Properties

Degree $4$
Conductor $3111696$
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  − 2·5-s + 2·11-s + 8·13-s − 6·17-s + 8·19-s − 6·23-s + 5·25-s + 20·29-s + 4·31-s − 6·37-s − 12·41-s + 8·43-s − 8·47-s + 2·53-s − 4·55-s + 4·59-s − 8·61-s − 16·65-s + 8·67-s + 20·71-s + 4·73-s − 4·79-s + 24·83-s + 12·85-s + 14·89-s − 16·95-s − 8·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.603·11-s + 2.21·13-s − 1.45·17-s + 1.83·19-s − 1.25·23-s + 25-s + 3.71·29-s + 0.718·31-s − 0.986·37-s − 1.87·41-s + 1.21·43-s − 1.16·47-s + 0.274·53-s − 0.539·55-s + 0.520·59-s − 1.02·61-s − 1.98·65-s + 0.977·67-s + 2.37·71-s + 0.468·73-s − 0.450·79-s + 2.63·83-s + 1.30·85-s + 1.48·89-s − 1.64·95-s − 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3111696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.635417944\)
\(L(\frac12)\) \(\approx\) \(2.635417944\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.360744082005070130876122216836, −8.929437560417049530649254143739, −8.701715136667653329038702155890, −8.279065044700320875082612490360, −8.055250166016411363199403881654, −7.71877403305596125776327766156, −6.90523684432299866215992131695, −6.55042227123339253103176996654, −6.52352173358036477463193674405, −6.12833013177810949718136754426, −5.23651417419348683409569018204, −5.03869372266193042569691126211, −4.47716268805571381575434138277, −4.06264070557333237350571060364, −3.49646891531374989788758422672, −3.35051073617788004427920111886, −2.67596701654423032832564116924, −1.92330405248765734572686974635, −1.11424805813388068093017046870, −0.75522123158939433778133858115, 0.75522123158939433778133858115, 1.11424805813388068093017046870, 1.92330405248765734572686974635, 2.67596701654423032832564116924, 3.35051073617788004427920111886, 3.49646891531374989788758422672, 4.06264070557333237350571060364, 4.47716268805571381575434138277, 5.03869372266193042569691126211, 5.23651417419348683409569018204, 6.12833013177810949718136754426, 6.52352173358036477463193674405, 6.55042227123339253103176996654, 6.90523684432299866215992131695, 7.71877403305596125776327766156, 8.055250166016411363199403881654, 8.279065044700320875082612490360, 8.701715136667653329038702155890, 8.929437560417049530649254143739, 9.360744082005070130876122216836

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