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Properties

Label 2-201-201.89-c0-0-0
Degree $2$
Conductor $201$
Sign $0.604 + 0.796i$
Analytic cond. $0.100312$
Root an. cond. $0.316720$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.604 + 0.796i$
Analytic conductor: \(0.100312\)
Root analytic conductor: \(0.316720\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :0),\ 0.604 + 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6981391720\)
\(L(\frac12)\) \(\approx\) \(0.6981391720\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
good2 \( 1 + (-0.841 + 0.540i)T^{2} \)
5 \( 1 + (0.654 + 0.755i)T^{2} \)
7 \( 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2} \)
11 \( 1 + (0.654 + 0.755i)T^{2} \)
13 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
17 \( 1 + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
23 \( 1 + (0.959 - 0.281i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \)
37 \( 1 + 1.30T + T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 + (0.959 - 0.281i)T^{2} \)
53 \( 1 + (-0.415 + 0.909i)T^{2} \)
59 \( 1 + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.415 + 0.909i)T^{2} \)
73 \( 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2} \)
79 \( 1 + (-0.857 - 0.989i)T + (-0.142 + 0.989i)T^{2} \)
83 \( 1 + (0.654 + 0.755i)T^{2} \)
89 \( 1 + (0.959 + 0.281i)T^{2} \)
97 \( 1 + 0.284T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line