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Properties

Label 2-1881-209.208-c1-0-30
Degree 22
Conductor 18811881
Sign −0.216−0.976i-0.216 - 0.976i
Analytic cond. 15.019815.0198
Root an. cond. 3.875543.87554
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  + 2.29·2-s + 3.24·4-s − 2.24·5-s + 2.78i·7-s + 2.85·8-s − 5.14·10-s + (3.31 + 0.0382i)11-s − 6.74·13-s + 6.38i·14-s + 0.0447·16-s + 6.10i·17-s + (0.994 + 4.24i)19-s − 7.29·20-s + (7.59 + 0.0876i)22-s + 5.56·23-s + ⋯
L(s)  = 1  + 1.61·2-s + 1.62·4-s − 1.00·5-s + 1.05i·7-s + 1.00·8-s − 1.62·10-s + (0.999 + 0.0115i)11-s − 1.87·13-s + 1.70i·14-s + 0.0111·16-s + 1.48i·17-s + (0.228 + 0.973i)19-s − 1.63·20-s + (1.61 + 0.0186i)22-s + 1.16·23-s + ⋯

Functional equation

Λ(s)=(1881s/2 ΓC(s) L(s)=((−0.216−0.976i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1881s/2 ΓC(s+1/2) L(s)=((−0.216−0.976i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18811881    =    32⋅11⋅193^{2} \cdot 11 \cdot 19
Sign: −0.216−0.976i-0.216 - 0.976i
Analytic conductor: 15.019815.0198
Root analytic conductor: 3.875543.87554
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1881(208,⋅)\chi_{1881} (208, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1881, ( :1/2), −0.216−0.976i)(2,\ 1881,\ (\ :1/2),\ -0.216 - 0.976i)

Particular Values

L(1)L(1) ≈\approx 2.8129272572.812927257
L(12)L(\frac12) ≈\approx 2.8129272572.812927257
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)
bad3 1 1
11 1+(−3.31−0.0382i)T 1 + (-3.31 - 0.0382i)T
19 1+(−0.994−4.24i)T 1 + (-0.994 - 4.24i)T
good2 1−2.29T+2T2 1 - 2.29T + 2T^{2}
5 1+2.24T+5T2 1 + 2.24T + 5T^{2}
7 1−2.78iT−7T2 1 - 2.78iT - 7T^{2}
13 1+6.74T+13T2 1 + 6.74T + 13T^{2}
17 1−6.10iT−17T2 1 - 6.10iT - 17T^{2}
23 1−5.56T+23T2 1 - 5.56T + 23T^{2}
29 1+3.42T+29T2 1 + 3.42T + 29T^{2}
31 1−6.66iT−31T2 1 - 6.66iT - 31T^{2}
37 1+1.10iT−37T2 1 + 1.10iT - 37T^{2}
41 1−4.69T+41T2 1 - 4.69T + 41T^{2}
43 1+7.61iT−43T2 1 + 7.61iT - 43T^{2}
47 1−5.34T+47T2 1 - 5.34T + 47T^{2}
53 1−11.4iT−53T2 1 - 11.4iT - 53T^{2}
59 1+11.9iT−59T2 1 + 11.9iT - 59T^{2}
61 1−9.96iT−61T2 1 - 9.96iT - 61T^{2}
67 1+6.39iT−67T2 1 + 6.39iT - 67T^{2}
71 1−5.29iT−71T2 1 - 5.29iT - 71T^{2}
73 1−2.27iT−73T2 1 - 2.27iT - 73T^{2}
79 1+7.84T+79T2 1 + 7.84T + 79T^{2}
83 1+15.4iT−83T2 1 + 15.4iT - 83T^{2}
89 1−4.77iT−89T2 1 - 4.77iT - 89T^{2}
97 1−14.8iT−97T2 1 - 14.8iT - 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

−9.338570018565173716721893306039, −8.637830392969651921416778116439, −7.55897327737333145457648156170, −6.95847100839101622319797650837, −5.95699861637794880488663142397, −5.35943197305851320757790556895, −4.43707174649335655853330833488, −3.79208494579440500767906471716, −2.93133549936345328858420476578, −1.88858575271740291916988706339, 0.56283687687746734952784776242, 2.48573000695453179265730838843, 3.29776208353696691874090248579, 4.27963906991284586426176836637, 4.60579829962267019289218339430, 5.45929856556949644553062692730, 6.77944398943514017472920401325, 7.18748349162981386708803482966, 7.66831796538312818373576794748, 9.178536055717067783936867203943

Graph of the ZZ-function along the critical line