Properties

Label 4-2394e2-1.1-c1e2-0-1
Degree $4$
Conductor $5731236$
Sign $1$
Analytic cond. $365.428$
Root an. cond. $4.37220$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·5-s − 2·7-s + 8-s + 3·10-s − 8·11-s − 3·13-s + 2·14-s − 16-s − 2·17-s − 19-s + 8·22-s + 23-s + 5·25-s + 3·26-s − 12·31-s + 2·34-s + 6·35-s + 8·37-s + 38-s − 3·40-s + 4·41-s + 6·43-s − 46-s − 2·47-s + 3·49-s − 5·50-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.34·5-s − 0.755·7-s + 0.353·8-s + 0.948·10-s − 2.41·11-s − 0.832·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s − 0.229·19-s + 1.70·22-s + 0.208·23-s + 25-s + 0.588·26-s − 2.15·31-s + 0.342·34-s + 1.01·35-s + 1.31·37-s + 0.162·38-s − 0.474·40-s + 0.624·41-s + 0.914·43-s − 0.147·46-s − 0.291·47-s + 3/7·49-s − 0.707·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5731236\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(365.428\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2394} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5731236,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1769026006\)
\(L(\frac12)\) \(\approx\) \(0.1769026006\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
19$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 4 T - 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 7 T - 22 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−9.133970303705775399907781801249, −8.703384850718296031286339079443, −8.380985307150271879563876966087, −7.903842742194320840529669973278, −7.65775925697095361332967329054, −7.47686521122505276793023086036, −6.98274740298103474054795386629, −6.78065127179588594301743962305, −5.87346395470072600320149432101, −5.71827095948449913959230065037, −5.22089256028457055065771119046, −4.72005707002373851847232298019, −4.37195260222924538857240390963, −3.96268734758562281318611646834, −3.18348530336096331408348201154, −3.10850891205835588942920262761, −2.23421797727027978624717093921, −2.17906795911070252714031494425, −0.814600805322982599120281158000, −0.21866548320714900126141631064, 0.21866548320714900126141631064, 0.814600805322982599120281158000, 2.17906795911070252714031494425, 2.23421797727027978624717093921, 3.10850891205835588942920262761, 3.18348530336096331408348201154, 3.96268734758562281318611646834, 4.37195260222924538857240390963, 4.72005707002373851847232298019, 5.22089256028457055065771119046, 5.71827095948449913959230065037, 5.87346395470072600320149432101, 6.78065127179588594301743962305, 6.98274740298103474054795386629, 7.47686521122505276793023086036, 7.65775925697095361332967329054, 7.903842742194320840529669973278, 8.380985307150271879563876966087, 8.703384850718296031286339079443, 9.133970303705775399907781801249

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