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Properties

Label 2-201-201.62-c0-0-0
Degree 22
Conductor 201201
Sign 0.977−0.209i0.977 - 0.209i
Analytic cond. 0.1003120.100312
Root an. cond. 0.3167200.316720
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  + (0.415 + 0.909i)3-s + (−0.142 − 0.989i)4-s + (0.186 − 0.215i)7-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)12-s + (−1.10 + 0.708i)13-s + (−0.959 + 0.281i)16-s + (−0.544 − 0.627i)19-s + (0.273 + 0.0801i)21-s + (0.841 − 0.540i)25-s + (−0.959 − 0.281i)27-s + (−0.239 − 0.153i)28-s + (−1.61 − 1.03i)31-s + (0.841 + 0.540i)36-s + 1.68·37-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)3-s + (−0.142 − 0.989i)4-s + (0.186 − 0.215i)7-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)12-s + (−1.10 + 0.708i)13-s + (−0.959 + 0.281i)16-s + (−0.544 − 0.627i)19-s + (0.273 + 0.0801i)21-s + (0.841 − 0.540i)25-s + (−0.959 − 0.281i)27-s + (−0.239 − 0.153i)28-s + (−1.61 − 1.03i)31-s + (0.841 + 0.540i)36-s + 1.68·37-s + ⋯

Functional equation

Λ(s)=(201s/2 ΓC(s) L(s)=((0.977−0.209i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(201s/2 ΓC(s) L(s)=((0.977−0.209i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 201201    =    3⋅673 \cdot 67
Sign: 0.977−0.209i0.977 - 0.209i
Analytic conductor: 0.1003120.100312
Root analytic conductor: 0.3167200.316720
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ201(62,⋅)\chi_{201} (62, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 201, ( :0), 0.977−0.209i)(2,\ 201,\ (\ :0),\ 0.977 - 0.209i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.73469346100.7346934610
L(12)L(\frac12) ≈\approx 0.73469346100.7346934610
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)
bad3 1+(−0.415−0.909i)T 1 + (-0.415 - 0.909i)T
67 1+(−0.841+0.540i)T 1 + (-0.841 + 0.540i)T
good2 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
5 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
7 1+(−0.186+0.215i)T+(−0.142−0.989i)T2 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2}
11 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
13 1+(1.10−0.708i)T+(0.415−0.909i)T2 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2}
17 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
19 1+(0.544+0.627i)T+(−0.142+0.989i)T2 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2}
23 1+(0.654−0.755i)T2 1 + (0.654 - 0.755i)T^{2}
29 1−T2 1 - T^{2}
31 1+(1.61+1.03i)T+(0.415+0.909i)T2 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2}
37 1−1.68T+T2 1 - 1.68T + T^{2}
41 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
43 1+(0.239−1.66i)T+(−0.959−0.281i)T2 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2}
47 1+(0.654−0.755i)T2 1 + (0.654 - 0.755i)T^{2}
53 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
59 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
61 1+(0.797+0.234i)T+(0.841+0.540i)T2 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2}
71 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
73 1+(−1.84−0.540i)T+(0.841+0.540i)T2 1 + (-1.84 - 0.540i)T + (0.841 + 0.540i)T^{2}
79 1+(−1.41+0.909i)T+(0.415−0.909i)T2 1 + (-1.41 + 0.909i)T + (0.415 - 0.909i)T^{2}
83 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
89 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
97 1−0.830T+T2 1 - 0.830T + 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

−12.87437398212227162990822184890, −11.34197574747998225003595613383, −10.73657278335023600728076315970, −9.618862533437738950827584256688, −9.188811016894397750878454323383, −7.79231444043136811084452426080, −6.38800713140840286698874402626, −5.04279795599805844043680312393, −4.29029707440931201342438965251, −2.38980598041041203657329163154, 2.35399982463227788155007059926, 3.58709128776213239155855266998, 5.27707622151570124240862432831, 6.82981130205455755479149753733, 7.65174021993838534597512107313, 8.461067397530067806453715320392, 9.397307942538483648845155435368, 10.90525113928465373630043900151, 12.12878107979351982848278764513, 12.57189597796151782298207190947

Graph of the ZZ-function along the critical line