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Properties

Label 2-201-201.29-c0-0-1
Degree $2$
Conductor $201$
Sign $-0.586 + 0.809i$
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  + (−0.866 − 0.5i)2-s − i·3-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + i·8-s − 9-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)13-s + 0.999i·14-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.866 + 0.5i)18-s + (0.5 − 0.866i)19-s + (−0.866 + 0.5i)21-s − 0.999·22-s + (0.866 + 0.5i)23-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s − i·3-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + i·8-s − 9-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)13-s + 0.999i·14-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.866 + 0.5i)18-s + (0.5 − 0.866i)19-s + (−0.866 + 0.5i)21-s − 0.999·22-s + (0.866 + 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.586 + 0.809i$
Analytic conductor: \(0.100312\)
Root analytic conductor: \(0.316720\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :0),\ -0.586 + 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4163892498\)
\(L(\frac12)\) \(\approx\) \(0.4163892498\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
67 \( 1 + T \)
good2 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−12.00635294644551483926887931755, −11.41071828953632805737585955614, −10.41683092778607604561645014316, −9.216889091870554163749756343538, −8.656001310041740246012097114734, −7.18369849732048065400585758462, −6.56762273525719702520866453237, −4.86565227648055727757135798497, −2.86926749676441456327653966422, −1.15003496418143021220767078789, 3.06862958808679531741982289930, 4.43400830213849649691932529700, 5.86192943407996183961397074661, 7.00874152878573842366442728339, 8.419824572179928034669012973985, 9.028142894516585149340121942550, 9.829723607099874443952264537757, 10.68161793487289808136445453498, 12.11494600182646605770811471076, 12.77128818781416612137199332616

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