \)\(47\!\cdots\!40\)\( T^{4} + \)\(65\!\cdots\!98\)\( p^{11} T^{5} + 320996210384389 p^{22} T^{6} + 1112482 p^{33} T^{7} + p^{44} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 26929326 T + 1425272952626381 T^{2} - \)\(35\!\cdots\!86\)\( T^{3} + \)\(13\!\cdots\!16\)\( T^{4} - \)\(35\!\cdots\!86\)\( p^{11} T^{5} + 1425272952626381 p^{22} T^{6} - 26929326 p^{33} T^{7} + p^{44} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 52628688 T + 9536538010179452 T^{2} + \)\(69\!\cdots\!44\)\( T^{3} + \)\(98\!\cdots\!34\)\( T^{4} + \)\(69\!\cdots\!44\)\( p^{11} T^{5} + 9536538010179452 p^{22} T^{6} - 52628688 p^{33} T^{7} + p^{44} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 403081484 T + 131550358039369648 T^{2} - \)\(29\!\cdots\!32\)\( T^{3} + \)\(53\!\cdots\!54\)\( T^{4} - \)\(29\!\cdots\!32\)\( p^{11} T^{5} + 131550358039369648 p^{22} T^{6} - 403081484 p^{33} T^{7} + p^{44} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 806679308 T + 598904178648660040 T^{2} - \)\(23\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!94\)\( T^{4} - \)\(23\!\cdots\!60\)\( p^{11} T^{5} + 598904178648660040 p^{22} T^{6} - 806679308 p^{33} T^{7} + p^{44} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 17234562 p T + 2118266434740710321 T^{2} - \)\(11\!\cdots\!98\)\( T^{3} + \)\(17\!\cdots\!76\)\( T^{4} - \)\(11\!\cdots\!98\)\( p^{11} T^{5} + 2118266434740710321 p^{22} T^{6} - 17234562 p^{34} T^{7} + p^{44} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 330299902 T + 2788523259571423525 T^{2} + \)\(95\!\cdots\!62\)\( T^{3} + \)\(35\!\cdots\!52\)\( T^{4} + \)\(95\!\cdots\!62\)\( p^{11} T^{5} + 2788523259571423525 p^{22} T^{6} + 330299902 p^{33} T^{7} + p^{44} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 2131455252 T + 8714465630423869712 T^{2} + \)\(14\!\cdots\!16\)\( T^{3} + \)\(30\!\cdots\!78\)\( T^{4} + \)\(14\!\cdots\!16\)\( p^{11} T^{5} + 8714465630423869712 p^{22} T^{6} + 2131455252 p^{33} T^{7} + p^{44} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 6330072456 T + 39552817676200729532 T^{2} + \)\(15\!\cdots\!72\)\( T^{3} + \)\(56\!\cdots\!98\)\( T^{4} + \)\(15\!\cdots\!72\)\( p^{11} T^{5} + 39552817676200729532 p^{22} T^{6} + 6330072456 p^{33} T^{7} + p^{44} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 19475328708 T + \)\(23\!\cdots\!28\)\( T^{2} + \)\(19\!\cdots\!36\)\( T^{3} + \)\(12\!\cdots\!42\)\( T^{4} + \)\(19\!\cdots\!36\)\( p^{11} T^{5} + \)\(23\!\cdots\!28\)\( p^{22} T^{6} + 19475328708 p^{33} T^{7} + p^{44} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 321345308 T + \)\(12\!\cdots\!16\)\( T^{2} - \)\(13\!\cdots\!92\)\( T^{3} + \)\(73\!\cdots\!26\)\( T^{4} - \)\(13\!\cdots\!92\)\( p^{11} T^{5} + \)\(12\!\cdots\!16\)\( p^{22} T^{6} - 321345308 p^{33} T^{7} + p^{44} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 12880263008 T + \)\(26\!\cdots\!20\)\( T^{2} - \)\(19\!\cdots\!12\)\( T^{3} + \)\(36\!\cdots\!70\)\( T^{4} - \)\(19\!\cdots\!12\)\( p^{11} T^{5} + \)\(26\!\cdots\!20\)\( p^{22} T^{6} - 12880263008 p^{33} T^{7} + p^{44} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 11505311496 T + \)\(78\!\cdots\!88\)\( T^{2} - \)\(83\!\cdots\!72\)\( T^{3} + \)\(25\!\cdots\!54\)\( T^{4} - \)\(83\!\cdots\!72\)\( p^{11} T^{5} + \)\(78\!