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Properties

Label 2-201-201.47-c2-0-41
Degree $2$
Conductor $201$
Sign $-0.837 + 0.546i$
Analytic cond. $5.47685$
Root an. cond. $2.34026$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.426 − 2.96i)3-s + (0.190 − 3.99i)4-s + (−1.36 − 5.64i)7-s + (−8.63 − 2.53i)9-s + (−11.7 − 2.27i)12-s + (6.10 + 17.6i)13-s + (−15.9 − 1.52i)16-s + (5.59 − 23.0i)19-s + (−17.3 + 1.65i)21-s + (−16.3 + 18.8i)25-s + (−11.2 + 24.5i)27-s + (−22.8 + 4.39i)28-s + (19.8 − 57.2i)31-s + (−11.7 + 34.0i)36-s + (34.6 − 60.0i)37-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)3-s + (0.0475 − 0.998i)4-s + (−0.195 − 0.806i)7-s + (−0.959 − 0.281i)9-s + (−0.981 − 0.189i)12-s + (0.469 + 1.35i)13-s + (−0.995 − 0.0950i)16-s + (0.294 − 1.21i)19-s + (−0.826 + 0.0788i)21-s + (−0.654 + 0.755i)25-s + (−0.415 + 0.909i)27-s + (−0.814 + 0.157i)28-s + (0.638 − 1.84i)31-s + (−0.327 + 0.945i)36-s + (0.937 − 1.62i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.837 + 0.546i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.837 + 0.546i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.381088 - 1.28209i\)
\(L(\frac12)\) \(\approx\) \(0.381088 - 1.28209i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.426 + 2.96i)T \)
67 \( 1 + (-65.1 + 15.6i)T \)
good2 \( 1 + (-0.190 + 3.99i)T^{2} \)
5 \( 1 + (16.3 - 18.8i)T^{2} \)
7 \( 1 + (1.36 + 5.64i)T + (-43.5 + 22.4i)T^{2} \)
11 \( 1 + (39.5 + 114. i)T^{2} \)
13 \( 1 + (-6.10 - 17.6i)T + (-132. + 104. i)T^{2} \)
17 \( 1 + (287. - 27.4i)T^{2} \)
19 \( 1 + (-5.59 + 23.0i)T + (-320. - 165. i)T^{2} \)
23 \( 1 + (-124. - 514. i)T^{2} \)
29 \( 1 + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-19.8 + 57.2i)T + (-755. - 594. i)T^{2} \)
37 \( 1 + (-34.6 + 60.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-975. - 1.36e3i)T^{2} \)
43 \( 1 + (4.93 - 3.17i)T + (768. - 1.68e3i)T^{2} \)
47 \( 1 + (-1.59e3 + 1.52e3i)T^{2} \)
53 \( 1 + (-1.16e3 - 2.55e3i)T^{2} \)
59 \( 1 + (495. - 3.44e3i)T^{2} \)
61 \( 1 + (-60.7 - 85.3i)T + (-1.21e3 + 3.51e3i)T^{2} \)
71 \( 1 + (5.01e3 + 479. i)T^{2} \)
73 \( 1 + (-18.9 - 26.5i)T + (-1.74e3 + 5.03e3i)T^{2} \)
79 \( 1 + (18.6 + 3.60i)T + (5.79e3 + 2.31e3i)T^{2} \)
83 \( 1 + (-6.76e3 - 1.30e3i)T^{2} \)
89 \( 1 + (7.60e3 - 2.23e3i)T^{2} \)
97 \( 1 + (-60.7 + 105. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.55801815182528819486801723193, −11.14776961718475790108993934542, −9.734651701317561470658531778977, −8.947282476921696836853166214074, −7.46744059127484833443909894848, −6.69995774886661222151742449522, −5.73380831625334087610632466709, −4.16577169358923502382120508390, −2.19785984475897998393334362312, −0.75970266555753019492444962997, 2.78898676173494927531726530102, 3.67999644965050053260566705030, 5.09348660371716021086514081231, 6.25392300310006787655425932730, 8.039743877978582992501414331331, 8.461667167868832472493206125629, 9.683240312916735887782027742142, 10.58112940995921287607910334547, 11.75563283796124999779025985917, 12.44058007007412681422311494352

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