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Properties

Label 2-209-19.11-c1-0-17
Degree 22
Conductor 209209
Sign −0.839+0.542i-0.839 + 0.542i
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  + (1.31 − 2.28i)2-s + (0.631 − 1.09i)3-s + (−2.47 − 4.28i)4-s + (−0.759 + 1.31i)5-s + (−1.66 − 2.88i)6-s + 1.94·7-s − 7.77·8-s + (0.702 + 1.21i)9-s + (2.00 + 3.46i)10-s − 11-s − 6.25·12-s + (1.57 + 2.72i)13-s + (2.56 − 4.44i)14-s + (0.959 + 1.66i)15-s + (−5.30 + 9.18i)16-s + (0.682 − 1.18i)17-s + ⋯
L(s)  = 1  + (0.932 − 1.61i)2-s + (0.364 − 0.631i)3-s + (−1.23 − 2.14i)4-s + (−0.339 + 0.588i)5-s + (−0.679 − 1.17i)6-s + 0.735·7-s − 2.75·8-s + (0.234 + 0.405i)9-s + (0.633 + 1.09i)10-s − 0.301·11-s − 1.80·12-s + (0.436 + 0.756i)13-s + (0.685 − 1.18i)14-s + (0.247 + 0.429i)15-s + (−1.32 + 2.29i)16-s + (0.165 − 0.286i)17-s + ⋯

Functional equation

Λ(s)=(209s/2 ΓC(s) L(s)=((−0.839+0.542i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(209s/2 ΓC(s+1/2) L(s)=((−0.839+0.542i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 209209    =    11⋅1911 \cdot 19
Sign: −0.839+0.542i-0.839 + 0.542i
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), −0.839+0.542i)(2,\ 209,\ (\ :1/2),\ -0.839 + 0.542i)

Particular Values

L(1)L(1) ≈\approx 0.545549−1.84959i0.545549 - 1.84959i
L(12)L(\frac12) ≈\approx 0.545549−1.84959i0.545549 - 1.84959i
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+(3.37+2.76i)T 1 + (3.37 + 2.76i)T
good2 1+(−1.31+2.28i)T+(−1−1.73i)T2 1 + (-1.31 + 2.28i)T + (-1 - 1.73i)T^{2}
3 1+(−0.631+1.09i)T+(−1.5−2.59i)T2 1 + (-0.631 + 1.09i)T + (-1.5 - 2.59i)T^{2}
5 1+(0.759−1.31i)T+(−2.5−4.33i)T2 1 + (0.759 - 1.31i)T + (-2.5 - 4.33i)T^{2}
7 1−1.94T+7T2 1 - 1.94T + 7T^{2}
13 1+(−1.57−2.72i)T+(−6.5+11.2i)T2 1 + (-1.57 - 2.72i)T + (-6.5 + 11.2i)T^{2}
17 1+(−0.682+1.18i)T+(−8.5−14.7i)T2 1 + (-0.682 + 1.18i)T + (-8.5 - 14.7i)T^{2}
23 1+(0.194+0.337i)T+(−11.5+19.9i)T2 1 + (0.194 + 0.337i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.909+1.57i)T+(−14.5+25.1i)T2 1 + (0.909 + 1.57i)T + (-14.5 + 25.1i)T^{2}
31 1+7.22T+31T2 1 + 7.22T + 31T^{2}
37 1−10.8T+37T2 1 - 10.8T + 37T^{2}
41 1+(5.02−8.69i)T+(−20.5−35.5i)T2 1 + (5.02 - 8.69i)T + (-20.5 - 35.5i)T^{2}
43 1+(−0.249+0.431i)T+(−21.5−37.2i)T2 1 + (-0.249 + 0.431i)T + (-21.5 - 37.2i)T^{2}
47 1+(5.73+9.93i)T+(−23.5+40.7i)T2 1 + (5.73 + 9.93i)T + (-23.5 + 40.7i)T^{2}
53 1+(0.490+0.849i)T+(−26.5+45.8i)T2 1 + (0.490 + 0.849i)T + (-26.5 + 45.8i)T^{2}
59 1+(2.98−5.17i)T+(−29.5−51.0i)T2 1 + (2.98 - 5.17i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2.22−3.85i)T+(−30.5+52.8i)T2 1 + (-2.22 - 3.85i)T + (-30.5 + 52.8i)T^{2}
67 1+(−6.90−11.9i)T+(−33.5+58.0i)T2 1 + (-6.90 - 11.9i)T + (-33.5 + 58.0i)T^{2}
71 1+(3.53−6.11i)T+(−35.5−61.4i)T2 1 + (3.53 - 6.11i)T + (-35.5 - 61.4i)T^{2}
73 1+(−4.27+7.39i)T+(−36.5−63.2i)T2 1 + (-4.27 + 7.39i)T + (-36.5 - 63.2i)T^{2}
79 1+(−7.58+13.1i)T+(−39.5−68.4i)T2 1 + (-7.58 + 13.1i)T + (-39.5 - 68.4i)T^{2}
83 1−7.70T+83T2 1 - 7.70T + 83T^{2}
89 1+(0.833+1.44i)T+(−44.5+77.0i)T2 1 + (0.833 + 1.44i)T + (-44.5 + 77.0i)T^{2}
97 1+(8.72−15.1i)T+(−48.5−84.0i)T2 1 + (8.72 - 15.1i)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

−11.86261274947850173889958684614, −11.20774388977915604109300089092, −10.56757022337613064817594335859, −9.323869190216158921255019613953, −8.054386482025759236674284035419, −6.72433135493316502458404438548, −5.15075898883313816316522547562, −4.11189680633162856379740932501, −2.74266725772947154777400046843, −1.66086715866769182826346391818, 3.54504113836738133443230260641, 4.42642598237463640529431996861, 5.34986550424181611025106557601, 6.47546339454052288909248350123, 7.88494780418990910548984297064, 8.322130661772869500969674245713, 9.383380137576990565149768541548, 10.89427921606373653120011526202, 12.46500432071863253801361239391, 12.81907311483737245924164326380

Graph of the ZZ-function along the critical line