Properties

Label 2-759-759.329-c0-0-1
Degree 22
Conductor 759759
Sign 0.985+0.167i0.985 + 0.167i
Analytic cond. 0.3787900.378790
Root an. cond. 0.6154590.615459
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(759s/2ΓC(s)L(s)=((0.985+0.167i)Λ(1s)\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}
Λ(s)=(759s/2ΓC(s)L(s)=((0.985+0.167i)Λ(1s)\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: 22
Conductor: 759759    =    311233 \cdot 11 \cdot 23
Sign: 0.985+0.167i0.985 + 0.167i
Analytic conductor: 0.3787900.378790
Root analytic conductor: 0.6154590.615459
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ759(329,)\chi_{759} (329, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 759, ( :0), 0.985+0.167i)(2,\ 759,\ (\ :0),\ 0.985 + 0.167i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.2597867611.259786761
L(12)L(\frac12) \approx 1.2597867611.259786761
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(0.959+0.281i)T 1 + (-0.959 + 0.281i)T
11 1+(0.6540.755i)T 1 + (-0.654 - 0.755i)T
23 1+(0.654+0.755i)T 1 + (0.654 + 0.755i)T
good2 1+(0.1420.989i)T2 1 + (-0.142 - 0.989i)T^{2}
5 1+(0.544+1.19i)T+(0.654+0.755i)T2 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2}
7 1+(0.8410.540i)T2 1 + (0.841 - 0.540i)T^{2}
13 1+(0.8410.540i)T2 1 + (-0.841 - 0.540i)T^{2}
17 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
19 1+(0.959+0.281i)T2 1 + (-0.959 + 0.281i)T^{2}
29 1+(0.9590.281i)T2 1 + (-0.959 - 0.281i)T^{2}
31 1+(0.6980.449i)T+(0.415+0.909i)T2 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2}
37 1+(1.80+0.822i)T+(0.654+0.755i)T2 1 + (1.80 + 0.822i)T + (0.654 + 0.755i)T^{2}
41 1+(0.654+0.755i)T2 1 + (-0.654 + 0.755i)T^{2}
43 1+(0.4150.909i)T2 1 + (0.415 - 0.909i)T^{2}
47 11.81iTT2 1 - 1.81iT - T^{2}
53 1+(0.2730.0801i)T+(0.8410.540i)T2 1 + (0.273 - 0.0801i)T + (0.841 - 0.540i)T^{2}
59 1+(0.5571.89i)T+(0.8410.540i)T2 1 + (0.557 - 1.89i)T + (-0.841 - 0.540i)T^{2}
61 1+(0.415+0.909i)T2 1 + (0.415 + 0.909i)T^{2}
67 1+(0.425+0.368i)T+(0.142+0.989i)T2 1 + (0.425 + 0.368i)T + (0.142 + 0.989i)T^{2}
71 1+(1.14+0.989i)T+(0.142+0.989i)T2 1 + (1.14 + 0.989i)T + (0.142 + 0.989i)T^{2}
73 1+(0.9590.281i)T2 1 + (0.959 - 0.281i)T^{2}
79 1+(0.841+0.540i)T2 1 + (0.841 + 0.540i)T^{2}
83 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
89 1+(1.61+1.03i)T+(0.4150.909i)T2 1 + (-1.61 + 1.03i)T + (0.415 - 0.909i)T^{2}
97 1+(1.650.755i)T+(0.6540.755i)T2 1 + (1.65 - 0.755i)T + (0.654 - 0.755i)T^{2}
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   L(s)=p j=12(1αj,pps)1L(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 ZZ-function along the critical line