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Properties

Label 2-936-104.77-c1-0-12
Degree 22
Conductor 936936
Sign −0.980+0.196i-0.980 + 0.196i
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 + (−1 − 5i)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 + (−0.196 − 0.980i)26-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: −0.980+0.196i-0.980 + 0.196i
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), −0.980+0.196i)(2,\ 936,\ (\ :1/2),\ -0.980 + 0.196i)

Particular Values

L(1)L(1) ≈\approx 0.116245−1.17396i0.116245 - 1.17396i
L(12)L(\frac12) ≈\approx 0.116245−1.17396i0.116245 - 1.17396i
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

−10.65612740236107695790796582103, −9.367058236941436946838956909401, −8.677899163356811975775206407300, −7.83881314973015279409779516984, −7.13176885768600318265574262979, −6.09090916221809287640824532885, −5.25323210754301293279961185377, −4.59174313634459510366590335032, −3.22520343473956141973312925965, −2.43280148417257110668833704133, 0.41005069647730060555325328768, 2.04571798562098403577891944973, 3.23257123115486763657558912311, 4.29353286187484130695075389112, 4.79578636324188286305997361737, 6.05299920579516037395756635088, 7.06301245441252675334246236615, 7.71697366843732888175956990579, 9.051255729462906967283285122444, 9.838685379266236219076443981276

Graph of the ZZ-function along the critical line