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Properties

Label 2-201-201.89-c0-0-0
Degree 22
Conductor 201201
Sign 0.604+0.796i0.604 + 0.796i
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  + (−0.142 − 0.989i)3-s + (0.841 − 0.540i)4-s + (−1.61 + 0.474i)7-s + (−0.959 + 0.281i)9-s + (−0.654 − 0.755i)12-s + (1.25 + 1.45i)13-s + (0.415 − 0.909i)16-s + (0.273 + 0.0801i)19-s + (0.698 + 1.53i)21-s + (−0.654 − 0.755i)25-s + (0.415 + 0.909i)27-s + (−1.10 + 1.27i)28-s + (−0.544 + 0.627i)31-s + (−0.654 + 0.755i)36-s − 1.30·37-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)3-s + (0.841 − 0.540i)4-s + (−1.61 + 0.474i)7-s + (−0.959 + 0.281i)9-s + (−0.654 − 0.755i)12-s + (1.25 + 1.45i)13-s + (0.415 − 0.909i)16-s + (0.273 + 0.0801i)19-s + (0.698 + 1.53i)21-s + (−0.654 − 0.755i)25-s + (0.415 + 0.909i)27-s + (−1.10 + 1.27i)28-s + (−0.544 + 0.627i)31-s + (−0.654 + 0.755i)36-s − 1.30·37-s + ⋯

Functional equation

Λ(s)=(201s/2 ΓC(s) L(s)=((0.604+0.796i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(201s/2 ΓC(s) L(s)=((0.604+0.796i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 201201    =    3⋅673 \cdot 67
Sign: 0.604+0.796i0.604 + 0.796i
Analytic conductor: 0.1003120.100312
Root analytic conductor: 0.3167200.316720
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ201(89,⋅)\chi_{201} (89, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 201, ( :0), 0.604+0.796i)(2,\ 201,\ (\ :0),\ 0.604 + 0.796i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.69813917200.6981391720
L(12)L(\frac12) ≈\approx 0.69813917200.6981391720
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+(0.142+0.989i)T 1 + (0.142 + 0.989i)T
67 1+(0.654+0.755i)T 1 + (0.654 + 0.755i)T
good2 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
5 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
7 1+(1.61−0.474i)T+(0.841−0.540i)T2 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2}
11 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
13 1+(−1.25−1.45i)T+(−0.142+0.989i)T2 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2}
17 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
19 1+(−0.273−0.0801i)T+(0.841+0.540i)T2 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2}
23 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
29 1−T2 1 - T^{2}
31 1+(0.544−0.627i)T+(−0.142−0.989i)T2 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2}
37 1+1.30T+T2 1 + 1.30T + T^{2}
41 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
43 1+(1.10+0.708i)T+(0.415+0.909i)T2 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2}
47 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
53 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
59 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
61 1+(0.118+0.258i)T+(−0.654+0.755i)T2 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2}
71 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
73 1+(−0.345−0.755i)T+(−0.654+0.755i)T2 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2}
79 1+(−0.857−0.989i)T+(−0.142+0.989i)T2 1 + (-0.857 - 0.989i)T + (-0.142 + 0.989i)T^{2}
83 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
89 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
97 1+0.284T+T2 1 + 0.284T + 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

−12.37316789988311214094465472473, −11.77742852179887133279016578852, −10.75911629619471249498339384613, −9.584435066882081977514518694399, −8.568699543251315869326137889269, −6.98180936821416719743331092434, −6.49447491935548997873910217474, −5.67468369759030647901819692801, −3.34961119824483874849099281838, −1.90019167931578965433003940820, 3.17567798050577701878043735311, 3.66784614225372011237558485805, 5.66117966350476714326642956885, 6.50703595247778087716436363889, 7.79182293091384234891211102079, 9.005040470679977833814445365941, 10.12255699316218524143192195233, 10.72573052314612602483091281341, 11.72283761235807029896309806241, 12.87256373979270582427048921721

Graph of the ZZ-function along the critical line