Properties

Label 2-1620-1.1-c1-0-8
Degree $2$
Conductor $1620$
Sign $1$
Analytic cond. $12.9357$
Root an. cond. $3.59663$
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  + 5-s + 0.732·7-s + 1.73·11-s − 1.46·13-s + 1.26·17-s + 2.46·19-s + 3.46·23-s + 25-s + 4.26·29-s − 7.92·31-s + 0.732·35-s + 4.19·37-s + 0.803·41-s + 6.73·43-s + 4.73·47-s − 6.46·49-s + 10.7·53-s + 1.73·55-s + 4.26·59-s − 4·61-s − 1.46·65-s − 14.3·67-s + 0.803·71-s + 10.1·73-s + 1.26·77-s + 6.39·79-s − 9.12·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.276·7-s + 0.522·11-s − 0.406·13-s + 0.307·17-s + 0.565·19-s + 0.722·23-s + 0.200·25-s + 0.792·29-s − 1.42·31-s + 0.123·35-s + 0.689·37-s + 0.125·41-s + 1.02·43-s + 0.690·47-s − 0.923·49-s + 1.47·53-s + 0.233·55-s + 0.555·59-s − 0.512·61-s − 0.181·65-s − 1.75·67-s + 0.0953·71-s + 1.19·73-s + 0.144·77-s + 0.719·79-s − 1.00·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $1$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.025638825\)
\(L(\frac12)\) \(\approx\) \(2.025638825\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
good7 \( 1 - 0.732T + 7T^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 - 1.26T + 17T^{2} \)
19 \( 1 - 2.46T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 4.26T + 29T^{2} \)
31 \( 1 + 7.92T + 31T^{2} \)
37 \( 1 - 4.19T + 37T^{2} \)
41 \( 1 - 0.803T + 41T^{2} \)
43 \( 1 - 6.73T + 43T^{2} \)
47 \( 1 - 4.73T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 - 4.26T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 - 0.803T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 6.39T + 79T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 + 2.73T + 97T^{2} \)
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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

−9.326599821164464428207386751854, −8.779425679424916360269217981607, −7.69249645127935090704980304523, −7.07640845622787122576374735369, −6.09294264949099454088106951846, −5.32404448871283120218357725714, −4.44869409446023593015511679432, −3.36063942565164379448368190773, −2.28133981595779567490894279524, −1.05917541111005367081889955987, 1.05917541111005367081889955987, 2.28133981595779567490894279524, 3.36063942565164379448368190773, 4.44869409446023593015511679432, 5.32404448871283120218357725714, 6.09294264949099454088106951846, 7.07640845622787122576374735369, 7.69249645127935090704980304523, 8.779425679424916360269217981607, 9.326599821164464428207386751854

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