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Properties

Label 2-18e2-4.3-c4-0-87
Degree $2$
Conductor $324$
Sign $-0.925 + 0.378i$
Analytic cond. $33.4918$
Root an. cond. $5.78721$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.92 − 0.772i)2-s + (14.8 − 6.06i)4-s − 21.1·5-s − 44.6i·7-s + (53.4 − 35.2i)8-s + (−83.0 + 16.3i)10-s − 67.7i·11-s − 29.1·13-s + (−34.4 − 175. i)14-s + (182. − 179. i)16-s − 402.·17-s + 644. i·19-s + (−313. + 128. i)20-s + (−52.3 − 265. i)22-s − 387. i·23-s + ⋯
L(s)  = 1  + (0.981 − 0.193i)2-s + (0.925 − 0.378i)4-s − 0.846·5-s − 0.910i·7-s + (0.834 − 0.550i)8-s + (−0.830 + 0.163i)10-s − 0.560i·11-s − 0.172·13-s + (−0.175 − 0.893i)14-s + (0.712 − 0.701i)16-s − 1.39·17-s + 1.78i·19-s + (−0.782 + 0.320i)20-s + (−0.108 − 0.549i)22-s − 0.732i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.925 + 0.378i$
Analytic conductor: \(33.4918\)
Root analytic conductor: \(5.78721\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :2),\ -0.925 + 0.378i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.697475418\)
\(L(\frac12)\) \(\approx\) \(1.697475418\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.92 + 0.772i)T \)
3 \( 1 \)
good5 \( 1 + 21.1T + 625T^{2} \)
7 \( 1 + 44.6iT - 2.40e3T^{2} \)
11 \( 1 + 67.7iT - 1.46e4T^{2} \)
13 \( 1 + 29.1T + 2.85e4T^{2} \)
17 \( 1 + 402.T + 8.35e4T^{2} \)
19 \( 1 - 644. iT - 1.30e5T^{2} \)
23 \( 1 + 387. iT - 2.79e5T^{2} \)
29 \( 1 + 724.T + 7.07e5T^{2} \)
31 \( 1 + 1.25e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.40e3T + 1.87e6T^{2} \)
41 \( 1 + 1.54e3T + 2.82e6T^{2} \)
43 \( 1 + 1.87e3iT - 3.41e6T^{2} \)
47 \( 1 + 4.16e3iT - 4.87e6T^{2} \)
53 \( 1 + 906.T + 7.89e6T^{2} \)
59 \( 1 - 4.52e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.62e3T + 1.38e7T^{2} \)
67 \( 1 + 67.7iT - 2.01e7T^{2} \)
71 \( 1 + 1.31e3iT - 2.54e7T^{2} \)
73 \( 1 - 9.47e3T + 2.83e7T^{2} \)
79 \( 1 - 4.36e3iT - 3.89e7T^{2} \)
83 \( 1 + 762. iT - 4.74e7T^{2} \)
89 \( 1 + 8.08e3T + 6.27e7T^{2} \)
97 \( 1 - 6.66e3T + 8.85e7T^{2} \)
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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

−10.79244651287169930403500708305, −10.04189311990037685826651272731, −8.451537313849835163541730785541, −7.48899615870080101718156586738, −6.62258927793904769240005398838, −5.46105014390535428453102148384, −4.09779421966819445986193763155, −3.69154065495586022562391089080, −2.01511125521932001144552502659, −0.33090443199650256754111572355, 2.04408342353128453326031808357, 3.19016853668829180034192365297, 4.46229334156339687118396321136, 5.21904242567715281104924092563, 6.56589823427426057198775973711, 7.30230606781921794967154225318, 8.413916988605105607348121153319, 9.394631744156219344354283596874, 10.99334310535335921651909628488, 11.46803744998676280667671090252

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