Properties

Label 2-3640-1.1-c1-0-29
Degree $2$
Conductor $3640$
Sign $1$
Analytic cond. $29.0655$
Root an. cond. $5.39124$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5-s + 7-s + 9-s + 6·11-s − 13-s − 2·15-s − 2·17-s + 2·19-s + 2·21-s − 2·23-s + 25-s − 4·27-s + 10·29-s − 6·31-s + 12·33-s − 35-s + 6·37-s − 2·39-s − 2·41-s + 10·43-s − 45-s + 49-s − 4·51-s + 14·53-s − 6·55-s + 4·57-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.277·13-s − 0.516·15-s − 0.485·17-s + 0.458·19-s + 0.436·21-s − 0.417·23-s + 1/5·25-s − 0.769·27-s + 1.85·29-s − 1.07·31-s + 2.08·33-s − 0.169·35-s + 0.986·37-s − 0.320·39-s − 0.312·41-s + 1.52·43-s − 0.149·45-s + 1/7·49-s − 0.560·51-s + 1.92·53-s − 0.809·55-s + 0.529·57-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: yes
Arithmetic: yes
Character: $\chi_{3640} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.049171595\)
\(L(\frac12)\) \(\approx\) \(3.049171595\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.609838383762858048247573806734, −7.925245387849671171025169939212, −7.19187209492735217735177440332, −6.50741785377801621029840038062, −5.54042150575646489689069819008, −4.32964042901436714897500918195, −3.95758346607943726243247521797, −3.00339694126723401472697619039, −2.13379787825443665400985068360, −1.02052801348837377914168880582, 1.02052801348837377914168880582, 2.13379787825443665400985068360, 3.00339694126723401472697619039, 3.95758346607943726243247521797, 4.32964042901436714897500918195, 5.54042150575646489689069819008, 6.50741785377801621029840038062, 7.19187209492735217735177440332, 7.925245387849671171025169939212, 8.609838383762858048247573806734

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