Properties

Degree 4
Conductor $ 2^{12} \cdot 3^{2} \cdot 7^{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 + 3·9-s − 4·11-s + 4·13-s − 8·17-s + 4·23-s + 2·25-s + 4·27-s − 4·29-s + 8·31-s − 8·33-s + 4·37-s + 8·39-s − 16·41-s + 16·43-s − 8·47-s − 16·51-s + 4·53-s − 16·59-s − 4·61-s − 8·67-s + 8·69-s + 12·71-s − 12·73-s + 4·75-s − 8·79-s + 5·81-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 1.20·11-s + 1.10·13-s − 1.94·17-s + 0.834·23-s + 2/5·25-s + 0.769·27-s − 0.742·29-s + 1.43·31-s − 1.39·33-s + 0.657·37-s + 1.28·39-s − 2.49·41-s + 2.43·43-s − 1.16·47-s − 2.24·51-s + 0.549·53-s − 2.08·59-s − 0.512·61-s − 0.977·67-s + 0.963·69-s + 1.42·71-s − 1.40·73-s + 0.461·75-s − 0.900·79-s + 5/9·81-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  =  \(0\)
Selberg data  =  \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.671070684\)
\(L(\frac12)\)  \(\approx\)  \(2.671070684\)
\(L(\frac{3}{2})\)   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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 166 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−7.81752263130246188765490064504, −7.65651325732566230463417525634, −7.29287451208356264095984422391, −6.81465441514980751787628056285, −6.53337898913859922368373281997, −6.41475120588702610241165444292, −5.70223318803946767535037767441, −5.59189357643917127795823790665, −5.00919488270935039219713240695, −4.61628337285320741375164272825, −4.25312523685626085477505613103, −4.23773742306480475982478718798, −3.46540363544261841324593555692, −3.16220555108758974624089982498, −2.70174786262735373283221458002, −2.68514697708499295060152238655, −1.93407079451544467341685431602, −1.65944002383093194328141439287, −1.10128092783868310464318061590, −0.33798999570611027605643578164, 0.33798999570611027605643578164, 1.10128092783868310464318061590, 1.65944002383093194328141439287, 1.93407079451544467341685431602, 2.68514697708499295060152238655, 2.70174786262735373283221458002, 3.16220555108758974624089982498, 3.46540363544261841324593555692, 4.23773742306480475982478718798, 4.25312523685626085477505613103, 4.61628337285320741375164272825, 5.00919488270935039219713240695, 5.59189357643917127795823790665, 5.70223318803946767535037767441, 6.41475120588702610241165444292, 6.53337898913859922368373281997, 6.81465441514980751787628056285, 7.29287451208356264095984422391, 7.65651325732566230463417525634, 7.81752263130246188765490064504

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