Properties

Label 2-2394-1.1-c1-0-40
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 $1$

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 + 2·13-s − 14-s + 16-s − 2·17-s − 19-s − 2·20-s − 8·23-s − 25-s + 2·26-s − 28-s − 2·29-s + 4·31-s + 32-s − 2·34-s + 2·35-s + 2·37-s − 38-s − 2·40-s − 6·41-s − 12·43-s − 8·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.229·19-s − 0.447·20-s − 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s + 0.338·35-s + 0.328·37-s − 0.162·38-s − 0.316·40-s − 0.937·41-s − 1.82·43-s − 1.17·46-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.281501563887625996464501642974, −7.912508640806354969373892342740, −6.85350902861062329638401661291, −6.28352954329121793129107491158, −5.40960412801561297413267643753, −4.33084839581572636702073752349, −3.84907283320357724315521311959, −2.94285784851423154213390802202, −1.74411672296421629506581859944, 0, 1.74411672296421629506581859944, 2.94285784851423154213390802202, 3.84907283320357724315521311959, 4.33084839581572636702073752349, 5.40960412801561297413267643753, 6.28352954329121793129107491158, 6.85350902861062329638401661291, 7.912508640806354969373892342740, 8.281501563887625996464501642974

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