Properties

Label 2-201-201.92-c0-0-0
Degree $2$
Conductor $201$
Sign $0.0231 - 0.999i$
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  + (−0.654 + 0.755i)3-s + (−0.959 + 0.281i)4-s + (0.273 + 1.89i)7-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)12-s + (−0.118 + 0.258i)13-s + (0.841 − 0.540i)16-s + (0.186 − 1.29i)19-s + (−1.61 − 1.03i)21-s + (0.415 − 0.909i)25-s + (0.841 + 0.540i)27-s + (−0.797 − 1.74i)28-s + (0.698 + 1.53i)31-s + (0.415 + 0.909i)36-s + 0.830·37-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)3-s + (−0.959 + 0.281i)4-s + (0.273 + 1.89i)7-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)12-s + (−0.118 + 0.258i)13-s + (0.841 − 0.540i)16-s + (0.186 − 1.29i)19-s + (−1.61 − 1.03i)21-s + (0.415 − 0.909i)25-s + (0.841 + 0.540i)27-s + (−0.797 − 1.74i)28-s + (0.698 + 1.53i)31-s + (0.415 + 0.909i)36-s + 0.830·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0231 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0231 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.0231 - 0.999i$
Analytic conductor: \(0.100312\)
Root analytic conductor: \(0.316720\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :0),\ 0.0231 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4854696331\)
\(L(\frac12)\) \(\approx\) \(0.4854696331\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.654 - 0.755i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
good2 \( 1 + (0.959 - 0.281i)T^{2} \)
5 \( 1 + (-0.415 + 0.909i)T^{2} \)
7 \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \)
11 \( 1 + (-0.415 + 0.909i)T^{2} \)
13 \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \)
17 \( 1 + (-0.841 - 0.540i)T^{2} \)
19 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
23 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
37 \( 1 - 0.830T + T^{2} \)
41 \( 1 + (-0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
47 \( 1 + (0.142 + 0.989i)T^{2} \)
53 \( 1 + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (0.654 - 0.755i)T^{2} \)
61 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (-0.841 + 0.540i)T^{2} \)
73 \( 1 + (-1.41 - 0.909i)T + (0.415 + 0.909i)T^{2} \)
79 \( 1 + (-0.345 + 0.755i)T + (-0.654 - 0.755i)T^{2} \)
83 \( 1 + (-0.415 + 0.909i)T^{2} \)
89 \( 1 + (0.142 - 0.989i)T^{2} \)
97 \( 1 + 1.30T + T^{2} \)
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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

−12.56454392178166689227560628310, −12.07019487061140072682560788389, −11.09821257820483302099802542808, −9.772470639335758558595474273014, −9.030112322810735742308218180929, −8.366334632644317498082118206967, −6.46649899546939721231958118624, −5.26967697284962259688452034107, −4.65588446674150082276570924847, −2.96749018349298029463062836871, 1.14623430820223783389319663343, 3.89623354500837733503281158679, 4.96842016623834166123314345611, 6.20322432546940861793431699772, 7.48134284303205171297777550253, 8.103434766751510728251708567167, 9.773069753235315718726476912626, 10.49383983029239418321288854902, 11.40336053937425555435787078790, 12.69117910975437728205348444729

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