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Properties

Label 2-209-209.75-c0-0-0
Degree 22
Conductor 209209
Sign 0.624+0.781i0.624 + 0.781i
Analytic cond. 0.1043040.104304
Root an. cond. 0.3229620.322962
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)4-s + (−0.5 − 1.53i)5-s + (1.30 + 0.951i)7-s + (0.309 − 0.951i)9-s + (−0.809 + 0.587i)11-s + (0.309 + 0.951i)16-s + (0.190 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (−0.5 + 1.53i)20-s + 0.618·23-s + (−1.30 + 0.951i)25-s + (−0.5 − 1.53i)28-s + (0.809 − 2.48i)35-s + (−0.809 + 0.587i)36-s + 0.618·43-s + 44-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)4-s + (−0.5 − 1.53i)5-s + (1.30 + 0.951i)7-s + (0.309 − 0.951i)9-s + (−0.809 + 0.587i)11-s + (0.309 + 0.951i)16-s + (0.190 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (−0.5 + 1.53i)20-s + 0.618·23-s + (−1.30 + 0.951i)25-s + (−0.5 − 1.53i)28-s + (0.809 − 2.48i)35-s + (−0.809 + 0.587i)36-s + 0.618·43-s + 44-s + ⋯

Functional equation

Λ(s)=(209s/2 ΓC(s) L(s)=((0.624+0.781i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(209s/2 ΓC(s) L(s)=((0.624+0.781i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 209209    =    11⋅1911 \cdot 19
Sign: 0.624+0.781i0.624 + 0.781i
Analytic conductor: 0.1043040.104304
Root analytic conductor: 0.3229620.322962
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ209(75,⋅)\chi_{209} (75, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 209, ( :0), 0.624+0.781i)(2,\ 209,\ (\ :0),\ 0.624 + 0.781i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.61963695100.6196369510
L(12)L(\frac12) ≈\approx 0.61963695100.6196369510
L(1)L(1) 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+(0.809−0.587i)T 1 + (0.809 - 0.587i)T
19 1+(0.809−0.587i)T 1 + (0.809 - 0.587i)T
good2 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
3 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
5 1+(0.5+1.53i)T+(−0.809+0.587i)T2 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2}
7 1+(−1.30−0.951i)T+(0.309+0.951i)T2 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2}
13 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
17 1+(−0.190−0.587i)T+(−0.809+0.587i)T2 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2}
23 1−0.618T+T2 1 - 0.618T + T^{2}
29 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
31 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
37 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
41 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
43 1−0.618T+T2 1 - 0.618T + T^{2}
47 1+(0.5−0.363i)T+(0.309−0.951i)T2 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2}
53 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
59 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
61 1+(−0.190−0.587i)T+(−0.809+0.587i)T2 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2}
67 1−T2 1 - T^{2}
71 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
73 1+(1.61+1.17i)T+(0.309+0.951i)T2 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2}
79 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
83 1+(0.5+1.53i)T+(−0.809+0.587i)T2 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)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

−12.55754108654927149206977030947, −11.77736733771123291824211224891, −10.44142081653990499334977380550, −9.234855821297590621975699829300, −8.631742236786995445683776810082, −7.87172071632455498761393167848, −5.84348882312015082626460189333, −4.94750417540238446597597982055, −4.23577991288248367378047185419, −1.52643915882307964783643299983, 2.76467273440989927742781032917, 4.12048781793688963050590855008, 5.11005384139615399801188318971, 7.08655780880078085318463662277, 7.69170128316693257239225794661, 8.441948033315951678875244942710, 10.14912427253826794641132597735, 10.92806231826180472754176426133, 11.40960441121386643588272252369, 12.97876538169174020162334207275

Graph of the ZZ-function along the critical line