Properties

Label 2-1620-1.1-c1-0-3
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 − 2.73·7-s − 1.73·11-s + 5.46·13-s + 4.73·17-s − 4.46·19-s − 3.46·23-s + 25-s + 7.73·29-s + 5.92·31-s − 2.73·35-s − 6.19·37-s + 11.1·41-s + 3.26·43-s + 1.26·47-s + 0.464·49-s + 7.26·53-s − 1.73·55-s + 7.73·59-s − 4·61-s + 5.46·65-s + 6.39·67-s + 11.1·71-s − 0.196·73-s + 4.73·77-s − 14.3·79-s + 15.1·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.03·7-s − 0.522·11-s + 1.51·13-s + 1.14·17-s − 1.02·19-s − 0.722·23-s + 0.200·25-s + 1.43·29-s + 1.06·31-s − 0.461·35-s − 1.01·37-s + 1.74·41-s + 0.498·43-s + 0.184·47-s + 0.0663·49-s + 0.998·53-s − 0.233·55-s + 1.00·59-s − 0.512·61-s + 0.677·65-s + 0.780·67-s + 1.32·71-s − 0.0229·73-s + 0.539·77-s − 1.61·79-s + 1.66·83-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

\(L(1)\) \(\approx\) \(1.702998947\)
\(L(\frac12)\) \(\approx\) \(1.702998947\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.73T + 7T^{2} \)
11 \( 1 + 1.73T + 11T^{2} \)
13 \( 1 - 5.46T + 13T^{2} \)
17 \( 1 - 4.73T + 17T^{2} \)
19 \( 1 + 4.46T + 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 7.73T + 29T^{2} \)
31 \( 1 - 5.92T + 31T^{2} \)
37 \( 1 + 6.19T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 - 3.26T + 43T^{2} \)
47 \( 1 - 1.26T + 47T^{2} \)
53 \( 1 - 7.26T + 53T^{2} \)
59 \( 1 - 7.73T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 - 6.39T + 67T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 + 0.196T + 73T^{2} \)
79 \( 1 + 14.3T + 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 - 0.732T + 97T^{2} \)
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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

−9.445793774507647510215971479441, −8.571275760361312750626586983059, −7.971780380203900873912842064569, −6.75770147679716341484852819185, −6.17415727635653731346811227811, −5.52591078784078507001710964719, −4.25332733345007527784652413659, −3.36814822788499321950969048033, −2.41348115153317722710224549998, −0.932769269361741768893001655107, 0.932769269361741768893001655107, 2.41348115153317722710224549998, 3.36814822788499321950969048033, 4.25332733345007527784652413659, 5.52591078784078507001710964719, 6.17415727635653731346811227811, 6.75770147679716341484852819185, 7.971780380203900873912842064569, 8.571275760361312750626586983059, 9.445793774507647510215971479441

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