Properties

Label 2-3640-1.1-c1-0-13
Degree $2$
Conductor $3640$
Sign $1$
Analytic cond. $29.0655$
Root an. cond. $5.39124$
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  − 2.36·3-s − 5-s + 7-s + 2.60·9-s + 5.04·11-s + 13-s + 2.36·15-s + 1.60·17-s − 3.29·19-s − 2.36·21-s + 2.31·23-s + 25-s + 0.928·27-s + 1.32·29-s − 4.36·31-s − 11.9·33-s − 35-s + 10.6·37-s − 2.36·39-s + 0.249·41-s + 3.68·43-s − 2.60·45-s − 3.21·47-s + 49-s − 3.80·51-s − 4.73·53-s − 5.04·55-s + ⋯
L(s)  = 1  − 1.36·3-s − 0.447·5-s + 0.377·7-s + 0.869·9-s + 1.52·11-s + 0.277·13-s + 0.611·15-s + 0.389·17-s − 0.756·19-s − 0.516·21-s + 0.481·23-s + 0.200·25-s + 0.178·27-s + 0.245·29-s − 0.784·31-s − 2.08·33-s − 0.169·35-s + 1.74·37-s − 0.379·39-s + 0.0390·41-s + 0.562·43-s − 0.388·45-s − 0.469·47-s + 0.142·49-s − 0.533·51-s − 0.650·53-s − 0.680·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(29.0655\)
Root analytic conductor: \(5.39124\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.136051375\)
\(L(\frac12)\) \(\approx\) \(1.136051375\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 - T \)
good3 \( 1 + 2.36T + 3T^{2} \)
11 \( 1 - 5.04T + 11T^{2} \)
17 \( 1 - 1.60T + 17T^{2} \)
19 \( 1 + 3.29T + 19T^{2} \)
23 \( 1 - 2.31T + 23T^{2} \)
29 \( 1 - 1.32T + 29T^{2} \)
31 \( 1 + 4.36T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 - 0.249T + 41T^{2} \)
43 \( 1 - 3.68T + 43T^{2} \)
47 \( 1 + 3.21T + 47T^{2} \)
53 \( 1 + 4.73T + 53T^{2} \)
59 \( 1 + 5.58T + 59T^{2} \)
61 \( 1 + 0.142T + 61T^{2} \)
67 \( 1 + 6.34T + 67T^{2} \)
71 \( 1 + 6.42T + 71T^{2} \)
73 \( 1 - 3.35T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + 4.73T + 83T^{2} \)
89 \( 1 + 7.27T + 89T^{2} \)
97 \( 1 - 3.52T + 97T^{2} \)
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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

−8.595823327269010297221524725790, −7.65449972395921680861396240594, −6.90478664043610417092164117916, −6.20400152426156582583501758008, −5.72336147666840447301418170827, −4.62729753820452156260178800075, −4.23432456636175315536530029595, −3.15035568591156924179030359768, −1.63299128221074507844215043857, −0.71108026784411981551783204096, 0.71108026784411981551783204096, 1.63299128221074507844215043857, 3.15035568591156924179030359768, 4.23432456636175315536530029595, 4.62729753820452156260178800075, 5.72336147666840447301418170827, 6.20400152426156582583501758008, 6.90478664043610417092164117916, 7.65449972395921680861396240594, 8.595823327269010297221524725790

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