Properties

Label 2-3640-1.1-c1-0-10
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  − 1.90·3-s − 5-s − 7-s + 0.631·9-s + 6.09·11-s + 13-s + 1.90·15-s + 7.44·17-s + 0.655·19-s + 1.90·21-s − 4.09·23-s + 25-s + 4.51·27-s − 8.00·29-s − 1.21·31-s − 11.6·33-s + 35-s − 1.75·37-s − 1.90·39-s − 8.81·41-s + 2.28·43-s − 0.631·45-s + 5.12·47-s + 49-s − 14.1·51-s − 1.12·53-s − 6.09·55-s + ⋯
L(s)  = 1  − 1.10·3-s − 0.447·5-s − 0.377·7-s + 0.210·9-s + 1.83·11-s + 0.277·13-s + 0.492·15-s + 1.80·17-s + 0.150·19-s + 0.415·21-s − 0.854·23-s + 0.200·25-s + 0.868·27-s − 1.48·29-s − 0.218·31-s − 2.02·33-s + 0.169·35-s − 0.289·37-s − 0.305·39-s − 1.37·41-s + 0.348·43-s − 0.0941·45-s + 0.747·47-s + 0.142·49-s − 1.98·51-s − 0.154·53-s − 0.822·55-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

\(L(1)\) \(\approx\) \(1.108342776\)
\(L(\frac12)\) \(\approx\) \(1.108342776\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.90T + 3T^{2} \)
11 \( 1 - 6.09T + 11T^{2} \)
17 \( 1 - 7.44T + 17T^{2} \)
19 \( 1 - 0.655T + 19T^{2} \)
23 \( 1 + 4.09T + 23T^{2} \)
29 \( 1 + 8.00T + 29T^{2} \)
31 \( 1 + 1.21T + 31T^{2} \)
37 \( 1 + 1.75T + 37T^{2} \)
41 \( 1 + 8.81T + 41T^{2} \)
43 \( 1 - 2.28T + 43T^{2} \)
47 \( 1 - 5.12T + 47T^{2} \)
53 \( 1 + 1.12T + 53T^{2} \)
59 \( 1 - 1.51T + 59T^{2} \)
61 \( 1 + 8.24T + 61T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 + 4.49T + 73T^{2} \)
79 \( 1 - 2.41T + 79T^{2} \)
83 \( 1 - 3.74T + 83T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−8.602030976640632600575055491058, −7.63263920173413915535502190352, −6.98046165376007469925203194450, −6.11421904193602056197510330097, −5.77817203900785957015262716269, −4.82759722654380059826067951943, −3.78852367788728738351178154260, −3.37498204257911760800811433131, −1.67840814059649002135743347666, −0.68162738238628253825191714430, 0.68162738238628253825191714430, 1.67840814059649002135743347666, 3.37498204257911760800811433131, 3.78852367788728738351178154260, 4.82759722654380059826067951943, 5.77817203900785957015262716269, 6.11421904193602056197510330097, 6.98046165376007469925203194450, 7.63263920173413915535502190352, 8.602030976640632600575055491058

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