Properties

Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·3-s − 8·5-s + 90·9-s + 40·11-s − 48·13-s + 96·15-s + 152·17-s + 224·19-s + 8·23-s − 232·25-s − 540·27-s − 144·29-s + 400·31-s − 480·33-s − 304·37-s + 576·39-s + 152·41-s − 160·43-s − 720·45-s + 544·47-s − 1.82e3·51-s − 1.32e3·53-s − 320·55-s − 2.68e3·57-s + 1.04e3·59-s − 896·61-s + 384·65-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.715·5-s + 10/3·9-s + 1.09·11-s − 1.02·13-s + 1.65·15-s + 2.16·17-s + 2.70·19-s + 0.0725·23-s − 1.85·25-s − 3.84·27-s − 0.922·29-s + 2.31·31-s − 2.53·33-s − 1.35·37-s + 2.36·39-s + 0.578·41-s − 0.567·43-s − 2.38·45-s + 1.68·47-s − 5.00·51-s − 3.42·53-s − 0.784·55-s − 6.24·57-s + 2.29·59-s − 1.88·61-s + 0.732·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.3792958004\)
\(L(\frac12)\) \(\approx\) \(0.3792958004\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + p T )^{4} \)
7 \( 1 \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 296 T^{2} + 2312 T^{3} + 48194 T^{4} + 2312 p^{3} T^{5} + 296 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 40 T + 2404 T^{2} - 79752 T^{3} + 4895510 T^{4} - 79752 p^{3} T^{5} + 2404 p^{6} T^{6} - 40 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 48 T + 4080 T^{2} + 163056 T^{3} + 7581746 T^{4} + 163056 p^{3} T^{5} + 4080 p^{6} T^{6} + 48 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 152 T + 15832 T^{2} - 1323320 T^{3} + 98297586 T^{4} - 1323320 p^{3} T^{5} + 15832 p^{6} T^{6} - 152 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 224 T + 42396 T^{2} - 4864224 T^{3} + 486114742 T^{4} - 4864224 p^{3} T^{5} + 42396 p^{6} T^{6} - 224 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 9780 T^{2} - 1509576 T^{3} + 54064134 T^{4} - 1509576 p^{3} T^{5} + 9780 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 144 T + 70628 T^{2} + 7042032 T^{3} + 2292651510 T^{4} + 7042032 p^{3} T^{5} + 70628 p^{6} T^{6} + 144 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 400 T + 171340 T^{2} - 38254288 T^{3} + 8472479270 T^{4} - 38254288 p^{3} T^{5} + 171340 p^{6} T^{6} - 400 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 304 T + 169172 T^{2} + 40635984 T^{3} + 11843462518 T^{4} + 40635984 p^{3} T^{5} + 169172 p^{6} T^{6} + 304 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 152 T + 250232 T^{2} - 28847096 T^{3} + 25157924114 T^{4} - 28847096 p^{3} T^{5} + 250232 p^{6} T^{6} - 152 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 160 T + 147340 T^{2} + 16282528 T^{3} + 15975380470 T^{4} + 16282528 p^{3} T^{5} + 147340 p^{6} T^{6} + 160 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 544 T + 378668 T^{2} - 154041760 T^{3} + 58194085286 T^{4} - 154041760 p^{3} T^{5} + 378668 p^{6} T^{6} - 544 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 1320 T + 959116 T^{2} + 462674104 T^{3} + 189646882294 T^{4} + 462674104 p^{3} T^{5} + 959116 p^{6} T^{6} + 1320 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 1040 T + 962396 T^{2} - 555546128 T^{3} + 297086579030 T^{4} - 555546128 p^{3} T^{5} + 962396 p^{6} T^{6} - 1040 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 896 T + 1188912 T^{2} + 644590848 T^{3} + 437823850738 T^{4} + 644590848 p^{3} T^{5} + 1188912 p^{6} T^{6} + 896 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 416 T + 995500 T^{2} - 335375264 T^{3} + 430142403478 T^{4} - 335375264 p^{3} T^{5} + 995500 p^{6} T^{6} - 416 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 248 T + 996916 T^{2} + 246307512 T^{3} + 457673544134 T^{4} + 246307512 p^{3} T^{5} + 996916 p^{6} T^{6} + 248 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 752 T + 1272832 T^{2} - 641462384 T^{3} + 655763261282 T^{4} - 641462384 p^{3} T^{5} + 1272832 p^{6} T^{6} - 752 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 864 T + 1864956 T^{2} + 1235423200 T^{3} + 1357772054598 T^{4} + 1235423200 p^{3} T^{5} + 1864956 p^{6} T^{6} + 864 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 1456 T + 2297228 T^{2} - 2130405040 T^{3} + 2028397026326 T^{4} - 2130405040 p^{3} T^{5} + 2297228 p^{6} T^{6} - 1456 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 2936 T + 5403960 T^{2} + 6890733528 T^{3} + 6651967591762 T^{4} + 6890733528 p^{3} T^{5} + 5403960 p^{6} T^{6} + 2936 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 144 T + 2591968 T^{2} + 824150160 T^{3} + 3033197727234 T^{4} + 824150160 p^{3} T^{5} + 2591968 p^{6} T^{6} + 144 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.15641393488103979745387859545, −5.60377573751039686331580395602, −5.55436363038400989744444122851, −5.54934240287517458452306505654, −5.27395102459675750887708131261, −4.85959379643837231747259212359, −4.85696469266478646857803304609, −4.75671281675162617510702793376, −4.49729600594054760345473952202, −3.90245381220939963250781435134, −3.85905180844342084976918265061, −3.75959851501576893486198366745, −3.72751067918108962341354364575, −3.25781663591593930184682379372, −2.82078673695492735060326749381, −2.78025405556676087704015210639, −2.66713983782461789902079437835, −1.81882822535050180288754829137, −1.80837738372235450866747629242, −1.46916245052952995455567339001, −1.30351176812353816387036055896, −0.942970035270922539394652223855, −0.75758484406899958127357751025, −0.50046460290877733154693984346, −0.097302972617646981771182896733, 0.097302972617646981771182896733, 0.50046460290877733154693984346, 0.75758484406899958127357751025, 0.942970035270922539394652223855, 1.30351176812353816387036055896, 1.46916245052952995455567339001, 1.80837738372235450866747629242, 1.81882822535050180288754829137, 2.66713983782461789902079437835, 2.78025405556676087704015210639, 2.82078673695492735060326749381, 3.25781663591593930184682379372, 3.72751067918108962341354364575, 3.75959851501576893486198366745, 3.85905180844342084976918265061, 3.90245381220939963250781435134, 4.49729600594054760345473952202, 4.75671281675162617510702793376, 4.85696469266478646857803304609, 4.85959379643837231747259212359, 5.27395102459675750887708131261, 5.54934240287517458452306505654, 5.55436363038400989744444122851, 5.60377573751039686331580395602, 6.15641393488103979745387859545

Graph of the $Z$-function along the critical line