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Properties

Label 2-18e2-36.11-c1-0-15
Degree $2$
Conductor $324$
Sign $0.819 - 0.573i$
Analytic cond. $2.58715$
Root an. cond. $1.60846$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (2.56 − 1.48i)5-s + 2.82i·8-s + 4.19·10-s + (−3.59 − 6.23i)13-s + (−2.00 + 3.46i)16-s + 7.20i·17-s + (5.13 + 2.96i)20-s + (1.90 − 3.29i)25-s − 10.1i·26-s + (1.10 + 0.637i)29-s + (−4.89 + 2.82i)32-s + (−5.09 + 8.83i)34-s − 11.3·37-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.499 + 0.866i)4-s + (1.14 − 0.663i)5-s + 0.999i·8-s + 1.32·10-s + (−0.997 − 1.72i)13-s + (−0.500 + 0.866i)16-s + 1.74i·17-s + (1.14 + 0.663i)20-s + (0.380 − 0.658i)25-s − 1.99i·26-s + (0.205 + 0.118i)29-s + (−0.866 + 0.499i)32-s + (−0.874 + 1.51i)34-s − 1.87·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.819 - 0.573i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.819 - 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27392 + 0.716965i\)
\(L(\frac12)\) \(\approx\) \(2.27392 + 0.716965i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
good5 \( 1 + (-2.56 + 1.48i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.59 + 6.23i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.20iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.10 - 0.637i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 + (-1.22 + 0.707i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.69 + 4.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.51iT - 89T^{2} \)
97 \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.18740364948102929643183318407, −10.70312910081430495984540364210, −9.965303123201932628954055421868, −8.651552014042071869377642897551, −7.88011707176696283945485912823, −6.58602841442879738343365667808, −5.58006016761676148662678916284, −5.03982505553591932495899511758, −3.49743207878092520574205188303, −2.04050369705806117574202086766, 1.94901573889275965182356239732, 2.88063652577383678518508707843, 4.48379095323540737101875717723, 5.42808694219137885982398363466, 6.61889981815513194980493633995, 7.15109358050425585223436216432, 9.268782588897504628903441962750, 9.720767442439953758021666152600, 10.70309379306097920714048740355, 11.65374520226725782582925023502

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