Properties

Label 2-2366-1.1-c1-0-23
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 − 3.30·3-s + 4-s + 0.376·5-s − 3.30·6-s + 7-s + 8-s + 7.92·9-s + 0.376·10-s + 5.68·11-s − 3.30·12-s + 14-s − 1.24·15-s + 16-s + 1.24·17-s + 7.92·18-s + 1.62·19-s + 0.376·20-s − 3.30·21-s + 5.68·22-s + 4.92·23-s − 3.30·24-s − 4.85·25-s − 16.2·27-s + 28-s − 4.61·29-s − 1.24·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.90·3-s + 0.5·4-s + 0.168·5-s − 1.34·6-s + 0.377·7-s + 0.353·8-s + 2.64·9-s + 0.119·10-s + 1.71·11-s − 0.954·12-s + 0.267·14-s − 0.321·15-s + 0.250·16-s + 0.302·17-s + 1.86·18-s + 0.372·19-s + 0.0842·20-s − 0.721·21-s + 1.21·22-s + 1.02·23-s − 0.674·24-s − 0.971·25-s − 3.13·27-s + 0.188·28-s − 0.856·29-s − 0.227·30-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

\(L(1)\) \(\approx\) \(1.893716372\)
\(L(\frac12)\) \(\approx\) \(1.893716372\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.30T + 3T^{2} \)
5 \( 1 - 0.376T + 5T^{2} \)
11 \( 1 - 5.68T + 11T^{2} \)
17 \( 1 - 1.24T + 17T^{2} \)
19 \( 1 - 1.62T + 19T^{2} \)
23 \( 1 - 4.92T + 23T^{2} \)
29 \( 1 + 4.61T + 29T^{2} \)
31 \( 1 + 5.68T + 31T^{2} \)
37 \( 1 - 6.92T + 37T^{2} \)
41 \( 1 + 1.07T + 41T^{2} \)
43 \( 1 + 3.24T + 43T^{2} \)
47 \( 1 - 2.31T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 - 5.30T + 61T^{2} \)
67 \( 1 - 8.92T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 0.317T + 73T^{2} \)
79 \( 1 - 2.17T + 79T^{2} \)
83 \( 1 - 9.74T + 83T^{2} \)
89 \( 1 + 9.24T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−9.330228903821700802838571582625, −7.88336525186266056407670575718, −6.92376247078996432905454173649, −6.59071624267587586238009960087, −5.63456710653650521273443694650, −5.28254029208122141489039151267, −4.26814079445525864522775297227, −3.69953076808292878029896517131, −1.83698409748401742657220602360, −0.948032633732754021203303713985, 0.948032633732754021203303713985, 1.83698409748401742657220602360, 3.69953076808292878029896517131, 4.26814079445525864522775297227, 5.28254029208122141489039151267, 5.63456710653650521273443694650, 6.59071624267587586238009960087, 6.92376247078996432905454173649, 7.88336525186266056407670575718, 9.330228903821700802838571582625

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