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Properties

Label 2-209-19.11-c1-0-12
Degree 22
Conductor 209209
Sign 0.845+0.534i0.845 + 0.534i
Analytic cond. 1.668871.66887
Root an. cond. 1.291841.29184
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  + (−0.0694 + 0.120i)2-s + (0.748 − 1.29i)3-s + (0.990 + 1.71i)4-s + (1.75 − 3.04i)5-s + (0.103 + 0.179i)6-s − 1.46·7-s − 0.552·8-s + (0.379 + 0.657i)9-s + (0.244 + 0.422i)10-s − 11-s + 2.96·12-s + (−1.19 − 2.07i)13-s + (0.101 − 0.176i)14-s + (−2.63 − 4.55i)15-s + (−1.94 + 3.36i)16-s + (1.62 − 2.80i)17-s + ⋯
L(s)  = 1  + (−0.0490 + 0.0850i)2-s + (0.432 − 0.748i)3-s + (0.495 + 0.857i)4-s + (0.786 − 1.36i)5-s + (0.0424 + 0.0734i)6-s − 0.555·7-s − 0.195·8-s + (0.126 + 0.219i)9-s + (0.0772 + 0.133i)10-s − 0.301·11-s + 0.855·12-s + (−0.331 − 0.574i)13-s + (0.0272 − 0.0472i)14-s + (−0.679 − 1.17i)15-s + (−0.485 + 0.841i)16-s + (0.392 − 0.680i)17-s + ⋯

Functional equation

Λ(s)=(209s/2 ΓC(s) L(s)=((0.845+0.534i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(209s/2 ΓC(s+1/2) L(s)=((0.845+0.534i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 209209    =    11⋅1911 \cdot 19
Sign: 0.845+0.534i0.845 + 0.534i
Analytic conductor: 1.668871.66887
Root analytic conductor: 1.291841.29184
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ209(144,⋅)\chi_{209} (144, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 209, ( :1/2), 0.845+0.534i)(2,\ 209,\ (\ :1/2),\ 0.845 + 0.534i)

Particular Values

L(1)L(1) ≈\approx 1.47345−0.426477i1.47345 - 0.426477i
L(12)L(\frac12) ≈\approx 1.47345−0.426477i1.47345 - 0.426477i
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)
bad11 1+T 1 + T
19 1+(1.08−4.22i)T 1 + (1.08 - 4.22i)T
good2 1+(0.0694−0.120i)T+(−1−1.73i)T2 1 + (0.0694 - 0.120i)T + (-1 - 1.73i)T^{2}
3 1+(−0.748+1.29i)T+(−1.5−2.59i)T2 1 + (-0.748 + 1.29i)T + (-1.5 - 2.59i)T^{2}
5 1+(−1.75+3.04i)T+(−2.5−4.33i)T2 1 + (-1.75 + 3.04i)T + (-2.5 - 4.33i)T^{2}
7 1+1.46T+7T2 1 + 1.46T + 7T^{2}
13 1+(1.19+2.07i)T+(−6.5+11.2i)T2 1 + (1.19 + 2.07i)T + (-6.5 + 11.2i)T^{2}
17 1+(−1.62+2.80i)T+(−8.5−14.7i)T2 1 + (-1.62 + 2.80i)T + (-8.5 - 14.7i)T^{2}
23 1+(−3.69−6.39i)T+(−11.5+19.9i)T2 1 + (-3.69 - 6.39i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.20+5.55i)T+(−14.5+25.1i)T2 1 + (3.20 + 5.55i)T + (-14.5 + 25.1i)T^{2}
31 1−0.866T+31T2 1 - 0.866T + 31T^{2}
37 1+2.78T+37T2 1 + 2.78T + 37T^{2}
41 1+(4.92−8.53i)T+(−20.5−35.5i)T2 1 + (4.92 - 8.53i)T + (-20.5 - 35.5i)T^{2}
43 1+(3.65−6.32i)T+(−21.5−37.2i)T2 1 + (3.65 - 6.32i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.67+2.90i)T+(−23.5+40.7i)T2 1 + (1.67 + 2.90i)T + (-23.5 + 40.7i)T^{2}
53 1+(−1.89−3.27i)T+(−26.5+45.8i)T2 1 + (-1.89 - 3.27i)T + (-26.5 + 45.8i)T^{2}
59 1+(4.20−7.27i)T+(−29.5−51.0i)T2 1 + (4.20 - 7.27i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.718−1.24i)T+(−30.5+52.8i)T2 1 + (-0.718 - 1.24i)T + (-30.5 + 52.8i)T^{2}
67 1+(3.24+5.61i)T+(−33.5+58.0i)T2 1 + (3.24 + 5.61i)T + (-33.5 + 58.0i)T^{2}
71 1+(−1.70+2.95i)T+(−35.5−61.4i)T2 1 + (-1.70 + 2.95i)T + (-35.5 - 61.4i)T^{2}
73 1+(1.40−2.43i)T+(−36.5−63.2i)T2 1 + (1.40 - 2.43i)T + (-36.5 - 63.2i)T^{2}
79 1+(−8.85+15.3i)T+(−39.5−68.4i)T2 1 + (-8.85 + 15.3i)T + (-39.5 - 68.4i)T^{2}
83 1−17.7T+83T2 1 - 17.7T + 83T^{2}
89 1+(6.43+11.1i)T+(−44.5+77.0i)T2 1 + (6.43 + 11.1i)T + (-44.5 + 77.0i)T^{2}
97 1+(−7.86+13.6i)T+(−48.5−84.0i)T2 1 + (-7.86 + 13.6i)T + (-48.5 - 84.0i)T^{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

−12.56195959610403738879505766991, −11.71909070172864031343387838052, −10.12107715390254869744165541683, −9.182152490816246564853076951052, −8.101510075364194136527822605373, −7.50777599526868171275936855449, −6.16596556918202143820499902329, −4.93735901672846743512611870063, −3.12225138242825851872560699139, −1.70939794489446019063378583610, 2.29304530911603406488447543962, 3.38533802087740995271757086415, 5.13670549679476856935241465549, 6.55547250320376770373367772903, 6.87737467881718996342260341429, 8.938611812982199169288625158907, 9.772808063422296310345557484253, 10.46231570414199819393008184762, 10.97267591150302607523758608945, 12.43758122197334734254146990657

Graph of the ZZ-function along the critical line