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Properties

Label 2-18e2-12.11-c1-0-0
Degree $2$
Conductor $324$
Sign $-0.0669 - 0.997i$
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  + (−1.03 − 0.965i)2-s + (0.133 + 1.99i)4-s + 0.517i·5-s + 2.92i·7-s + (1.78 − 2.19i)8-s + (0.499 − 0.534i)10-s − 5.64·11-s − 4.46·13-s + (2.82 − 3.01i)14-s + (−3.96 + 0.534i)16-s + 2.31i·17-s + 5.06i·19-s + (−1.03 + 0.0693i)20-s + (5.83 + 5.45i)22-s − 1.51·23-s + ⋯
L(s)  = 1  + (−0.730 − 0.683i)2-s + (0.0669 + 0.997i)4-s + 0.231i·5-s + 1.10i·7-s + (0.632 − 0.774i)8-s + (0.158 − 0.169i)10-s − 1.70·11-s − 1.23·13-s + (0.754 − 0.806i)14-s + (−0.991 + 0.133i)16-s + 0.560i·17-s + 1.16i·19-s + (−0.230 + 0.0155i)20-s + (1.24 + 1.16i)22-s − 0.315·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0669 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0669 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.0669 - 0.997i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ -0.0669 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.345872 + 0.369872i\)
\(L(\frac12)\) \(\approx\) \(0.345872 + 0.369872i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.03 + 0.965i)T \)
3 \( 1 \)
good5 \( 1 - 0.517iT - 5T^{2} \)
7 \( 1 - 2.92iT - 7T^{2} \)
11 \( 1 + 5.64T + 11T^{2} \)
13 \( 1 + 4.46T + 13T^{2} \)
17 \( 1 - 2.31iT - 17T^{2} \)
19 \( 1 - 5.06iT - 19T^{2} \)
23 \( 1 + 1.51T + 23T^{2} \)
29 \( 1 - 4.76iT - 29T^{2} \)
31 \( 1 + 7.98iT - 31T^{2} \)
37 \( 1 - 0.267T + 37T^{2} \)
41 \( 1 - 8.10iT - 41T^{2} \)
43 \( 1 - 2.92iT - 43T^{2} \)
47 \( 1 - 4.13T + 47T^{2} \)
53 \( 1 - 4.52iT - 53T^{2} \)
59 \( 1 + 4.13T + 59T^{2} \)
61 \( 1 + 2.26T + 61T^{2} \)
67 \( 1 + 2.92iT - 67T^{2} \)
71 \( 1 + 9.77T + 71T^{2} \)
73 \( 1 - 4.66T + 73T^{2} \)
79 \( 1 + 10.9iT - 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 + 13.5iT - 89T^{2} \)
97 \( 1 + 0.535T + 97T^{2} \)
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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

−11.87101486904054599990572472790, −10.72852214348896214362636489690, −10.11119395631389131624095989621, −9.171671314616952065309950926218, −8.112544212776657160605112288014, −7.50875653517560697198781500464, −5.97851360003165703883279122921, −4.76305405798207977857441001278, −3.04192445635136276950692817894, −2.17603027309688124409642496843, 0.42863723013195417110561148250, 2.55143431722386095096380326296, 4.66842547333559660448373565010, 5.34571813830144037488844771158, 6.97313773017316464398301157234, 7.42062090429542895728843522513, 8.413684177201404706868402914623, 9.517648844829063785226639905226, 10.38840864995902476802014870832, 10.89722768832016996692969071143

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