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Properties

Label 2-820-820.423-c0-0-0
Degree 22
Conductor 820820
Sign 0.656+0.754i0.656 + 0.754i
Analytic cond. 0.4092330.409233
Root an. cond. 0.6397130.639713
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  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.453 − 0.891i)5-s + (−0.587 + 0.809i)8-s + (0.707 − 0.707i)9-s + (−0.156 + 0.987i)10-s + (−1.26 − 1.47i)13-s + (0.309 − 0.951i)16-s + (0.678 + 1.10i)17-s + (−0.453 + 0.891i)18-s + (−0.156 − 0.987i)20-s + (−0.587 − 0.809i)25-s + (1.65 + 1.01i)26-s + (−1.79 + 0.431i)29-s + i·32-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.453 − 0.891i)5-s + (−0.587 + 0.809i)8-s + (0.707 − 0.707i)9-s + (−0.156 + 0.987i)10-s + (−1.26 − 1.47i)13-s + (0.309 − 0.951i)16-s + (0.678 + 1.10i)17-s + (−0.453 + 0.891i)18-s + (−0.156 − 0.987i)20-s + (−0.587 − 0.809i)25-s + (1.65 + 1.01i)26-s + (−1.79 + 0.431i)29-s + i·32-s + ⋯

Functional equation

Λ(s)=(820s/2 ΓC(s) L(s)=((0.656+0.754i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(820s/2 ΓC(s) L(s)=((0.656+0.754i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 820820    =    22⋅5⋅412^{2} \cdot 5 \cdot 41
Sign: 0.656+0.754i0.656 + 0.754i
Analytic conductor: 0.4092330.409233
Root analytic conductor: 0.6397130.639713
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ820(423,⋅)\chi_{820} (423, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 820, ( :0), 0.656+0.754i)(2,\ 820,\ (\ :0),\ 0.656 + 0.754i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.68812328690.6881232869
L(12)L(\frac12) ≈\approx 0.68812328690.6881232869
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.951−0.309i)T 1 + (0.951 - 0.309i)T
5 1+(−0.453+0.891i)T 1 + (-0.453 + 0.891i)T
41 1+(−0.951+0.309i)T 1 + (-0.951 + 0.309i)T
good3 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
7 1+(−0.987+0.156i)T2 1 + (-0.987 + 0.156i)T^{2}
11 1+(0.453−0.891i)T2 1 + (0.453 - 0.891i)T^{2}
13 1+(1.26+1.47i)T+(−0.156+0.987i)T2 1 + (1.26 + 1.47i)T + (-0.156 + 0.987i)T^{2}
17 1+(−0.678−1.10i)T+(−0.453+0.891i)T2 1 + (-0.678 - 1.10i)T + (-0.453 + 0.891i)T^{2}
19 1+(0.987−0.156i)T2 1 + (0.987 - 0.156i)T^{2}
23 1+(−0.587−0.809i)T2 1 + (-0.587 - 0.809i)T^{2}
29 1+(1.79−0.431i)T+(0.891−0.453i)T2 1 + (1.79 - 0.431i)T + (0.891 - 0.453i)T^{2}
31 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
37 1+(−0.183+1.16i)T+(−0.951−0.309i)T2 1 + (-0.183 + 1.16i)T + (-0.951 - 0.309i)T^{2}
43 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
47 1+(0.987+0.156i)T2 1 + (0.987 + 0.156i)T^{2}
53 1+(−1.70−1.04i)T+(0.453+0.891i)T2 1 + (-1.70 - 1.04i)T + (0.453 + 0.891i)T^{2}
59 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
61 1+(−0.896−1.76i)T+(−0.587+0.809i)T2 1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)T^{2}
67 1+(−0.891+0.453i)T2 1 + (-0.891 + 0.453i)T^{2}
71 1+(−0.453+0.891i)T2 1 + (-0.453 + 0.891i)T^{2}
73 1−0.312T+T2 1 - 0.312T + T^{2}
79 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
83 1−iT2 1 - iT^{2}
89 1+(0.497−0.581i)T+(−0.156−0.987i)T2 1 + (0.497 - 0.581i)T + (-0.156 - 0.987i)T^{2}
97 1+(−0.243−1.01i)T+(−0.891+0.453i)T2 1 + (-0.243 - 1.01i)T + (-0.891 + 0.453i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−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

−10.12298131762112309605626592796, −9.464537311467726271453096533869, −8.752172708876996250341137054578, −7.74854822364332257036662299603, −7.19962477926097944745602597369, −5.80241024018568866816505008234, −5.44639143479321480822509161035, −3.93856776655871429629794651315, −2.35211641019870796640135412539, −1.00929201697721347311449467585, 1.87359634500319362106795455995, 2.63438486039748067285325631525, 4.00207716841804440636961539106, 5.32198172807158643050512940730, 6.66438809018034084003023796861, 7.23521624406168455709189176994, 7.82463370168145983569588110117, 9.263280268752989656464578734842, 9.700069663212240697619981452404, 10.32517196871777397097905464674

Graph of the ZZ-function along the critical line