Properties

Label 2-3640-1.1-c1-0-7
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  + 0.512·3-s − 5-s − 7-s − 2.73·9-s − 3.81·11-s + 13-s − 0.512·15-s − 0.761·17-s − 1.04·19-s − 0.512·21-s + 5.81·23-s + 25-s − 2.93·27-s + 4.32·29-s + 4.61·31-s − 1.95·33-s + 35-s − 5.92·37-s + 0.512·39-s − 7.43·41-s − 2.78·43-s + 2.73·45-s − 3.12·47-s + 49-s − 0.390·51-s + 7.12·53-s + 3.81·55-s + ⋯
L(s)  = 1  + 0.295·3-s − 0.447·5-s − 0.377·7-s − 0.912·9-s − 1.14·11-s + 0.277·13-s − 0.132·15-s − 0.184·17-s − 0.240·19-s − 0.111·21-s + 1.21·23-s + 0.200·25-s − 0.565·27-s + 0.802·29-s + 0.828·31-s − 0.339·33-s + 0.169·35-s − 0.973·37-s + 0.0819·39-s − 1.16·41-s − 0.425·43-s + 0.408·45-s − 0.455·47-s + 0.142·49-s − 0.0546·51-s + 0.978·53-s + 0.513·55-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: $\chi_{3640} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.260936348\)
\(L(\frac12)\) \(\approx\) \(1.260936348\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.512T + 3T^{2} \)
11 \( 1 + 3.81T + 11T^{2} \)
17 \( 1 + 0.761T + 17T^{2} \)
19 \( 1 + 1.04T + 19T^{2} \)
23 \( 1 - 5.81T + 23T^{2} \)
29 \( 1 - 4.32T + 29T^{2} \)
31 \( 1 - 4.61T + 31T^{2} \)
37 \( 1 + 5.92T + 37T^{2} \)
41 \( 1 + 7.43T + 41T^{2} \)
43 \( 1 + 2.78T + 43T^{2} \)
47 \( 1 + 3.12T + 47T^{2} \)
53 \( 1 - 7.12T + 53T^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 - 8.24T + 61T^{2} \)
67 \( 1 - 2.21T + 67T^{2} \)
71 \( 1 - 4.58T + 71T^{2} \)
73 \( 1 + 3.07T + 73T^{2} \)
79 \( 1 - 9.11T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 - 17.3T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−8.513395120477472370396689386882, −7.989711906123809179302647851853, −7.04575440582848409809558102924, −6.43194712137990054132741293418, −5.38697606373562795889220758544, −4.88788294310038060175131097773, −3.68489666453329605537091603861, −3.02112980269086870654108432361, −2.24864501631302080887951834213, −0.61835439471621721868960779322, 0.61835439471621721868960779322, 2.24864501631302080887951834213, 3.02112980269086870654108432361, 3.68489666453329605537091603861, 4.88788294310038060175131097773, 5.38697606373562795889220758544, 6.43194712137990054132741293418, 7.04575440582848409809558102924, 7.989711906123809179302647851853, 8.513395120477472370396689386882

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