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Properties

Label 2-18e2-9.2-c4-0-10
Degree $2$
Conductor $324$
Sign $0.984 + 0.173i$
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  + (−11.5 + 19.9i)7-s + (−95.5 − 165. i)13-s + 647·19-s + (−312.5 + 541. i)25-s + (−97 − 168. i)31-s + 2.59e3·37-s + (1.60e3 − 2.78e3i)43-s + (936 + 1.62e3i)49-s + (2.61e3 − 4.53e3i)61-s + (4.40e3 + 7.62e3i)67-s + 9.79e3·73-s + (6.18e3 − 1.07e4i)79-s + 4.39e3·91-s + (−4.87e3 + 8.43e3i)97-s + (−1.71e3 − 2.97e3i)103-s + ⋯
L(s)  = 1  + (−0.234 + 0.406i)7-s + (−0.565 − 0.978i)13-s + 1.79·19-s + (−0.5 + 0.866i)25-s + (−0.100 − 0.174i)31-s + 1.89·37-s + (0.869 − 1.50i)43-s + (0.389 + 0.675i)49-s + (0.703 − 1.21i)61-s + (0.981 + 1.69i)67-s + 1.83·73-s + (0.990 − 1.71i)79-s + 0.530·91-s + (−0.517 + 0.896i)97-s + (−0.161 − 0.280i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(33.4918\)
Root analytic conductor: \(5.78721\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :2),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.859489669\)
\(L(\frac12)\) \(\approx\) \(1.859489669\)
\(L(3)\) 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 \( 1 \)
good5 \( 1 + (312.5 - 541. i)T^{2} \)
7 \( 1 + (11.5 - 19.9i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (95.5 + 165. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 8.35e4T^{2} \)
19 \( 1 - 647T + 1.30e5T^{2} \)
23 \( 1 + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (97 + 168. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 2.59e3T + 1.87e6T^{2} \)
41 \( 1 + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.60e3 + 2.78e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 7.89e6T^{2} \)
59 \( 1 + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-2.61e3 + 4.53e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-4.40e3 - 7.62e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 2.54e7T^{2} \)
73 \( 1 - 9.79e3T + 2.83e7T^{2} \)
79 \( 1 + (-6.18e3 + 1.07e4i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 6.27e7T^{2} \)
97 \( 1 + (4.87e3 - 8.43e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−11.00805170551163166664566939325, −9.860322905715721756821036967348, −9.276426029516073399808786000862, −7.973368198120343489119614863787, −7.25958048298423462165230775750, −5.87272240618940916972546492789, −5.12950135553836154172595099068, −3.60321538832587221365575747731, −2.49305937577561699623507325351, −0.78441914411268765061571758560, 0.899484892134494880909182299002, 2.48518751708662994692253580319, 3.84557459404577389657553353586, 4.93154076845340244045882595047, 6.18040194328737237664020213915, 7.19605100137889176665628576847, 8.031590562615429078988023600050, 9.410513765090091291423950710267, 9.844665600494469759064933175680, 11.12177847785443696740505880072

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