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Properties

Label 2-18e2-9.7-c3-0-9
Degree $2$
Conductor $324$
Sign $-0.173 + 0.984i$
Analytic cond. $19.1166$
Root an. cond. $4.37225$
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  + (18.5 − 32.0i)7-s + (9.5 + 16.4i)13-s − 163·19-s + (62.5 − 108. i)25-s + (−154 − 266. i)31-s + 323·37-s + (260 − 450. i)43-s + (−513. − 888. i)49-s + (−359.5 + 622. i)61-s + (63.5 + 109. i)67-s − 919·73-s + (693.5 − 1.20e3i)79-s + 703·91-s + (261.5 − 452. i)97-s + (900.5 + 1.55e3i)103-s + ⋯
L(s)  = 1  + (0.998 − 1.73i)7-s + (0.202 + 0.351i)13-s − 1.96·19-s + (0.5 − 0.866i)25-s + (−0.892 − 1.54i)31-s + 1.43·37-s + (0.922 − 1.59i)43-s + (−1.49 − 2.59i)49-s + (−0.754 + 1.30i)61-s + (0.115 + 0.200i)67-s − 1.47·73-s + (0.987 − 1.71i)79-s + 0.809·91-s + (0.273 − 0.474i)97-s + (0.861 + 1.49i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.658433184\)
\(L(\frac12)\) \(\approx\) \(1.658433184\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-18.5 + 32.0i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-9.5 - 16.4i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 + 163T + 6.85e3T^{2} \)
23 \( 1 + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (154 + 266. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 323T + 5.06e4T^{2} \)
41 \( 1 + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-260 + 450. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (359.5 - 622. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-63.5 - 109. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + 919T + 3.89e5T^{2} \)
79 \( 1 + (-693.5 + 1.20e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + (-261.5 + 452. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.78299548409709470952709914213, −10.33162794386494669412965372292, −8.953043385444201245210453513865, −7.967399725724804926506902896000, −7.18499353061306969646189713767, −6.10680941510975930185433914235, −4.51022269081200820312581021148, −4.02180062955347489553927996165, −2.05761083367984261881213541140, −0.59966876441746043954477880949, 1.69588153939007361935906085505, 2.83484577463696515932765821180, 4.53321806167653875795497413445, 5.49998564915415249301575703055, 6.40720083733342314152904971510, 7.88243040241093718532034804722, 8.640209554342365264113812422406, 9.322112633772919277707789870379, 10.79858361250154796072058737060, 11.31389043468278016591972427802

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