Properties

Label 2-201-67.29-c3-0-12
Degree $2$
Conductor $201$
Sign $0.762 - 0.646i$
Analytic cond. $11.8593$
Root an. cond. $3.44374$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.55 + 2.69i)2-s − 3·3-s + (−0.854 − 1.47i)4-s − 13.0·5-s + (4.67 − 8.09i)6-s + (−12.0 − 20.7i)7-s − 19.6·8-s + 9·9-s + (20.3 − 35.3i)10-s + (18.0 + 31.2i)11-s + (2.56 + 4.43i)12-s + (5.53 − 9.59i)13-s + 74.8·14-s + 39.2·15-s + (37.3 − 64.7i)16-s + (−10.8 + 18.7i)17-s + ⋯
L(s)  = 1  + (−0.550 + 0.954i)2-s − 0.577·3-s + (−0.106 − 0.184i)4-s − 1.17·5-s + (0.318 − 0.550i)6-s + (−0.648 − 1.12i)7-s − 0.866·8-s + 0.333·9-s + (0.644 − 1.11i)10-s + (0.494 + 0.855i)11-s + (0.0616 + 0.106i)12-s + (0.118 − 0.204i)13-s + 1.42·14-s + 0.675·15-s + (0.583 − 1.01i)16-s + (−0.154 + 0.267i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.762 - 0.646i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ 0.762 - 0.646i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.561930 + 0.206180i\)
\(L(\frac12)\) \(\approx\) \(0.561930 + 0.206180i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
67 \( 1 + (-130. + 532. i)T \)
good2 \( 1 + (1.55 - 2.69i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 13.0T + 125T^{2} \)
7 \( 1 + (12.0 + 20.7i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-18.0 - 31.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-5.53 + 9.59i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (10.8 - 18.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-0.126 + 0.218i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-41.8 + 72.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-77.2 - 133. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-96.9 - 167. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-138. + 239. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-20.2 - 35.0i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 4.82T + 7.95e4T^{2} \)
47 \( 1 + (-152. - 264. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 38.6T + 1.48e5T^{2} \)
59 \( 1 - 193.T + 2.05e5T^{2} \)
61 \( 1 + (-244. + 424. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (107. + 185. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-107. + 186. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (165. + 286. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-696. + 1.20e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 359.T + 7.04e5T^{2} \)
97 \( 1 + (-387. + 671. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.18206672743113124556221928470, −11.07792251756340203486590369074, −10.09493972672888893416629193741, −8.906632748441099561576356930580, −7.74007784014238032657150046054, −7.07725042271037524510191994931, −6.34567398318537377710569642188, −4.60353406733077330958669714403, −3.47631181863551220353949233102, −0.55126659601550784855340089877, 0.75179010202665934388126339916, 2.67148934670230282318699132243, 3.89516277261358924724097531793, 5.65739933472998501795940534365, 6.59397577542726428667877824435, 8.180010107459269850117680477698, 9.104421923419192405583671399377, 9.989969594526524088604328912699, 11.29663933488334426018484591518, 11.63999368361887866357429386502

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