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Properties

Label 2-820-820.403-c0-0-0
Degree 22
Conductor 820820
Sign 0.884−0.466i0.884 - 0.466i
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 + (0.891 − 0.453i)5-s + (0.587 + 0.809i)8-s + (−0.707 + 0.707i)9-s + (0.987 − 0.156i)10-s + (−1.70 + 0.133i)13-s + (0.309 + 0.951i)16-s + (−0.465 − 1.93i)17-s + (−0.891 + 0.453i)18-s + (0.987 + 0.156i)20-s + (0.587 − 0.809i)25-s + (−1.65 − 0.398i)26-s + (−0.652 + 0.399i)29-s + i·32-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (0.891 − 0.453i)5-s + (0.587 + 0.809i)8-s + (−0.707 + 0.707i)9-s + (0.987 − 0.156i)10-s + (−1.70 + 0.133i)13-s + (0.309 + 0.951i)16-s + (−0.465 − 1.93i)17-s + (−0.891 + 0.453i)18-s + (0.987 + 0.156i)20-s + (0.587 − 0.809i)25-s + (−1.65 − 0.398i)26-s + (−0.652 + 0.399i)29-s + i·32-s + ⋯

Functional equation

Λ(s)=(820s/2 ΓC(s) L(s)=((0.884−0.466i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(820s/2 ΓC(s) L(s)=((0.884−0.466i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 820820    =    22⋅5⋅412^{2} \cdot 5 \cdot 41
Sign: 0.884−0.466i0.884 - 0.466i
Analytic conductor: 0.4092330.409233
Root analytic conductor: 0.6397130.639713
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ820(403,⋅)\chi_{820} (403, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 820, ( :0), 0.884−0.466i)(2,\ 820,\ (\ :0),\ 0.884 - 0.466i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7641659351.764165935
L(12)L(\frac12) ≈\approx 1.7641659351.764165935
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+(−0.891+0.453i)T 1 + (-0.891 + 0.453i)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+(0.156−0.987i)T2 1 + (0.156 - 0.987i)T^{2}
11 1+(0.891−0.453i)T2 1 + (0.891 - 0.453i)T^{2}
13 1+(1.70−0.133i)T+(0.987−0.156i)T2 1 + (1.70 - 0.133i)T + (0.987 - 0.156i)T^{2}
17 1+(0.465+1.93i)T+(−0.891+0.453i)T2 1 + (0.465 + 1.93i)T + (-0.891 + 0.453i)T^{2}
19 1+(−0.156+0.987i)T2 1 + (-0.156 + 0.987i)T^{2}
23 1+(0.587−0.809i)T2 1 + (0.587 - 0.809i)T^{2}
29 1+(0.652−0.399i)T+(0.453−0.891i)T2 1 + (0.652 - 0.399i)T + (0.453 - 0.891i)T^{2}
31 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
37 1+(−1.16+0.183i)T+(0.951−0.309i)T2 1 + (-1.16 + 0.183i)T + (0.951 - 0.309i)T^{2}
43 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
47 1+(−0.156−0.987i)T2 1 + (-0.156 - 0.987i)T^{2}
53 1+(−1.26−0.303i)T+(0.891+0.453i)T2 1 + (-1.26 - 0.303i)T + (0.891 + 0.453i)T^{2}
59 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
61 1+(0.278+0.142i)T+(0.587+0.809i)T2 1 + (0.278 + 0.142i)T + (0.587 + 0.809i)T^{2}
67 1+(−0.453+0.891i)T2 1 + (-0.453 + 0.891i)T^{2}
71 1+(−0.891+0.453i)T2 1 + (-0.891 + 0.453i)T^{2}
73 1+1.97T+T2 1 + 1.97T + T^{2}
79 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
83 1−iT2 1 - iT^{2}
89 1+(−1.84−0.144i)T+(0.987+0.156i)T2 1 + (-1.84 - 0.144i)T + (0.987 + 0.156i)T^{2}
97 1+(0.243+0.398i)T+(−0.453+0.891i)T2 1 + (0.243 + 0.398i)T + (-0.453 + 0.891i)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

−10.60534140686924225230210172433, −9.617490583677095092752040612835, −8.839352857403180271254440199571, −7.64445767631221318197941533858, −7.03608920432862563890404580962, −5.88022342445863992484146836607, −5.08293460131620082574140364525, −4.62706554447727705716843188130, −2.84569026988039599268067648681, −2.21494520710589251224715152495, 1.92236376821973836403390783812, 2.82787762865579891425296423490, 3.92163995766779642830131788382, 5.10651809989965872761109302190, 5.96477763368944218224452867811, 6.55325447524849129514630465555, 7.55749876280028280883146102711, 8.848196484358762008591879625050, 9.900077158430581779960072402003, 10.32176864790115156935984242747

Graph of the ZZ-function along the critical line