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Properties

Label 2-201-201.161-c3-0-53
Degree $2$
Conductor $201$
Sign $-0.832 + 0.554i$
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  + (−4.72 − 2.15i)3-s + (1.13 − 7.91i)4-s + (18.3 − 15.8i)7-s + (17.6 + 20.4i)9-s + (−22.4 + 34.9i)12-s + (50.6 − 78.8i)13-s + (−61.4 − 18.0i)16-s + (−77.6 + 89.5i)19-s + (−120. + 35.4i)21-s + (−105. − 67.5i)25-s + (−39.5 − 134. i)27-s + (−104. − 163. i)28-s + (−33.9 − 52.8i)31-s + (181. − 116. i)36-s − 328.·37-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)3-s + (0.142 − 0.989i)4-s + (0.988 − 0.856i)7-s + (0.654 + 0.755i)9-s + (−0.540 + 0.841i)12-s + (1.08 − 1.68i)13-s + (−0.959 − 0.281i)16-s + (−0.937 + 1.08i)19-s + (−1.25 + 0.368i)21-s + (−0.841 − 0.540i)25-s + (−0.281 − 0.959i)27-s + (−0.707 − 1.10i)28-s + (−0.196 − 0.306i)31-s + (0.841 − 0.540i)36-s − 1.45·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.354441 - 1.17082i\)
\(L(\frac12)\) \(\approx\) \(0.354441 - 1.17082i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.72 + 2.15i)T \)
67 \( 1 + (-547. + 37.5i)T \)
good2 \( 1 + (-1.13 + 7.91i)T^{2} \)
5 \( 1 + (105. + 67.5i)T^{2} \)
7 \( 1 + (-18.3 + 15.8i)T + (48.8 - 339. i)T^{2} \)
11 \( 1 + (1.11e3 + 719. i)T^{2} \)
13 \( 1 + (-50.6 + 78.8i)T + (-912. - 1.99e3i)T^{2} \)
17 \( 1 + (4.71e3 - 1.38e3i)T^{2} \)
19 \( 1 + (77.6 - 89.5i)T + (-976. - 6.78e3i)T^{2} \)
23 \( 1 + (7.96e3 + 9.19e3i)T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (33.9 + 52.8i)T + (-1.23e4 + 2.70e4i)T^{2} \)
37 \( 1 + 328.T + 5.06e4T^{2} \)
41 \( 1 + (-6.61e4 + 1.94e4i)T^{2} \)
43 \( 1 + (-459. + 66.1i)T + (7.62e4 - 2.23e4i)T^{2} \)
47 \( 1 + (6.79e4 + 7.84e4i)T^{2} \)
53 \( 1 + (-1.42e5 - 4.19e4i)T^{2} \)
59 \( 1 + (-8.53e4 + 1.86e5i)T^{2} \)
61 \( 1 + (-62.8 - 213. i)T + (-1.90e5 + 1.22e5i)T^{2} \)
71 \( 1 + (3.43e5 + 1.00e5i)T^{2} \)
73 \( 1 + (994. - 291. i)T + (3.27e5 - 2.10e5i)T^{2} \)
79 \( 1 + (-237. + 370. i)T + (-2.04e5 - 4.48e5i)T^{2} \)
83 \( 1 + (-4.81e5 - 3.09e5i)T^{2} \)
89 \( 1 + (4.61e5 - 5.32e5i)T^{2} \)
97 \( 1 - 639. iT - 9.12e5T^{2} \)
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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

−11.33747742370877721705367541445, −10.60351871005723210748626838704, −10.22609268949381566334100885267, −8.356454303546839728683618757254, −7.43193435393827626968868922750, −6.12368801586794591429143433124, −5.45992632878194604516760722191, −4.18463839331450860159980128794, −1.71841465154480833360097771709, −0.61066264076637889210987216359, 1.94110494646768899267088129419, 3.86931022046184290435224256113, 4.81892811887775113817718187418, 6.17246704720988616113106710972, 7.17188177470014025358965638044, 8.611952210751670723728903325568, 9.169461542568916348873808612573, 10.90926119070661149382677357664, 11.45187888057720080376803246022, 12.07608294332857809712403239020

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