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Properties

Label 2-936-104.77-c1-0-21
Degree 22
Conductor 936936
Sign −0.196−0.980i-0.196 - 0.980i
Analytic cond. 7.473997.47399
Root an. cond. 2.733862.73386
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 + i)2-s − 2i·4-s + 5-s + 3i·7-s + (2 + 2i)8-s + (−1 + i)10-s + 2·11-s + (3 + 2i)13-s + (−3 − 3i)14-s − 4·16-s − 3·17-s − 2i·20-s + (−2 + 2i)22-s + 6·23-s − 4·25-s + (−5 + i)26-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − i·4-s + 0.447·5-s + 1.13i·7-s + (0.707 + 0.707i)8-s + (−0.316 + 0.316i)10-s + 0.603·11-s + (0.832 + 0.554i)13-s + (−0.801 − 0.801i)14-s − 16-s − 0.727·17-s − 0.447i·20-s + (−0.426 + 0.426i)22-s + 1.25·23-s − 0.800·25-s + (−0.980 + 0.196i)26-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((−0.196−0.980i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((−0.196−0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: −0.196−0.980i-0.196 - 0.980i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(181,⋅)\chi_{936} (181, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), −0.196−0.980i)(2,\ 936,\ (\ :1/2),\ -0.196 - 0.980i)

Particular Values

L(1)L(1) ≈\approx 0.747922+0.912318i0.747922 + 0.912318i
L(12)L(\frac12) ≈\approx 0.747922+0.912318i0.747922 + 0.912318i
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−i)T 1 + (1 - i)T
3 1 1
13 1+(−3−2i)T 1 + (-3 - 2i)T
good5 1−T+5T2 1 - T + 5T^{2}
7 1−3iT−7T2 1 - 3iT - 7T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
17 1+3T+17T2 1 + 3T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1−6T+23T2 1 - 6T + 23T^{2}
29 1+6iT−29T2 1 + 6iT - 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1+3T+37T2 1 + 3T + 37T^{2}
41 1−10iT−41T2 1 - 10iT - 41T^{2}
43 1−9iT−43T2 1 - 9iT - 43T^{2}
47 1−7iT−47T2 1 - 7iT - 47T^{2}
53 1−6iT−53T2 1 - 6iT - 53T^{2}
59 1−10T+59T2 1 - 10T + 59T^{2}
61 1+10iT−61T2 1 + 10iT - 61T^{2}
67 1−12T+67T2 1 - 12T + 67T^{2}
71 1−5iT−71T2 1 - 5iT - 71T^{2}
73 1−6iT−73T2 1 - 6iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+16T+83T2 1 + 16T + 83T^{2}
89 1+4iT−89T2 1 + 4iT - 89T^{2}
97 1−18iT−97T2 1 - 18iT - 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

−9.896031036831754310614677895701, −9.318769480271139168092393886138, −8.720224063261848675880045306886, −7.953424199830714189751546535025, −6.69002984096417571674870515759, −6.21658840685772715266108046972, −5.37050853715856357629698359101, −4.28217468648720941394710763754, −2.56668023260271273998744267207, −1.41932102265810360053244028427, 0.77671772229382756802961712828, 1.95342585048035458596849589691, 3.41304881685524990367262677816, 4.07455294406761503892822765197, 5.39988461629087636259830217889, 6.82376974626189334107760792119, 7.21038028289056137379673801635, 8.489778513412188484963770881819, 8.963966574754652868723293127536, 9.992638986484286305177440986783

Graph of the ZZ-function along the critical line