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Properties

Label 2-368-16.13-c1-0-34
Degree 22
Conductor 368368
Sign 0.936+0.350i0.936 + 0.350i
Analytic cond. 2.938492.93849
Root an. cond. 1.714201.71420
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.38 − 0.287i)2-s + (0.339 + 0.339i)3-s + (1.83 − 0.796i)4-s + (1.46 − 1.46i)5-s + (0.568 + 0.372i)6-s + 2.63i·7-s + (2.31 − 1.63i)8-s − 2.76i·9-s + (1.61 − 2.45i)10-s + (−2.99 + 2.99i)11-s + (0.893 + 0.352i)12-s + (−0.749 − 0.749i)13-s + (0.757 + 3.64i)14-s + 0.998·15-s + (2.72 − 2.92i)16-s − 3.64·17-s + ⋯
L(s)  = 1  + (0.979 − 0.203i)2-s + (0.196 + 0.196i)3-s + (0.917 − 0.398i)4-s + (0.657 − 0.657i)5-s + (0.231 + 0.152i)6-s + 0.994i·7-s + (0.816 − 0.576i)8-s − 0.923i·9-s + (0.509 − 0.777i)10-s + (−0.901 + 0.901i)11-s + (0.258 + 0.101i)12-s + (−0.207 − 0.207i)13-s + (0.202 + 0.973i)14-s + 0.257·15-s + (0.682 − 0.730i)16-s − 0.885·17-s + ⋯

Functional equation

Λ(s)=(368s/2 ΓC(s) L(s)=((0.936+0.350i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(368s/2 ΓC(s+1/2) L(s)=((0.936+0.350i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 368368    =    24⋅232^{4} \cdot 23
Sign: 0.936+0.350i0.936 + 0.350i
Analytic conductor: 2.938492.93849
Root analytic conductor: 1.714201.71420
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ368(93,⋅)\chi_{368} (93, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 368, ( :1/2), 0.936+0.350i)(2,\ 368,\ (\ :1/2),\ 0.936 + 0.350i)

Particular Values

L(1)L(1) ≈\approx 2.63665−0.477676i2.63665 - 0.477676i
L(12)L(\frac12) ≈\approx 2.63665−0.477676i2.63665 - 0.477676i
L(32)L(\frac{3}{2}) 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+(−1.38+0.287i)T 1 + (-1.38 + 0.287i)T
23 1+iT 1 + iT
good3 1+(−0.339−0.339i)T+3iT2 1 + (-0.339 - 0.339i)T + 3iT^{2}
5 1+(−1.46+1.46i)T−5iT2 1 + (-1.46 + 1.46i)T - 5iT^{2}
7 1−2.63iT−7T2 1 - 2.63iT - 7T^{2}
11 1+(2.99−2.99i)T−11iT2 1 + (2.99 - 2.99i)T - 11iT^{2}
13 1+(0.749+0.749i)T+13iT2 1 + (0.749 + 0.749i)T + 13iT^{2}
17 1+3.64T+17T2 1 + 3.64T + 17T^{2}
19 1+(−1.76−1.76i)T+19iT2 1 + (-1.76 - 1.76i)T + 19iT^{2}
29 1+(−0.0790−0.0790i)T+29iT2 1 + (-0.0790 - 0.0790i)T + 29iT^{2}
31 1+2.07T+31T2 1 + 2.07T + 31T^{2}
37 1+(3.64−3.64i)T−37iT2 1 + (3.64 - 3.64i)T - 37iT^{2}
41 1−1.90iT−41T2 1 - 1.90iT - 41T^{2}
43 1+(−6.84+6.84i)T−43iT2 1 + (-6.84 + 6.84i)T - 43iT^{2}
47 1+8.55T+47T2 1 + 8.55T + 47T^{2}
53 1+(6.62−6.62i)T−53iT2 1 + (6.62 - 6.62i)T - 53iT^{2}
59 1+(5.45−5.45i)T−59iT2 1 + (5.45 - 5.45i)T - 59iT^{2}
61 1+(−2.13−2.13i)T+61iT2 1 + (-2.13 - 2.13i)T + 61iT^{2}
67 1+(−3.51−3.51i)T+67iT2 1 + (-3.51 - 3.51i)T + 67iT^{2}
71 1+1.61iT−71T2 1 + 1.61iT - 71T^{2}
73 1+14.4iT−73T2 1 + 14.4iT - 73T^{2}
79 1+12.5T+79T2 1 + 12.5T + 79T^{2}
83 1+(−1.26−1.26i)T+83iT2 1 + (-1.26 - 1.26i)T + 83iT^{2}
89 1+12.3iT−89T2 1 + 12.3iT - 89T^{2}
97 1−7.54T+97T2 1 - 7.54T + 97T^{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

−11.67331046134537991926051670959, −10.43055051161851484355695852987, −9.596587694967829241646983811956, −8.790614164973288290668028109915, −7.40141690428137971701284600117, −6.19792187276212811289909293263, −5.37176151774524990851548993173, −4.51571447069317620394101415623, −3.02976387776582137422094243857, −1.89314453486748937723889002260, 2.14647611322891899249383761149, 3.17806668114539975544731516127, 4.57073523551011673659973577738, 5.58455115885075479121458773326, 6.67541779815116205496210586145, 7.43952672869535936440227599559, 8.326986679551943293536942114158, 9.934659212313522878571839524724, 10.96201940963414219599887160547, 11.10960710558376953664746241946

Graph of the ZZ-function along the critical line