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Properties

Label 2-201-201.104-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: $\chi_{201} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :0),\ 0.586 + 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9148019240\)
\(L(\frac12)\) \(\approx\) \(0.9148019240\)
\(L(1)\) 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

−12.52850858340813686363197555039, −12.03619230109981386841385096944, −11.03612746028804515040337747260, −9.631241605773465811531850408459, −8.244543117062685754216292349168, −7.63639696742692720184301075209, −5.86237701988828560970330625284, −5.35801835167399409429081676615, −3.33240206634039053429583645498, −2.45801410805512419283023089293, 3.25837212886460724700325050683, 4.43329433667706348483214198311, 5.16449676002361227003296168042, 6.44965633872900794959990974483, 7.50380806710069990355949851386, 9.105257492640903248615743272686, 10.06209924086214089659810549732, 10.58392425370295767773492326349, 12.06300494364004050974075833294, 13.06396372837765419088243433711

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