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Properties

Label 2-3168-44.43-c1-0-41
Degree 22
Conductor 31683168
Sign 0.762+0.647i0.762 + 0.647i
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  − 1.86·5-s + 3.91·7-s + (3.30 − 0.269i)11-s − 3.53i·13-s − 1.77i·17-s + 3.02·19-s + 7.41i·23-s − 1.50·25-s − 5.55i·29-s − 9.85i·31-s − 7.32·35-s + 9.85·37-s + 8.38i·41-s − 2.26·43-s + 3.13i·47-s + ⋯
L(s)  = 1  − 0.836·5-s + 1.48·7-s + (0.996 − 0.0813i)11-s − 0.980i·13-s − 0.429i·17-s + 0.694·19-s + 1.54i·23-s − 0.300·25-s − 1.03i·29-s − 1.76i·31-s − 1.23·35-s + 1.62·37-s + 1.30i·41-s − 0.345·43-s + 0.456i·47-s + ⋯

Functional equation

Λ(s)=(3168s/2 ΓC(s) L(s)=((0.762+0.647i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3168s/2 ΓC(s+1/2) L(s)=((0.762+0.647i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31683168    =    25⋅32⋅112^{5} \cdot 3^{2} \cdot 11
Sign: 0.762+0.647i0.762 + 0.647i
Analytic conductor: 25.296625.2966
Root analytic conductor: 5.029575.02957
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3168(703,⋅)\chi_{3168} (703, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3168, ( :1/2), 0.762+0.647i)(2,\ 3168,\ (\ :1/2),\ 0.762 + 0.647i)

Particular Values

L(1)L(1) ≈\approx 2.0052195272.005219527
L(12)L(\frac12) ≈\approx 2.0052195272.005219527
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+(−3.30+0.269i)T 1 + (-3.30 + 0.269i)T
good5 1+1.86T+5T2 1 + 1.86T + 5T^{2}
7 1−3.91T+7T2 1 - 3.91T + 7T^{2}
13 1+3.53iT−13T2 1 + 3.53iT - 13T^{2}
17 1+1.77iT−17T2 1 + 1.77iT - 17T^{2}
19 1−3.02T+19T2 1 - 3.02T + 19T^{2}
23 1−7.41iT−23T2 1 - 7.41iT - 23T^{2}
29 1+5.55iT−29T2 1 + 5.55iT - 29T^{2}
31 1+9.85iT−31T2 1 + 9.85iT - 31T^{2}
37 1−9.85T+37T2 1 - 9.85T + 37T^{2}
41 1−8.38iT−41T2 1 - 8.38iT - 41T^{2}
43 1+2.26T+43T2 1 + 2.26T + 43T^{2}
47 1−3.13iT−47T2 1 - 3.13iT - 47T^{2}
53 1+8.13T+53T2 1 + 8.13T + 53T^{2}
59 1+10.5iT−59T2 1 + 10.5iT - 59T^{2}
61 1+8.82iT−61T2 1 + 8.82iT - 61T^{2}
67 1+4.84iT−67T2 1 + 4.84iT - 67T^{2}
71 1−0.607iT−71T2 1 - 0.607iT - 71T^{2}
73 1−13.8iT−73T2 1 - 13.8iT - 73T^{2}
79 1+3.15T+79T2 1 + 3.15T + 79T^{2}
83 1−6.37T+83T2 1 - 6.37T + 83T^{2}
89 1+15.9T+89T2 1 + 15.9T + 89T^{2}
97 1−9.34T+97T2 1 - 9.34T + 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

−8.227080748424149228816906404499, −7.87584111094597633039372179067, −7.48182888417748494482584660989, −6.25153102112154930740022135641, −5.49946663743303762436535294097, −4.62625408326843188053845777664, −3.98399586179645150977347442664, −3.06629861735314785612471765593, −1.78823534438991200898584245996, −0.75433269867036962013398481923, 1.12347621292684529418623564779, 1.97508278213573388080110008019, 3.30861883301933583140866181703, 4.33240787224152817041795406732, 4.58727988162399793591738599159, 5.66348831904745439863575430831, 6.71344637996032918960852856646, 7.26509933053507258707203957887, 8.092445127386432068838319138669, 8.692217222326032466396264582002

Graph of the ZZ-function along the critical line