\cdots\!88\)\( p^{22} T^{6} - 11505311496 p^{33} T^{7} + p^{44} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 19599103928 T + \)\(98\!\cdots\!64\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(41\!\cdots\!46\)\( T^{4} - \)\(14\!\cdots\!20\)\( p^{11} T^{5} + \)\(98\!\cdots\!64\)\( p^{22} T^{6} - 19599103928 p^{33} T^{7} + p^{44} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 13222352492 T + \)\(62\!\cdots\!96\)\( T^{2} + \)\(24\!\cdots\!72\)\( T^{3} - \)\(47\!\cdots\!34\)\( T^{4} + \)\(24\!\cdots\!72\)\( p^{11} T^{5} + \)\(62\!\cdots\!96\)\( p^{22} T^{6} - 13222352492 p^{33} T^{7} + p^{44} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 97025189940 T + \)\(72\!\cdots\!24\)\( T^{2} - \)\(34\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(34\!\cdots\!80\)\( p^{11} T^{5} + \)\(72\!\cdots\!24\)\( p^{22} T^{6} - 97025189940 p^{33} T^{7} + p^{44} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 74750287524 T + \)\(10\!\cdots\!88\)\( T^{2} - \)\(58\!\cdots\!64\)\( T^{3} + \)\(43\!\cdots\!94\)\( T^{4} - \)\(58\!\cdots\!64\)\( p^{11} T^{5} + \)\(10\!\cdots\!88\)\( p^{22} T^{6} - 74750287524 p^{33} T^{7} + p^{44} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 221782308572 T + \)\(22\!\cdots\!00\)\( T^{2} - \)\(58\!\cdots\!52\)\( T^{3} - \)\(17\!\cdots\!58\)\( T^{4} - \)\(58\!\cdots\!52\)\( p^{11} T^{5} + \)\(22\!\cdots\!00\)\( p^{22} T^{6} - 221782308572 p^{33} T^{7} + p^{44} T^{8} \) | |
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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
−8.114468204228823928286322633166, −7.76312882270499822567359918789, −7.74619197779294168836758653896, −7.58699291525585734002436167244, −7.32790592804371592441599702591, −6.41829648176529639745737236451, −6.35756614221128269858722992166, −6.27227469533749897025392703250, −6.08880582904825238149326231291, −5.03771052832354032243514498755, −4.93218269225018189582902719237, −4.65124065966678608552607044523, −4.33810383533621211592043769961, −3.70678175798704785548182576480, −3.27086733560916375525302848262, −3.11614484162222447642445556194, −3.01030515606241439098682030213, −2.13289570913450489488752921161, −2.04617731387336825895744037214, −1.86744453565487554632802442570, −1.71490449859181009235283286752, −1.11260477371908371072701900530, −0.965395421786701999817296576551, −0.955050593197082612911594317979, −0.56335013764084730534936706225, 0.56335013764084730534936706225, 0.955050593197082612911594317979, 0.965395421786701999817296576551, 1.11260477371908371072701900530, 1.71490449859181009235283286752, 1.86744453565487554632802442570, 2.04617731387336825895744037214, 2.13289570913450489488752921161, 3.01030515606241439098682030213, 3.11614484162222447642445556194, 3.27086733560916375525302848262, 3.70678175798704785548182576480, 4.33810383533621211592043769961, 4.65124065966678608552607044523, 4.93218269225018189582902719237, 5.03771052832354032243514498755, 6.08880582904825238149326231291, 6.27227469533749897025392703250, 6.35756614221128269858722992166, 6.41829648176529639745737236451, 7.32790592804371592441599702591, 7.58699291525585734002436167244, 7.74619197779294168836758653896, 7.76312882270499822567359918789, 8.114468204228823928286322633166