Properties

Degree $4$
Conductor $3111696$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 2·5-s + 6·9-s − 4·11-s − 3·13-s − 6·15-s + 14·17-s + 10·19-s − 4·23-s + 5·25-s + 9·27-s + 29-s + 3·31-s − 12·33-s + 22·37-s − 9·39-s + 9·41-s − 5·43-s − 12·45-s − 3·47-s + 42·51-s + 6·53-s + 8·55-s + 30·57-s + 7·59-s − 3·61-s + 6·65-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.894·5-s + 2·9-s − 1.20·11-s − 0.832·13-s − 1.54·15-s + 3.39·17-s + 2.29·19-s − 0.834·23-s + 25-s + 1.73·27-s + 0.185·29-s + 0.538·31-s − 2.08·33-s + 3.61·37-s − 1.44·39-s + 1.40·41-s − 0.762·43-s − 1.78·45-s − 0.437·47-s + 5.88·51-s + 0.824·53-s + 1.07·55-s + 3.97·57-s + 0.911·59-s − 0.384·61-s + 0.744·65-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

\(L(1)\) \(\approx\) \(5.057026891\)
\(L(\frac12)\) \(\approx\) \(5.057026891\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
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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

−9.494369520059240105349093439046, −9.289373618114066976767908533012, −8.439381825879307724344873397904, −8.247968132402511121733142599034, −7.69929798365836044307133322510, −7.62200855230161455202228512280, −7.56525048000470160776068205357, −7.21508148579139680569296874029, −6.13151671014564510412031210945, −5.96383061153251407913677904124, −5.30774630498182352102861261643, −4.89091267378441973098772403982, −4.55742412940866144273103127435, −3.82924845079674152275958423076, −3.32918717017753630730606040679, −3.26810381607533865412834763725, −2.56852784842004776780560795607, −2.43113924164211559633267557438, −1.09449135929903545031904738485, −0.983440521401001918670678879355, 0.983440521401001918670678879355, 1.09449135929903545031904738485, 2.43113924164211559633267557438, 2.56852784842004776780560795607, 3.26810381607533865412834763725, 3.32918717017753630730606040679, 3.82924845079674152275958423076, 4.55742412940866144273103127435, 4.89091267378441973098772403982, 5.30774630498182352102861261643, 5.96383061153251407913677904124, 6.13151671014564510412031210945, 7.21508148579139680569296874029, 7.56525048000470160776068205357, 7.62200855230161455202228512280, 7.69929798365836044307133322510, 8.247968132402511121733142599034, 8.439381825879307724344873397904, 9.289373618114066976767908533012, 9.494369520059240105349093439046

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