Properties

Degree 4
Conductor $ 2^{12} \cdot 3^{2} \cdot 7^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·5-s + 3·9-s + 4·11-s − 8·13-s + 8·15-s + 4·17-s − 4·23-s + 4·25-s − 4·27-s + 8·29-s − 8·31-s − 8·33-s + 8·37-s + 16·39-s + 4·41-s − 12·45-s − 8·51-s + 4·53-s − 16·55-s + 8·59-s − 16·61-s + 32·65-s + 8·69-s − 4·71-s + 8·73-s − 8·75-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 9-s + 1.20·11-s − 2.21·13-s + 2.06·15-s + 0.970·17-s − 0.834·23-s + 4/5·25-s − 0.769·27-s + 1.48·29-s − 1.43·31-s − 1.39·33-s + 1.31·37-s + 2.56·39-s + 0.624·41-s − 1.78·45-s − 1.12·51-s + 0.549·53-s − 2.15·55-s + 1.04·59-s − 2.04·61-s + 3.96·65-s + 0.963·69-s − 0.474·71-s + 0.936·73-s − 0.923·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(88510464\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{9408} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 168 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 64 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 208 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.44580358884736283410009563597, −7.37926055663323740349445626231, −6.74289247590207632968240427264, −6.72179868179206086224145312982, −6.12698372246636125989589359703, −5.87595854259471568082728533687, −5.38897811734563400694683718008, −5.15438545557841502301353931146, −4.59039069376921842145555086532, −4.47709280091476783126997238757, −3.99191189864677436049537304978, −3.94490002001419325246999053548, −3.21949935623726062744663923486, −3.07834121522320874497585029493, −2.17419224357951494962394437670, −2.12944986438998220534557064232, −1.10897008304573366247725079464, −0.925649264892962053697896084650, 0, 0, 0.925649264892962053697896084650, 1.10897008304573366247725079464, 2.12944986438998220534557064232, 2.17419224357951494962394437670, 3.07834121522320874497585029493, 3.21949935623726062744663923486, 3.94490002001419325246999053548, 3.99191189864677436049537304978, 4.47709280091476783126997238757, 4.59039069376921842145555086532, 5.15438545557841502301353931146, 5.38897811734563400694683718008, 5.87595854259471568082728533687, 6.12698372246636125989589359703, 6.72179868179206086224145312982, 6.74289247590207632968240427264, 7.37926055663323740349445626231, 7.44580358884736283410009563597

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