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Properties

Label 2-931-931.436-c0-0-0
Degree 22
Conductor 931931
Sign 0.481+0.876i0.481 + 0.876i
Analytic cond. 0.4646290.464629
Root an. cond. 0.6816370.681637
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  + (0.955 − 0.294i)4-s + (0.142 − 1.90i)5-s + (0.0747 − 0.997i)7-s + (0.365 + 0.930i)9-s + (−0.722 + 1.84i)11-s + (0.826 − 0.563i)16-s + (0.326 − 0.302i)17-s + (−0.5 + 0.866i)19-s + (−0.425 − 1.86i)20-s + (−0.535 − 0.496i)23-s + (−2.62 − 0.395i)25-s + (−0.222 − 0.974i)28-s + (−1.88 − 0.284i)35-s + (0.623 + 0.781i)36-s + (1.32 + 0.636i)43-s + (−0.147 + 1.97i)44-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)4-s + (0.142 − 1.90i)5-s + (0.0747 − 0.997i)7-s + (0.365 + 0.930i)9-s + (−0.722 + 1.84i)11-s + (0.826 − 0.563i)16-s + (0.326 − 0.302i)17-s + (−0.5 + 0.866i)19-s + (−0.425 − 1.86i)20-s + (−0.535 − 0.496i)23-s + (−2.62 − 0.395i)25-s + (−0.222 − 0.974i)28-s + (−1.88 − 0.284i)35-s + (0.623 + 0.781i)36-s + (1.32 + 0.636i)43-s + (−0.147 + 1.97i)44-s + ⋯

Functional equation

Λ(s)=(931s/2 ΓC(s) L(s)=((0.481+0.876i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(931s/2 ΓC(s) L(s)=((0.481+0.876i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 931931    =    72⋅197^{2} \cdot 19
Sign: 0.481+0.876i0.481 + 0.876i
Analytic conductor: 0.4646290.464629
Root analytic conductor: 0.6816370.681637
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ931(436,⋅)\chi_{931} (436, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 931, ( :0), 0.481+0.876i)(2,\ 931,\ (\ :0),\ 0.481 + 0.876i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2448472651.244847265
L(12)L(\frac12) ≈\approx 1.2448472651.244847265
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)
bad7 1+(−0.0747+0.997i)T 1 + (-0.0747 + 0.997i)T
19 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good2 1+(−0.955+0.294i)T2 1 + (-0.955 + 0.294i)T^{2}
3 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
5 1+(−0.142+1.90i)T+(−0.988−0.149i)T2 1 + (-0.142 + 1.90i)T + (-0.988 - 0.149i)T^{2}
11 1+(0.722−1.84i)T+(−0.733−0.680i)T2 1 + (0.722 - 1.84i)T + (-0.733 - 0.680i)T^{2}
13 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
17 1+(−0.326+0.302i)T+(0.0747−0.997i)T2 1 + (-0.326 + 0.302i)T + (0.0747 - 0.997i)T^{2}
23 1+(0.535+0.496i)T+(0.0747+0.997i)T2 1 + (0.535 + 0.496i)T + (0.0747 + 0.997i)T^{2}
29 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
41 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
43 1+(−1.32−0.636i)T+(0.623+0.781i)T2 1 + (-1.32 - 0.636i)T + (0.623 + 0.781i)T^{2}
47 1+(1.63−0.246i)T+(0.955−0.294i)T2 1 + (1.63 - 0.246i)T + (0.955 - 0.294i)T^{2}
53 1+(−0.826+0.563i)T2 1 + (-0.826 + 0.563i)T^{2}
59 1+(0.988−0.149i)T2 1 + (0.988 - 0.149i)T^{2}
61 1+(−0.698−0.215i)T+(0.826+0.563i)T2 1 + (-0.698 - 0.215i)T + (0.826 + 0.563i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
73 1+(−0.988−0.149i)T+(0.955+0.294i)T2 1 + (-0.988 - 0.149i)T + (0.955 + 0.294i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+(0.914−1.14i)T+(−0.222−0.974i)T2 1 + (0.914 - 1.14i)T + (-0.222 - 0.974i)T^{2}
89 1+(0.733−0.680i)T2 1 + (0.733 - 0.680i)T^{2}
97 1−T2 1 - 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

−10.00998974299057213813340430827, −9.640550434923029321259604974228, −8.077315066411742335466464989295, −7.80991362831868491878477835764, −6.86408239301547854348197130832, −5.58711090708542263541373768980, −4.80143418963005399798868329223, −4.20550379886040510390486990879, −2.18147832742595872276909544889, −1.41498514178854086254503295525, 2.20921680240799776125907303548, 3.05578402586603668746804653317, 3.56510916960175904583355203195, 5.72208223500396974406671604544, 6.17333753764184944769059383947, 6.86263496252385146540491260706, 7.77788784388352566434682172579, 8.613686116832009583081906777426, 9.780251030726493296002586335987, 10.62074879254784528359023380258

Graph of the ZZ-function along the critical line