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Properties

Label 2-3640-3640.1133-c0-0-2
Degree $2$
Conductor $3640$
Sign $0.836 + 0.547i$
Analytic cond. $1.81659$
Root an. cond. $1.34781$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.133 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.133 − 0.5i)6-s + (−0.5 + 0.866i)7-s − 0.999i·8-s + (0.633 + 0.366i)9-s + (−0.499 + 0.866i)10-s + (−0.366 − 0.366i)12-s + (−0.5 + 0.866i)13-s + 0.999i·14-s + (0.133 + 0.5i)15-s + (−0.5 − 0.866i)16-s + 0.732·18-s + (1.36 − 0.366i)19-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.133 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.133 − 0.5i)6-s + (−0.5 + 0.866i)7-s − 0.999i·8-s + (0.633 + 0.366i)9-s + (−0.499 + 0.866i)10-s + (−0.366 − 0.366i)12-s + (−0.5 + 0.866i)13-s + 0.999i·14-s + (0.133 + 0.5i)15-s + (−0.5 − 0.866i)16-s + 0.732·18-s + (1.36 − 0.366i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $0.836 + 0.547i$
Analytic conductor: \(1.81659\)
Root analytic conductor: \(1.34781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3640} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :0),\ 0.836 + 0.547i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.991073222\)
\(L(\frac12)\) \(\approx\) \(1.991073222\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−8.717967489952568568986571799017, −7.57212902729520347396553925855, −7.02544204189259829306201939011, −6.61624478748902133413017496417, −5.45498025893815433150383053226, −4.83827661315099925152514471691, −3.92548378967361137827125617708, −3.01372674701942156584736153058, −2.46665718454008616739226298522, −1.25122296366582932250520249932, 1.04598909295945391975339183167, 3.00664253371041397887106363949, 3.43886257287635480778530708650, 4.25106382033136616748791690260, 4.87896992815248673363254349775, 5.54310721329136171442441513469, 6.78759809666205527081490156802, 7.21293781084562682189842220215, 7.79771733088551849053997028943, 8.652197512657043992131295539128

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