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Properties

Label 2-209-11.3-c1-0-0
Degree 22
Conductor 209209
Sign −0.660+0.750i-0.660 + 0.750i
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  + (−1.28 + 0.936i)2-s + (0.935 + 2.87i)3-s + (0.166 − 0.511i)4-s + (−2.91 − 2.11i)5-s + (−3.90 − 2.83i)6-s + (0.0705 − 0.217i)7-s + (−0.719 − 2.21i)8-s + (−4.98 + 3.61i)9-s + 5.74·10-s + (−3.21 + 0.802i)11-s + 1.62·12-s + (−1.03 + 0.753i)13-s + (0.112 + 0.345i)14-s + (3.37 − 10.3i)15-s + (3.87 + 2.81i)16-s + (5.03 + 3.65i)17-s + ⋯
L(s)  = 1  + (−0.911 + 0.662i)2-s + (0.539 + 1.66i)3-s + (0.0831 − 0.255i)4-s + (−1.30 − 0.947i)5-s + (−1.59 − 1.15i)6-s + (0.0266 − 0.0820i)7-s + (−0.254 − 0.783i)8-s + (−1.66 + 1.20i)9-s + 1.81·10-s + (−0.970 + 0.242i)11-s + 0.470·12-s + (−0.287 + 0.209i)13-s + (0.0300 + 0.0924i)14-s + (0.870 − 2.67i)15-s + (0.968 + 0.703i)16-s + (1.22 + 0.887i)17-s + ⋯

Functional equation

Λ(s)=(209s/2 ΓC(s) L(s)=((−0.660+0.750i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(209s/2 ΓC(s+1/2) L(s)=((−0.660+0.750i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 209209    =    11⋅1911 \cdot 19
Sign: −0.660+0.750i-0.660 + 0.750i
Analytic conductor: 1.668871.66887
Root analytic conductor: 1.291841.29184
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ209(58,⋅)\chi_{209} (58, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 209, ( :1/2), −0.660+0.750i)(2,\ 209,\ (\ :1/2),\ -0.660 + 0.750i)

Particular Values

L(1)L(1) ≈\approx 0.144097−0.318825i0.144097 - 0.318825i
L(12)L(\frac12) ≈\approx 0.144097−0.318825i0.144097 - 0.318825i
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+(3.21−0.802i)T 1 + (3.21 - 0.802i)T
19 1+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
good2 1+(1.28−0.936i)T+(0.618−1.90i)T2 1 + (1.28 - 0.936i)T + (0.618 - 1.90i)T^{2}
3 1+(−0.935−2.87i)T+(−2.42+1.76i)T2 1 + (-0.935 - 2.87i)T + (-2.42 + 1.76i)T^{2}
5 1+(2.91+2.11i)T+(1.54+4.75i)T2 1 + (2.91 + 2.11i)T + (1.54 + 4.75i)T^{2}
7 1+(−0.0705+0.217i)T+(−5.66−4.11i)T2 1 + (-0.0705 + 0.217i)T + (-5.66 - 4.11i)T^{2}
13 1+(1.03−0.753i)T+(4.01−12.3i)T2 1 + (1.03 - 0.753i)T + (4.01 - 12.3i)T^{2}
17 1+(−5.03−3.65i)T+(5.25+16.1i)T2 1 + (-5.03 - 3.65i)T + (5.25 + 16.1i)T^{2}
23 1+7.86T+23T2 1 + 7.86T + 23T^{2}
29 1+(−0.00257+0.00791i)T+(−23.4−17.0i)T2 1 + (-0.00257 + 0.00791i)T + (-23.4 - 17.0i)T^{2}
31 1+(6.07−4.41i)T+(9.57−29.4i)T2 1 + (6.07 - 4.41i)T + (9.57 - 29.4i)T^{2}
37 1+(−1.24+3.82i)T+(−29.9−21.7i)T2 1 + (-1.24 + 3.82i)T + (-29.9 - 21.7i)T^{2}
41 1+(−1.87−5.76i)T+(−33.1+24.0i)T2 1 + (-1.87 - 5.76i)T + (-33.1 + 24.0i)T^{2}
43 1+2.42T+43T2 1 + 2.42T + 43T^{2}
47 1+(0.516+1.58i)T+(−38.0+27.6i)T2 1 + (0.516 + 1.58i)T + (-38.0 + 27.6i)T^{2}
53 1+(−0.124+0.0902i)T+(16.3−50.4i)T2 1 + (-0.124 + 0.0902i)T + (16.3 - 50.4i)T^{2}
59 1+(−1.70+5.23i)T+(−47.7−34.6i)T2 1 + (-1.70 + 5.23i)T + (-47.7 - 34.6i)T^{2}
61 1+(−7.70−5.59i)T+(18.8+58.0i)T2 1 + (-7.70 - 5.59i)T + (18.8 + 58.0i)T^{2}
67 1+12.8T+67T2 1 + 12.8T + 67T^{2}
71 1+(−3.08−2.24i)T+(21.9+67.5i)T2 1 + (-3.08 - 2.24i)T + (21.9 + 67.5i)T^{2}
73 1+(2.11−6.52i)T+(−59.0−42.9i)T2 1 + (2.11 - 6.52i)T + (-59.0 - 42.9i)T^{2}
79 1+(−8.87+6.44i)T+(24.4−75.1i)T2 1 + (-8.87 + 6.44i)T + (24.4 - 75.1i)T^{2}
83 1+(−0.136−0.0989i)T+(25.6+78.9i)T2 1 + (-0.136 - 0.0989i)T + (25.6 + 78.9i)T^{2}
89 1+1.97T+89T2 1 + 1.97T + 89T^{2}
97 1+(−0.0190+0.0138i)T+(29.9−92.2i)T2 1 + (-0.0190 + 0.0138i)T + (29.9 - 92.2i)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

−12.82701912707860302798441598162, −11.95194207340306658749266016817, −10.55529756326075435262975299750, −9.860146960597007733833334055602, −8.916108242829931740715091314693, −8.118639824527110591514646610299, −7.66471480565363960355127297392, −5.50366175334161531231603335943, −4.27705939552346878204024169958, −3.53237525374661053024848961624, 0.36430052517561302172726081913, 2.29030345221075216978373069044, 3.24601747575296114073070847422, 5.74188382817937322479465567258, 7.27685381709797315370975265312, 7.76594054998996381906054626785, 8.433539664110856089907361930907, 9.846840075920903888485957897785, 10.94358467143805940793968075795, 11.82432771271426749023855879819

Graph of the ZZ-function along the critical line