Properties

Label 2-2394-1.1-c1-0-16
Degree $2$
Conductor $2394$
Sign $1$
Analytic cond. $19.1161$
Root an. cond. $4.37220$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 4·5-s − 7-s − 8-s − 4·10-s + 6·11-s − 4·13-s + 14-s + 16-s + 4·17-s − 19-s + 4·20-s − 6·22-s − 2·23-s + 11·25-s + 4·26-s − 28-s − 2·29-s + 4·31-s − 32-s − 4·34-s − 4·35-s + 2·37-s + 38-s − 4·40-s − 6·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.377·7-s − 0.353·8-s − 1.26·10-s + 1.80·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.229·19-s + 0.894·20-s − 1.27·22-s − 0.417·23-s + 11/5·25-s + 0.784·26-s − 0.188·28-s − 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.685·34-s − 0.676·35-s + 0.328·37-s + 0.162·38-s − 0.632·40-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2394} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.962236938\)
\(L(\frac12)\) \(\approx\) \(1.962236938\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

Imaginary part of the first few zeros on the critical line

−9.170766396680054577994306068589, −8.478463677722098593043536921802, −7.26142226179841058711952374724, −6.66923383553576609964750752019, −5.99160831439827741765439496257, −5.32705438536939705343146893129, −4.07413511690756283882706586153, −2.85126478970855653424985240380, −1.96927041097654768116323530907, −1.07395817415686514733421458233, 1.07395817415686514733421458233, 1.96927041097654768116323530907, 2.85126478970855653424985240380, 4.07413511690756283882706586153, 5.32705438536939705343146893129, 5.99160831439827741765439496257, 6.66923383553576609964750752019, 7.26142226179841058711952374724, 8.478463677722098593043536921802, 9.170766396680054577994306068589

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