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Properties

Label 2-960-960.389-c0-0-1
Degree 22
Conductor 960960
Sign 0.956+0.290i0.956 + 0.290i
Analytic cond. 0.4791020.479102
Root an. cond. 0.6921720.692172
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  + (−0.195 − 0.980i)2-s + (0.831 + 0.555i)3-s + (−0.923 + 0.382i)4-s + (0.980 + 0.195i)5-s + (0.382 − 0.923i)6-s + (0.555 + 0.831i)8-s + (0.382 + 0.923i)9-s − i·10-s + (−0.980 − 0.195i)12-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)16-s + (−1.38 + 1.38i)17-s + (0.831 − 0.555i)18-s + (−0.216 − 1.08i)19-s + (−0.980 + 0.195i)20-s + ⋯
L(s)  = 1  + (−0.195 − 0.980i)2-s + (0.831 + 0.555i)3-s + (−0.923 + 0.382i)4-s + (0.980 + 0.195i)5-s + (0.382 − 0.923i)6-s + (0.555 + 0.831i)8-s + (0.382 + 0.923i)9-s − i·10-s + (−0.980 − 0.195i)12-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)16-s + (−1.38 + 1.38i)17-s + (0.831 − 0.555i)18-s + (−0.216 − 1.08i)19-s + (−0.980 + 0.195i)20-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((0.956+0.290i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s) L(s)=((0.956+0.290i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.956+0.290i0.956 + 0.290i
Analytic conductor: 0.4791020.479102
Root analytic conductor: 0.6921720.692172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ960(389,⋅)\chi_{960} (389, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :0), 0.956+0.290i)(2,\ 960,\ (\ :0),\ 0.956 + 0.290i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2485783801.248578380
L(12)L(\frac12) ≈\approx 1.2485783801.248578380
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+(0.195+0.980i)T 1 + (0.195 + 0.980i)T
3 1+(−0.831−0.555i)T 1 + (-0.831 - 0.555i)T
5 1+(−0.980−0.195i)T 1 + (-0.980 - 0.195i)T
good7 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
11 1+(0.382−0.923i)T2 1 + (0.382 - 0.923i)T^{2}
13 1+(−0.923+0.382i)T2 1 + (-0.923 + 0.382i)T^{2}
17 1+(1.38−1.38i)T−iT2 1 + (1.38 - 1.38i)T - iT^{2}
19 1+(0.216+1.08i)T+(−0.923+0.382i)T2 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2}
23 1+(0.360−0.149i)T+(0.707−0.707i)T2 1 + (0.360 - 0.149i)T + (0.707 - 0.707i)T^{2}
29 1+(0.382+0.923i)T2 1 + (0.382 + 0.923i)T^{2}
31 1+1.84iT−T2 1 + 1.84iT - T^{2}
37 1+(0.923+0.382i)T2 1 + (0.923 + 0.382i)T^{2}
41 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
43 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
47 1+(−1.17+1.17i)T−iT2 1 + (-1.17 + 1.17i)T - iT^{2}
53 1+(0.425+0.636i)T+(−0.382+0.923i)T2 1 + (0.425 + 0.636i)T + (-0.382 + 0.923i)T^{2}
59 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)T^{2}
61 1+(−0.923−0.617i)T+(0.382+0.923i)T2 1 + (-0.923 - 0.617i)T + (0.382 + 0.923i)T^{2}
67 1+(−0.382−0.923i)T2 1 + (-0.382 - 0.923i)T^{2}
71 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
73 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
79 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
83 1+(−0.149−0.750i)T+(−0.923+0.382i)T2 1 + (-0.149 - 0.750i)T + (-0.923 + 0.382i)T^{2}
89 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)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

−10.16203082594990455231089736433, −9.444196495218162976854878256473, −8.812113211353761383045791250777, −8.135381427092924445131068106835, −6.89042203985823997692921466181, −5.65857110441289903918475932802, −4.56082797500166697892004162595, −3.78605830574531027549749174681, −2.52641905046569448820966043719, −1.93930814312629708338747869783, 1.45832731178677814512245356744, 2.75969476923195816037271619409, 4.21354241241027332760214610515, 5.21592302356782620196135050640, 6.28576833617730507630132936869, 6.83787603842275751557321520410, 7.74430618963198388583800835426, 8.652858141990657970756101795716, 9.178514027191747004828402249546, 9.853707363287675848902197133692

Graph of the ZZ-function along the critical line