Properties

Label 2-3640-1.1-c1-0-66
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.24·3-s − 5-s + 7-s + 2.02·9-s − 5.09·11-s − 13-s − 2.24·15-s − 2.63·17-s + 6.11·19-s + 2.24·21-s − 5.44·23-s + 25-s − 2.17·27-s − 6.47·29-s − 3.50·31-s − 11.4·33-s − 35-s + 0.458·37-s − 2.24·39-s − 7.15·41-s + 2.62·43-s − 2.02·45-s − 8.09·47-s + 49-s − 5.91·51-s + 3.74·53-s + 5.09·55-s + ⋯
L(s)  = 1  + 1.29·3-s − 0.447·5-s + 0.377·7-s + 0.676·9-s − 1.53·11-s − 0.277·13-s − 0.578·15-s − 0.639·17-s + 1.40·19-s + 0.489·21-s − 1.13·23-s + 0.200·25-s − 0.419·27-s − 1.20·29-s − 0.629·31-s − 1.98·33-s − 0.169·35-s + 0.0753·37-s − 0.359·39-s − 1.11·41-s + 0.399·43-s − 0.302·45-s − 1.18·47-s + 0.142·49-s − 0.828·51-s + 0.515·53-s + 0.686·55-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: \(1\)
Selberg data: \((2,\ 3640,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.24T + 3T^{2} \)
11 \( 1 + 5.09T + 11T^{2} \)
17 \( 1 + 2.63T + 17T^{2} \)
19 \( 1 - 6.11T + 19T^{2} \)
23 \( 1 + 5.44T + 23T^{2} \)
29 \( 1 + 6.47T + 29T^{2} \)
31 \( 1 + 3.50T + 31T^{2} \)
37 \( 1 - 0.458T + 37T^{2} \)
41 \( 1 + 7.15T + 41T^{2} \)
43 \( 1 - 2.62T + 43T^{2} \)
47 \( 1 + 8.09T + 47T^{2} \)
53 \( 1 - 3.74T + 53T^{2} \)
59 \( 1 - 9.17T + 59T^{2} \)
61 \( 1 - 9.41T + 61T^{2} \)
67 \( 1 + 5.80T + 67T^{2} \)
71 \( 1 + 5.83T + 71T^{2} \)
73 \( 1 + 8.27T + 73T^{2} \)
79 \( 1 - 3.86T + 79T^{2} \)
83 \( 1 + 8.61T + 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 - 8.15T + 97T^{2} \)
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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

−8.270048394468544072467375550938, −7.49707831284595023010415911747, −7.19413229125968558718428953911, −5.73874195854876828266030128154, −5.12793516326234278790497335556, −4.12591475985970370471108407296, −3.34094770951071722910562684366, −2.58325519144815968624140872546, −1.79061753565348062261397577800, 0, 1.79061753565348062261397577800, 2.58325519144815968624140872546, 3.34094770951071722910562684366, 4.12591475985970370471108407296, 5.12793516326234278790497335556, 5.73874195854876828266030128154, 7.19413229125968558718428953911, 7.49707831284595023010415911747, 8.270048394468544072467375550938

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