Properties

Label 2-3640-1.1-c1-0-44
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s − 2·9-s + 5·11-s − 13-s + 15-s − 2·17-s − 6·19-s + 21-s + 9·23-s + 25-s + 5·27-s + 6·29-s − 5·31-s − 5·33-s + 35-s + 37-s + 39-s + 3·41-s + 6·43-s + 2·45-s − 7·47-s + 49-s + 2·51-s − 12·53-s − 5·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 1.50·11-s − 0.277·13-s + 0.258·15-s − 0.485·17-s − 1.37·19-s + 0.218·21-s + 1.87·23-s + 1/5·25-s + 0.962·27-s + 1.11·29-s − 0.898·31-s − 0.870·33-s + 0.169·35-s + 0.164·37-s + 0.160·39-s + 0.468·41-s + 0.914·43-s + 0.298·45-s − 1.02·47-s + 1/7·49-s + 0.280·51-s − 1.64·53-s − 0.674·55-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: yes
Arithmetic: yes
Character: Trivial
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 \( 1 + T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−8.278283368893772609223079196613, −7.27211159080710657510784057376, −6.44361966934794791680955647047, −6.26750413669250201515588865349, −5.02961690112152856334551402167, −4.42637788908097299066150393977, −3.49249730310852852773224541883, −2.61652812405999217097848225758, −1.23503925829553907016295769157, 0, 1.23503925829553907016295769157, 2.61652812405999217097848225758, 3.49249730310852852773224541883, 4.42637788908097299066150393977, 5.02961690112152856334551402167, 6.26750413669250201515588865349, 6.44361966934794791680955647047, 7.27211159080710657510784057376, 8.278283368893772609223079196613

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