Properties

Degree 8
Conductor $ 3^{4} \cdot 5^{8} \cdot 7^{4} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 14·4-s − 18·9-s − 32·11-s + 51·16-s + 192·19-s − 376·29-s − 240·31-s − 252·36-s + 200·41-s − 448·44-s − 98·49-s − 208·59-s − 936·61-s − 420·64-s − 272·71-s + 2.68e3·76-s + 864·79-s + 243·81-s + 2.80e3·89-s + 576·99-s − 328·101-s − 280·109-s − 5.26e3·116-s + 1.71e3·121-s − 3.36e3·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 7/4·4-s − 2/3·9-s − 0.877·11-s + 0.796·16-s + 2.31·19-s − 2.40·29-s − 1.39·31-s − 7/6·36-s + 0.761·41-s − 1.53·44-s − 2/7·49-s − 0.458·59-s − 1.96·61-s − 0.820·64-s − 0.454·71-s + 4.05·76-s + 1.23·79-s + 1/3·81-s + 3.34·89-s + 0.584·99-s − 0.323·101-s − 0.246·109-s − 4.21·116-s + 1.28·121-s − 2.43·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{4} \cdot 5^{8} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{525} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)
\(L(2)\)  \(\approx\)  \(0.3598166946\)
\(L(\frac12)\)  \(\approx\)  \(0.3598166946\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
5 \( 1 \)
7$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
good2$D_4\times C_2$ \( 1 - 7 p T^{2} + 145 T^{4} - 7 p^{7} T^{6} + p^{12} T^{8} \)
11$D_{4}$ \( ( 1 + 16 T - 474 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 5836 T^{2} + 17983510 T^{4} - 5836 p^{6} T^{6} + p^{12} T^{8} \)
17$D_4\times C_2$ \( 1 - 700 p T^{2} + 83185606 T^{4} - 700 p^{7} T^{6} + p^{12} T^{8} \)
19$D_{4}$ \( ( 1 - 96 T + 13974 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 30044 T^{2} + 519364774 T^{4} - 30044 p^{6} T^{6} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 188 T + 34286 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 120 T + 60590 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 124204 T^{2} + 8380919670 T^{4} - 124204 p^{6} T^{6} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 100 T + 89142 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 112876 T^{2} + 6993047350 T^{4} - 112876 p^{6} T^{6} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 + 43524 T^{2} + 9878986694 T^{4} + 43524 p^{6} T^{6} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 139244 T^{2} + 23569194934 T^{4} - 139244 p^{6} T^{6} + p^{12} T^{8} \)
59$D_{4}$ \( ( 1 + 104 T + 80534 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 468 T + 494606 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 244724 T^{2} + 162973601494 T^{4} + 244724 p^{6} T^{6} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 + 136 T + 540446 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 231908 T^{2} + 102056920486 T^{4} + 231908 p^{6} T^{6} + p^{12} T^{8} \)
79$D_{4}$ \( ( 1 - 432 T + 602142 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 2003724 T^{2} + 1638356284694 T^{4} - 2003724 p^{6} T^{6} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 - 1404 T + 1802390 T^{2} - 1404 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 30012 p T^{2} + 3760771737350 T^{4} - 30012 p^{7} T^{6} + p^{12} T^{8} \)
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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

−7.42352238183458281935175914184, −7.38587612940060796665089327720, −7.03224188546199446463504659677, −6.53901162618826522119014304450, −6.52098345512566227270857767548, −6.21190783877143353400978157624, −6.01219921610400689026852646202, −5.61380325139987437745669345677, −5.43645285244259870090673784417, −5.34928425951295273483486321233, −5.04721798697159176931049261675, −4.73575836325827677520876230848, −4.34444997278207465325877110384, −3.99460517101096099218866406134, −3.58594363017692517235905469043, −3.35557173768033699089864519082, −3.21729990458931298399777354219, −2.80115695381813566034433924974, −2.60844670117902808385764646680, −2.12008348659027572228565817607, −1.95636227347155551477786825347, −1.68404896701510478572898027533, −1.17559999782369849778827296974, −0.74753662772118604501513033014, −0.07683288953441671884117443745, 0.07683288953441671884117443745, 0.74753662772118604501513033014, 1.17559999782369849778827296974, 1.68404896701510478572898027533, 1.95636227347155551477786825347, 2.12008348659027572228565817607, 2.60844670117902808385764646680, 2.80115695381813566034433924974, 3.21729990458931298399777354219, 3.35557173768033699089864519082, 3.58594363017692517235905469043, 3.99460517101096099218866406134, 4.34444997278207465325877110384, 4.73575836325827677520876230848, 5.04721798697159176931049261675, 5.34928425951295273483486321233, 5.43645285244259870090673784417, 5.61380325139987437745669345677, 6.01219921610400689026852646202, 6.21190783877143353400978157624, 6.52098345512566227270857767548, 6.53901162618826522119014304450, 7.03224188546199446463504659677, 7.38587612940060796665089327720, 7.42352238183458281935175914184

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