Properties

Label 2-201-201.143-c0-0-0
Degree $2$
Conductor $201$
Sign $0.922 - 0.385i$
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  + (−0.959 − 0.281i)3-s + (0.415 + 0.909i)4-s + (0.698 + 0.449i)7-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)12-s + (−0.239 − 1.66i)13-s + (−0.654 + 0.755i)16-s + (−1.61 + 1.03i)19-s + (−0.544 − 0.627i)21-s + (−0.142 − 0.989i)25-s + (−0.654 − 0.755i)27-s + (−0.118 + 0.822i)28-s + (0.186 − 1.29i)31-s + (−0.142 + 0.989i)36-s − 0.284·37-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)3-s + (0.415 + 0.909i)4-s + (0.698 + 0.449i)7-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)12-s + (−0.239 − 1.66i)13-s + (−0.654 + 0.755i)16-s + (−1.61 + 1.03i)19-s + (−0.544 − 0.627i)21-s + (−0.142 − 0.989i)25-s + (−0.654 − 0.755i)27-s + (−0.118 + 0.822i)28-s + (0.186 − 1.29i)31-s + (−0.142 + 0.989i)36-s − 0.284·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.922 - 0.385i$
Analytic conductor: \(0.100312\)
Root analytic conductor: \(0.316720\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :0),\ 0.922 - 0.385i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6058731517\)
\(L(\frac12)\) \(\approx\) \(0.6058731517\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
good2 \( 1 + (-0.415 - 0.909i)T^{2} \)
5 \( 1 + (0.142 + 0.989i)T^{2} \)
7 \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \)
11 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
23 \( 1 + (-0.841 - 0.540i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + 0.284T + T^{2} \)
41 \( 1 + (0.654 + 0.755i)T^{2} \)
43 \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (-0.841 - 0.540i)T^{2} \)
53 \( 1 + (0.654 - 0.755i)T^{2} \)
59 \( 1 + (0.959 + 0.281i)T^{2} \)
61 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (0.654 - 0.755i)T^{2} \)
73 \( 1 + (-0.857 - 0.989i)T + (-0.142 + 0.989i)T^{2} \)
79 \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \)
83 \( 1 + (0.142 + 0.989i)T^{2} \)
89 \( 1 + (-0.841 + 0.540i)T^{2} \)
97 \( 1 + 1.91T + T^{2} \)
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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

−12.55640639689542369176864540021, −11.91060328658130514310492137473, −10.95625719586983058202825656582, −10.16714868730782212709426247814, −8.310512894255061027012426427526, −7.84053137072750613844789626342, −6.50757408678597100604034450616, −5.50950457490169342739397703299, −4.14287885964806557761753126274, −2.29956614588298711166077894075, 1.74606183655345839181257615387, 4.33583321075802183371986404210, 5.13560550894647523254990130729, 6.50042205016937012463619097690, 7.07038311488843842856600957351, 8.900441216588631705890431366014, 9.904453240060501194748925808107, 10.99765077691192919185943307672, 11.24883087285052313886764350612, 12.36837185488386301143517757923

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