↑

Properties

Label 2-201-201.158-c0-0-0
Degree $2$
Conductor $201$
Sign $0.811 + 0.584i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.841 − 0.540i)3-s + (−0.654 − 0.755i)4-s + (−0.544 + 1.19i)7-s + (0.415 − 0.909i)9-s + (−0.959 − 0.281i)12-s + (−0.797 − 0.234i)13-s + (−0.142 + 0.989i)16-s + (0.698 + 1.53i)19-s + (0.186 + 1.29i)21-s + (−0.959 − 0.281i)25-s + (−0.142 − 0.989i)27-s + (1.25 − 0.368i)28-s + (0.273 − 0.0801i)31-s + (−0.959 + 0.281i)36-s − 1.91·37-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)3-s + (−0.654 − 0.755i)4-s + (−0.544 + 1.19i)7-s + (0.415 − 0.909i)9-s + (−0.959 − 0.281i)12-s + (−0.797 − 0.234i)13-s + (−0.142 + 0.989i)16-s + (0.698 + 1.53i)19-s + (0.186 + 1.29i)21-s + (−0.959 − 0.281i)25-s + (−0.142 − 0.989i)27-s + (1.25 − 0.368i)28-s + (0.273 − 0.0801i)31-s + (−0.959 + 0.281i)36-s − 1.91·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.811 + 0.584i$
Analytic conductor: \(0.100312\)
Root analytic conductor: \(0.316720\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :0),\ 0.811 + 0.584i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7437706953\)
\(L(\frac12)\) \(\approx\) \(0.7437706953\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (0.959 + 0.281i)T \)
good2 \( 1 + (0.654 + 0.755i)T^{2} \)
5 \( 1 + (0.959 + 0.281i)T^{2} \)
7 \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \)
11 \( 1 + (0.959 + 0.281i)T^{2} \)
13 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
17 \( 1 + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
23 \( 1 + (-0.415 + 0.909i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
37 \( 1 + 1.91T + T^{2} \)
41 \( 1 + (0.142 + 0.989i)T^{2} \)
43 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (-0.415 + 0.909i)T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (0.142 - 0.989i)T^{2} \)
73 \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \)
79 \( 1 + (-1.84 - 0.540i)T + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.959 + 0.281i)T^{2} \)
89 \( 1 + (-0.415 - 0.909i)T^{2} \)
97 \( 1 - 1.68T + T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.50664016261030240315213819473, −12.08811043558521405543568459023, −10.25684391880811394923001164313, −9.518091533839374753802678579190, −8.748964354752458884457020914744, −7.70192932962233064746346531502, −6.27081290457826050001988564834, −5.34087881942035109478968310011, −3.62113893298499218577679697460, −2.07000219317826501699000640031, 2.92504428679052955048683596281, 3.98953302729703698693914251767, 4.91295105019490803429707273156, 7.09869186400720539811729541316, 7.71043854129635949808234621352, 8.997955898834019969226381736490, 9.633924983554843862988903682081, 10.59282056517344995095577954704, 11.91551020452528617770171466420, 13.17010816475812628104990323503

Graph of the $Z$-function along the critical line