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Properties

Label 2-3168-88.43-c1-0-38
Degree 22
Conductor 31683168
Sign 0.517+0.855i0.517 + 0.855i
Analytic cond. 25.296625.2966
Root an. cond. 5.029575.02957
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  − 2i·5-s + 3.02·7-s + (2.33 + 2.35i)11-s − 1.32·13-s + 1.69i·17-s − 7.04i·19-s − 3.12i·23-s + 25-s + 1.54·29-s − 8.30i·31-s − 6.04i·35-s − 4.66i·37-s + 7.73i·41-s + 3.95i·43-s + 6i·47-s + ⋯
L(s)  = 1  − 0.894i·5-s + 1.14·7-s + (0.703 + 0.711i)11-s − 0.367·13-s + 0.411i·17-s − 1.61i·19-s − 0.651i·23-s + 0.200·25-s + 0.286·29-s − 1.49i·31-s − 1.02i·35-s − 0.766i·37-s + 1.20i·41-s + 0.603i·43-s + 0.875i·47-s + ⋯

Functional equation

Λ(s)=(3168s/2 ΓC(s) L(s)=((0.517+0.855i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3168s/2 ΓC(s+1/2) L(s)=((0.517+0.855i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31683168    =    25⋅32⋅112^{5} \cdot 3^{2} \cdot 11
Sign: 0.517+0.855i0.517 + 0.855i
Analytic conductor: 25.296625.2966
Root analytic conductor: 5.029575.02957
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3168(2287,⋅)\chi_{3168} (2287, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3168, ( :1/2), 0.517+0.855i)(2,\ 3168,\ (\ :1/2),\ 0.517 + 0.855i)

Particular Values

L(1)L(1) ≈\approx 2.1904559272.190455927
L(12)L(\frac12) ≈\approx 2.1904559272.190455927
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
3 1 1
11 1+(−2.33−2.35i)T 1 + (-2.33 - 2.35i)T
good5 1+2iT−5T2 1 + 2iT - 5T^{2}
7 1−3.02T+7T2 1 - 3.02T + 7T^{2}
13 1+1.32T+13T2 1 + 1.32T + 13T^{2}
17 1−1.69iT−17T2 1 - 1.69iT - 17T^{2}
19 1+7.04iT−19T2 1 + 7.04iT - 19T^{2}
23 1+3.12iT−23T2 1 + 3.12iT - 23T^{2}
29 1−1.54T+29T2 1 - 1.54T + 29T^{2}
31 1+8.30iT−31T2 1 + 8.30iT - 31T^{2}
37 1+4.66iT−37T2 1 + 4.66iT - 37T^{2}
41 1−7.73iT−41T2 1 - 7.73iT - 41T^{2}
43 1−3.95iT−43T2 1 - 3.95iT - 43T^{2}
47 1−6iT−47T2 1 - 6iT - 47T^{2}
53 1−8.24iT−53T2 1 - 8.24iT - 53T^{2}
59 1−11.9T+59T2 1 - 11.9T + 59T^{2}
61 1−8.10T+61T2 1 - 8.10T + 61T^{2}
67 1+6.24T+67T2 1 + 6.24T + 67T^{2}
71 1+12.2iT−71T2 1 + 12.2iT - 71T^{2}
73 1+3.08iT−73T2 1 + 3.08iT - 73T^{2}
79 1−12.4T+79T2 1 - 12.4T + 79T^{2}
83 1+4.71iT−83T2 1 + 4.71iT - 83T^{2}
89 1+16.6T+89T2 1 + 16.6T + 89T^{2}
97 1+13.3T+97T2 1 + 13.3T + 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

−8.559265819530087211100029781322, −7.904024132828787331343861794117, −7.14622931142315751394809937059, −6.30974419787907786704196805414, −5.30215273185009316684525105932, −4.51799917464475203238237895549, −4.31756285043683327286969381693, −2.72635489573020620194601770187, −1.77353539771034145792166302182, −0.77131389546558420690160240205, 1.19826718851821747748104164374, 2.18018774917416114806963016768, 3.31464885647746528500727501516, 3.94570305511394097409104244136, 5.11737731406286715163862818677, 5.65507844009595802964616520789, 6.77314763134082511865433207179, 7.13444881332669220639219705353, 8.254192759282531533467104100442, 8.491978372856094870018480473118

Graph of the ZZ-function along the critical line