Properties

Degree 8
Conductor $ 2^{8} \cdot 7^{8} \cdot 41^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·5-s + 2·9-s + 4·11-s + 8·15-s + 4·17-s − 6·19-s − 12·23-s + 4·25-s + 4·27-s − 4·29-s + 8·31-s − 8·33-s + 16·37-s + 4·41-s + 4·43-s − 8·45-s + 6·47-s − 8·51-s − 16·53-s − 16·55-s + 12·57-s − 12·59-s − 24·61-s + 28·67-s + 24·69-s − 2·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 2/3·9-s + 1.20·11-s + 2.06·15-s + 0.970·17-s − 1.37·19-s − 2.50·23-s + 4/5·25-s + 0.769·27-s − 0.742·29-s + 1.43·31-s − 1.39·33-s + 2.63·37-s + 0.624·41-s + 0.609·43-s − 1.19·45-s + 0.875·47-s − 1.12·51-s − 2.19·53-s − 2.15·55-s + 1.58·57-s − 1.56·59-s − 3.07·61-s + 3.42·67-s + 2.88·69-s − 0.237·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{8} \cdot 7^{8} \cdot 41^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8036} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{8} \cdot 7^{8} \cdot 41^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(0.5626123297\)
\(L(\frac12)\)  \(\approx\)  \(0.5626123297\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;41\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41$C_1$ \( ( 1 - T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + 2 T + 2 T^{2} - 4 T^{3} - 8 T^{4} - 4 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 4 T + 12 T^{2} + 16 T^{3} + 34 T^{4} + 16 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 4 T + 26 T^{2} - 114 T^{3} + 384 T^{4} - 114 p T^{5} + 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 12 T^{2} + 48 T^{3} + 118 T^{4} + 48 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 4 T + 20 T^{2} - 124 T^{3} + 534 T^{4} - 124 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 6 T + 62 T^{2} + 208 T^{3} + 1448 T^{4} + 208 p T^{5} + 62 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 12 T + 108 T^{2} + 700 T^{3} + 3718 T^{4} + 700 p T^{5} + 108 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 4 T + 76 T^{2} + 12 p T^{3} + 2870 T^{4} + 12 p^{2} T^{5} + 76 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 8 T + 92 T^{2} - 712 T^{3} + 3846 T^{4} - 712 p T^{5} + 92 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 16 T + 212 T^{2} - 1740 T^{3} + 12626 T^{4} - 1740 p T^{5} + 212 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 4 T + 124 T^{2} - 244 T^{3} + 6678 T^{4} - 244 p T^{5} + 124 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 6 T + 126 T^{2} - 640 T^{3} + 8608 T^{4} - 640 p T^{5} + 126 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 16 T + 4 p T^{2} + 1824 T^{3} + 15558 T^{4} + 1824 p T^{5} + 4 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 12 T + 252 T^{2} + 1996 T^{3} + 22582 T^{4} + 1996 p T^{5} + 252 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 24 T + 420 T^{2} + 4824 T^{3} + 44086 T^{4} + 4824 p T^{5} + 420 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 28 T + 538 T^{2} - 6638 T^{3} + 64208 T^{4} - 6638 p T^{5} + 538 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 2 T + 98 T^{2} - 268 T^{3} + 48 p T^{4} - 268 p T^{5} + 98 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 8 T + 212 T^{2} + 1060 T^{3} + 19890 T^{4} + 1060 p T^{5} + 212 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 18 T + 366 T^{2} + 4224 T^{3} + 45328 T^{4} + 4224 p T^{5} + 366 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 12 T + 252 T^{2} - 1644 T^{3} + 24598 T^{4} - 1644 p T^{5} + 252 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 4 T + 228 T^{2} + 796 T^{3} + 326 p T^{4} + 796 p T^{5} + 228 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 16 T + 268 T^{2} + 3376 T^{3} + 38118 T^{4} + 3376 p T^{5} + 268 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.76341423448914135704479225613, −5.24270651171888901579667815582, −4.96874127198491779714891165446, −4.91977417285033395412179726190, −4.57334345666562291311843201601, −4.51347622807082402932696548503, −4.40418347434474417132724448852, −4.39802973805081363137438000904, −3.98902580672859115147281611352, −3.71194421171989158426874434892, −3.66923624399222566219068459732, −3.64791273370711140922520327103, −3.45990619207554907656513300565, −2.84378768197958473543549434162, −2.81540700855864372681098927662, −2.60792281870034595819969482731, −2.48510818608992314013434872527, −2.00755783222049729545104591313, −1.90035878920286947355075789376, −1.45194768314063447941970447733, −1.36080858171967737786895083088, −0.972711304184212734527438129061, −0.911941813392380263609007453269, −0.26591607078661686952370277140, −0.24327616662904040255591992902, 0.24327616662904040255591992902, 0.26591607078661686952370277140, 0.911941813392380263609007453269, 0.972711304184212734527438129061, 1.36080858171967737786895083088, 1.45194768314063447941970447733, 1.90035878920286947355075789376, 2.00755783222049729545104591313, 2.48510818608992314013434872527, 2.60792281870034595819969482731, 2.81540700855864372681098927662, 2.84378768197958473543549434162, 3.45990619207554907656513300565, 3.64791273370711140922520327103, 3.66923624399222566219068459732, 3.71194421171989158426874434892, 3.98902580672859115147281611352, 4.39802973805081363137438000904, 4.40418347434474417132724448852, 4.51347622807082402932696548503, 4.57334345666562291311843201601, 4.91977417285033395412179726190, 4.96874127198491779714891165446, 5.24270651171888901579667815582, 5.76341423448914135704479225613

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