Properties

Label 2-2394-1.1-c1-0-22
Degree $2$
Conductor $2394$
Sign $1$
Analytic cond. $19.1161$
Root an. cond. $4.37220$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·5-s + 7-s − 8-s − 2·10-s + 4.82·11-s + 5.65·13-s − 14-s + 16-s + 6.82·17-s − 19-s + 2·20-s − 4.82·22-s + 4·23-s − 25-s − 5.65·26-s + 28-s + 2·29-s − 4.82·31-s − 32-s − 6.82·34-s + 2·35-s − 8.48·37-s + 38-s − 2·40-s − 3.65·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.894·5-s + 0.377·7-s − 0.353·8-s − 0.632·10-s + 1.45·11-s + 1.56·13-s − 0.267·14-s + 0.250·16-s + 1.65·17-s − 0.229·19-s + 0.447·20-s − 1.02·22-s + 0.834·23-s − 0.200·25-s − 1.10·26-s + 0.188·28-s + 0.371·29-s − 0.867·31-s − 0.176·32-s − 1.17·34-s + 0.338·35-s − 1.39·37-s + 0.162·38-s − 0.316·40-s − 0.571·41-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: no
Arithmetic: yes
Character: $\chi_{2394} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.053449934\)
\(L(\frac12)\) \(\approx\) \(2.053449934\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T + 5T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 4.82T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 + 3.65T + 41T^{2} \)
43 \( 1 - 9.65T + 43T^{2} \)
47 \( 1 + 4.48T + 47T^{2} \)
53 \( 1 + 7.65T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 + 6.82T + 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 7.65T + 73T^{2} \)
79 \( 1 - 7.65T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−9.091235061615110192500441976884, −8.367850581834467909623872376838, −7.51564378683913989797805091718, −6.58411837978721256143834293048, −6.01953831112159033340689169029, −5.27580575356770397780687947488, −3.93433865226781707539049283785, −3.15970766137001751001675268124, −1.66239463988881447462814017537, −1.21822951447109040137952411219, 1.21822951447109040137952411219, 1.66239463988881447462814017537, 3.15970766137001751001675268124, 3.93433865226781707539049283785, 5.27580575356770397780687947488, 6.01953831112159033340689169029, 6.58411837978721256143834293048, 7.51564378683913989797805091718, 8.367850581834467909623872376838, 9.091235061615110192500441976884

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