Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{8} \cdot 7^{2} $
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 − 7-s − 8-s + 4·13-s − 14-s − 16-s + 12·17-s + 4·19-s + 5·25-s + 4·26-s + 6·29-s + 4·31-s + 12·34-s + 4·37-s + 4·38-s − 6·41-s − 8·43-s + 12·47-s + 5·50-s + 12·53-s + 56-s + 6·58-s + 6·59-s − 8·61-s + 4·62-s + 64-s + 4·67-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.377·7-s − 0.353·8-s + 1.10·13-s − 0.267·14-s − 1/4·16-s + 2.91·17-s + 0.917·19-s + 25-s + 0.784·26-s + 1.11·29-s + 0.718·31-s + 2.05·34-s + 0.657·37-s + 0.648·38-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 0.707·50-s + 1.64·53-s + 0.133·56-s + 0.787·58-s + 0.781·59-s − 1.02·61-s + 0.508·62-s + 1/8·64-s + 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1134} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1285956,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $3.925715946$
$L(\frac12)$  $\approx$  $3.925715946$
$L(\frac{3}{2})$   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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−9.965363900033757056542797857688, −9.896832065968700209378951010560, −8.981262373361538721293537345588, −8.974194983815212186385065008631, −8.294617437855192102100987822451, −8.031293774802245118875663112020, −7.48655076274356680955745994755, −7.13726025543389462193913449045, −6.51934795054915739147487627171, −6.20692714034726877009450547907, −5.63891753560402551551713279035, −5.40485379215751779468256095872, −4.95184129186987004969716310671, −4.38023679322460222737079841298, −3.67027232218535153757676445394, −3.45173581884622919931806138410, −3.01118174722901517185218105187, −2.43237795903141877385950354742, −1.10167326553198431223830906085, −1.06526607354851973657984272848, 1.06526607354851973657984272848, 1.10167326553198431223830906085, 2.43237795903141877385950354742, 3.01118174722901517185218105187, 3.45173581884622919931806138410, 3.67027232218535153757676445394, 4.38023679322460222737079841298, 4.95184129186987004969716310671, 5.40485379215751779468256095872, 5.63891753560402551551713279035, 6.20692714034726877009450547907, 6.51934795054915739147487627171, 7.13726025543389462193913449045, 7.48655076274356680955745994755, 8.031293774802245118875663112020, 8.294617437855192102100987822451, 8.974194983815212186385065008631, 8.981262373361538721293537345588, 9.896832065968700209378951010560, 9.965363900033757056542797857688

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