Properties

Degree 8
Conductor $ 3^{4} \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-s + 4·3-s − 2·4-s + 3·5-s + 4·6-s − 2·8-s + 10·9-s + 3·10-s − 5·11-s − 8·12-s + 5·13-s + 12·15-s − 16-s − 5·17-s + 10·18-s + 6·19-s − 6·20-s − 5·22-s + 3·23-s − 8·24-s − 8·25-s + 5·26-s + 20·27-s + 2·29-s + 12·30-s + 11·31-s − 3·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.30·3-s − 4-s + 1.34·5-s + 1.63·6-s − 0.707·8-s + 10/3·9-s + 0.948·10-s − 1.50·11-s − 2.30·12-s + 1.38·13-s + 3.09·15-s − 1/4·16-s − 1.21·17-s + 2.35·18-s + 1.37·19-s − 1.34·20-s − 1.06·22-s + 0.625·23-s − 1.63·24-s − 8/5·25-s + 0.980·26-s + 3.84·27-s + 0.371·29-s + 2.19·30-s + 1.97·31-s − 0.530·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{4} \cdot 7^{8} \cdot 41^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6027} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 3^{4} \cdot 7^{8} \cdot 41^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $28.28457818$
$L(\frac12)$  $\approx$  $28.28457818$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
41$C_1$ \( ( 1 - T )^{4} \)
good2$C_2 \wr S_4$ \( 1 - T + 3 T^{2} - 3 T^{3} + p^{3} T^{4} - 3 p T^{5} + 3 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 3 T + 17 T^{2} - 38 T^{3} + 122 T^{4} - 38 p T^{5} + 17 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 5 T + 42 T^{2} + 145 T^{3} + 690 T^{4} + 145 p T^{5} + 42 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 5 T + 3 p T^{2} - 148 T^{3} + 764 T^{4} - 148 p T^{5} + 3 p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 5 T + 58 T^{2} + 247 T^{3} + 1398 T^{4} + 247 p T^{5} + 58 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 6 T + 72 T^{2} - 330 T^{3} + 2022 T^{4} - 330 p T^{5} + 72 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 3 T + 47 T^{2} - 262 T^{3} + 1088 T^{4} - 262 p T^{5} + 47 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 2 T + 54 T^{2} - 130 T^{3} + 1941 T^{4} - 130 p T^{5} + 54 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 11 T + 150 T^{2} - 967 T^{3} + 7190 T^{4} - 967 p T^{5} + 150 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 4 T + 110 T^{2} - 508 T^{3} + 5379 T^{4} - 508 p T^{5} + 110 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 19 T + 230 T^{2} + 1807 T^{3} + 12846 T^{4} + 1807 p T^{5} + 230 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 4 T + 38 T^{2} - 156 T^{3} - 1117 T^{4} - 156 p T^{5} + 38 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 9 T + 177 T^{2} - 1048 T^{3} + 12638 T^{4} - 1048 p T^{5} + 177 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 16 T + 288 T^{2} - 2884 T^{3} + 27038 T^{4} - 2884 p T^{5} + 288 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 27 T + 440 T^{2} - 4789 T^{3} + 42114 T^{4} - 4789 p T^{5} + 440 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 13 T + 261 T^{2} + 2600 T^{3} + 25998 T^{4} + 2600 p T^{5} + 261 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - T + 108 T^{2} - 729 T^{3} + 5138 T^{4} - 729 p T^{5} + 108 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 25 T + 490 T^{2} - 6051 T^{3} + 61430 T^{4} - 6051 p T^{5} + 490 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + T + 113 T^{2} + 424 T^{3} + 9198 T^{4} + 424 p T^{5} + 113 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 268 T^{2} + 144 T^{3} + 30726 T^{4} + 144 p T^{5} + 268 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 10 T + 316 T^{2} - 2606 T^{3} + 40470 T^{4} - 2606 p T^{5} + 316 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 15 T + 319 T^{2} + 3494 T^{3} + 45974 T^{4} + 3494 p T^{5} + 319 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.66097074611877612734837244178, −5.28373734483245046359074436086, −5.26274685436588270753086327861, −5.20940055477422901691465283945, −4.93254017753831758949180927288, −4.55064061691604467599709184044, −4.51917019306950125394227445550, −4.30784490054688919712410742117, −4.20502828972309407945557062413, −3.83231112671185332548974236389, −3.62742135017520801864893846074, −3.57039981056327012749947034668, −3.55183377681472187776939238284, −2.95568067188973707092916207759, −2.90221058852194751988939217289, −2.79849874987979233633920068320, −2.35233810266798006277017958435, −2.26228361192809710772194810035, −2.21157660179966866005681968225, −1.82293284791531446133331586147, −1.72342363493978399200393971813, −1.34199826485016237070942909885, −0.808217620795406463893468902869, −0.792560062031241744548811686836, −0.43968734951597055735551428549, 0.43968734951597055735551428549, 0.792560062031241744548811686836, 0.808217620795406463893468902869, 1.34199826485016237070942909885, 1.72342363493978399200393971813, 1.82293284791531446133331586147, 2.21157660179966866005681968225, 2.26228361192809710772194810035, 2.35233810266798006277017958435, 2.79849874987979233633920068320, 2.90221058852194751988939217289, 2.95568067188973707092916207759, 3.55183377681472187776939238284, 3.57039981056327012749947034668, 3.62742135017520801864893846074, 3.83231112671185332548974236389, 4.20502828972309407945557062413, 4.30784490054688919712410742117, 4.51917019306950125394227445550, 4.55064061691604467599709184044, 4.93254017753831758949180927288, 5.20940055477422901691465283945, 5.26274685436588270753086327861, 5.28373734483245046359074436086, 5.66097074611877612734837244178

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