Properties

Label 2-2366-1.1-c1-0-59
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 + 2.85·3-s + 4-s + 0.532·5-s + 2.85·6-s + 7-s + 8-s + 5.14·9-s + 0.532·10-s + 2.71·11-s + 2.85·12-s + 14-s + 1.51·15-s + 16-s − 2.94·17-s + 5.14·18-s + 5.83·19-s + 0.532·20-s + 2.85·21-s + 2.71·22-s − 5.52·23-s + 2.85·24-s − 4.71·25-s + 6.11·27-s + 28-s − 8.19·29-s + 1.51·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.64·3-s + 0.5·4-s + 0.237·5-s + 1.16·6-s + 0.377·7-s + 0.353·8-s + 1.71·9-s + 0.168·10-s + 0.818·11-s + 0.823·12-s + 0.267·14-s + 0.392·15-s + 0.250·16-s − 0.713·17-s + 1.21·18-s + 1.33·19-s + 0.118·20-s + 0.622·21-s + 0.578·22-s − 1.15·23-s + 0.582·24-s − 0.943·25-s + 1.17·27-s + 0.188·28-s − 1.52·29-s + 0.277·30-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

\(L(1)\) \(\approx\) \(5.645552937\)
\(L(\frac12)\) \(\approx\) \(5.645552937\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.85T + 3T^{2} \)
5 \( 1 - 0.532T + 5T^{2} \)
11 \( 1 - 2.71T + 11T^{2} \)
17 \( 1 + 2.94T + 17T^{2} \)
19 \( 1 - 5.83T + 19T^{2} \)
23 \( 1 + 5.52T + 23T^{2} \)
29 \( 1 + 8.19T + 29T^{2} \)
31 \( 1 - 3.42T + 31T^{2} \)
37 \( 1 - 6.00T + 37T^{2} \)
41 \( 1 + 7.76T + 41T^{2} \)
43 \( 1 + 9.13T + 43T^{2} \)
47 \( 1 + 9.98T + 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 + 1.24T + 59T^{2} \)
61 \( 1 - 1.36T + 61T^{2} \)
67 \( 1 + 9.49T + 67T^{2} \)
71 \( 1 - 1.34T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 - 7.62T + 79T^{2} \)
83 \( 1 + 3.49T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 - 7.63T + 97T^{2} \)
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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

−8.925843510482283031480323060294, −8.147091295575915315188973309302, −7.56387565620167676926935509875, −6.75666615639916406994883872858, −5.82607560415971956864905926088, −4.78179818682738036376226575287, −3.87168810688954574872618716492, −3.36829189168726227403589618691, −2.24398704396652079212291353105, −1.60836176932259246917542507243, 1.60836176932259246917542507243, 2.24398704396652079212291353105, 3.36829189168726227403589618691, 3.87168810688954574872618716492, 4.78179818682738036376226575287, 5.82607560415971956864905926088, 6.75666615639916406994883872858, 7.56387565620167676926935509875, 8.147091295575915315188973309302, 8.925843510482283031480323060294

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