Properties

Label 2-2366-1.1-c1-0-39
Degree $2$
Conductor $2366$
Sign $1$
Analytic cond. $18.8926$
Root an. cond. $4.34656$
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 + 0.252·3-s + 4-s + 1.14·5-s + 0.252·6-s + 7-s + 8-s − 2.93·9-s + 1.14·10-s + 4.44·11-s + 0.252·12-s + 14-s + 0.290·15-s + 16-s + 2.70·17-s − 2.93·18-s + 6.56·19-s + 1.14·20-s + 0.252·21-s + 4.44·22-s − 2.07·23-s + 0.252·24-s − 3.68·25-s − 1.50·27-s + 28-s + 7.19·29-s + 0.290·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.145·3-s + 0.5·4-s + 0.513·5-s + 0.103·6-s + 0.377·7-s + 0.353·8-s − 0.978·9-s + 0.362·10-s + 1.34·11-s + 0.0729·12-s + 0.267·14-s + 0.0749·15-s + 0.250·16-s + 0.657·17-s − 0.692·18-s + 1.50·19-s + 0.256·20-s + 0.0551·21-s + 0.947·22-s − 0.433·23-s + 0.0516·24-s − 0.736·25-s − 0.288·27-s + 0.188·28-s + 1.33·29-s + 0.0529·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2366\)    =    \(2 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(18.8926\)
Root analytic conductor: \(4.34656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2366,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.616834550\)
\(L(\frac12)\) \(\approx\) \(3.616834550\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 \)
good3 \( 1 - 0.252T + 3T^{2} \)
5 \( 1 - 1.14T + 5T^{2} \)
11 \( 1 - 4.44T + 11T^{2} \)
17 \( 1 - 2.70T + 17T^{2} \)
19 \( 1 - 6.56T + 19T^{2} \)
23 \( 1 + 2.07T + 23T^{2} \)
29 \( 1 - 7.19T + 29T^{2} \)
31 \( 1 + 7.90T + 31T^{2} \)
37 \( 1 + 9.64T + 37T^{2} \)
41 \( 1 - 9.49T + 41T^{2} \)
43 \( 1 + 3.40T + 43T^{2} \)
47 \( 1 + 1.67T + 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 + 0.0677T + 59T^{2} \)
61 \( 1 - 8.10T + 61T^{2} \)
67 \( 1 - 0.513T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 8.02T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 - 2.12T + 97T^{2} \)
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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

−9.014538429411604651595019347875, −8.187988419444336982525294555597, −7.32616759152327662636439731216, −6.52522683549684448591997433516, −5.62754231898247012560681358826, −5.27349824229311006955856083379, −3.99780094304221561441179651022, −3.34768799854453957149637518661, −2.27441578623463540866532557438, −1.20750747207444022522903627562, 1.20750747207444022522903627562, 2.27441578623463540866532557438, 3.34768799854453957149637518661, 3.99780094304221561441179651022, 5.27349824229311006955856083379, 5.62754231898247012560681358826, 6.52522683549684448591997433516, 7.32616759152327662636439731216, 8.187988419444336982525294555597, 9.014538429411604651595019347875

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