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Properties

Label 2-759-759.329-c0-0-1
Degree $2$
Conductor $759$
Sign $0.985 + 0.167i$
Analytic cond. $0.378790$
Root an. cond. $0.615459$
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  + (0.959 − 0.281i)3-s + (0.142 + 0.989i)4-s + (−0.544 − 1.19i)5-s + (0.841 − 0.540i)9-s + (0.654 + 0.755i)11-s + (0.415 + 0.909i)12-s + (−0.857 − 0.989i)15-s + (−0.959 + 0.281i)16-s + (1.10 − 0.708i)20-s + (−0.654 − 0.755i)23-s + (−0.468 + 0.540i)25-s + (0.654 − 0.755i)27-s + (0.698 + 0.449i)31-s + (0.841 + 0.540i)33-s + (0.654 + 0.755i)36-s + (−1.80 − 0.822i)37-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)3-s + (0.142 + 0.989i)4-s + (−0.544 − 1.19i)5-s + (0.841 − 0.540i)9-s + (0.654 + 0.755i)11-s + (0.415 + 0.909i)12-s + (−0.857 − 0.989i)15-s + (−0.959 + 0.281i)16-s + (1.10 − 0.708i)20-s + (−0.654 − 0.755i)23-s + (−0.468 + 0.540i)25-s + (0.654 − 0.755i)27-s + (0.698 + 0.449i)31-s + (0.841 + 0.540i)33-s + (0.654 + 0.755i)36-s + (−1.80 − 0.822i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(759\)    =    \(3 \cdot 11 \cdot 23\)
Sign: $0.985 + 0.167i$
Analytic conductor: \(0.378790\)
Root analytic conductor: \(0.615459\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{759} (329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 759,\ (\ :0),\ 0.985 + 0.167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.259786761\)
\(L(\frac12)\) \(\approx\) \(1.259786761\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.959 + 0.281i)T \)
11 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
good2 \( 1 + (-0.142 - 0.989i)T^{2} \)
5 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
7 \( 1 + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (-0.841 - 0.540i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (-0.959 + 0.281i)T^{2} \)
29 \( 1 + (-0.959 - 0.281i)T^{2} \)
31 \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \)
37 \( 1 + (1.80 + 0.822i)T + (0.654 + 0.755i)T^{2} \)
41 \( 1 + (-0.654 + 0.755i)T^{2} \)
43 \( 1 + (0.415 - 0.909i)T^{2} \)
47 \( 1 - 1.81iT - T^{2} \)
53 \( 1 + (0.273 - 0.0801i)T + (0.841 - 0.540i)T^{2} \)
59 \( 1 + (0.557 - 1.89i)T + (-0.841 - 0.540i)T^{2} \)
61 \( 1 + (0.415 + 0.909i)T^{2} \)
67 \( 1 + (0.425 + 0.368i)T + (0.142 + 0.989i)T^{2} \)
71 \( 1 + (1.14 + 0.989i)T + (0.142 + 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.654 + 0.755i)T^{2} \)
89 \( 1 + (-1.61 + 1.03i)T + (0.415 - 0.909i)T^{2} \)
97 \( 1 + (1.65 - 0.755i)T + (0.654 - 0.755i)T^{2} \)
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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

−10.39907543668218576076302340619, −9.190106398187520352676815178396, −8.818802799117354863527613955680, −7.988931455894788268670224082279, −7.39085575890482624132304394274, −6.41483789461924228818846850239, −4.62953953977017547443132414986, −4.13169368230537108057237909656, −3.01332629440400930167828346057, −1.66851182074270606768332203879, 1.82068606928102639317806062344, 3.10427308500204328268688528516, 3.86755288993258236671664488487, 5.16186917082149028258544852879, 6.39971705183491492453100326513, 7.01978389951912351677596866016, 8.043249648737376402116090207370, 8.913790189027861216309503947952, 9.887718951952190899246854234660, 10.39318469500331951905401857812

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