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Properties

Label 2-18e2-108.47-c1-0-7
Degree $2$
Conductor $324$
Sign $0.919 + 0.393i$
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  + (−0.438 + 1.34i)2-s + (−1.61 − 1.17i)4-s + (−3.32 + 0.585i)5-s + (1.72 − 2.05i)7-s + (2.29 − 1.65i)8-s + (0.669 − 4.72i)10-s + (0.506 − 2.87i)11-s + (0.552 + 0.201i)13-s + (2.00 + 3.22i)14-s + (1.21 + 3.81i)16-s + (6.36 − 3.67i)17-s + (−2.82 − 1.63i)19-s + (6.05 + 2.97i)20-s + (3.64 + 1.94i)22-s + (0.365 − 0.306i)23-s + ⋯
L(s)  = 1  + (−0.310 + 0.950i)2-s + (−0.807 − 0.589i)4-s + (−1.48 + 0.262i)5-s + (0.652 − 0.777i)7-s + (0.811 − 0.584i)8-s + (0.211 − 1.49i)10-s + (0.152 − 0.866i)11-s + (0.153 + 0.0557i)13-s + (0.536 + 0.861i)14-s + (0.304 + 0.952i)16-s + (1.54 − 0.890i)17-s + (−0.648 − 0.374i)19-s + (1.35 + 0.664i)20-s + (0.776 + 0.414i)22-s + (0.0761 − 0.0638i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.919 + 0.393i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.919 + 0.393i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.727449 - 0.149020i\)
\(L(\frac12)\) \(\approx\) \(0.727449 - 0.149020i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.438 - 1.34i)T \)
3 \( 1 \)
good5 \( 1 + (3.32 - 0.585i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (-1.72 + 2.05i)T + (-1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.506 + 2.87i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-0.552 - 0.201i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-6.36 + 3.67i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.82 + 1.63i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.365 + 0.306i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (1.49 + 4.10i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.73 + 4.45i)T + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (1.59 + 2.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.09 + 3.01i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.79 - 0.316i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.940 + 0.789i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 7.37iT - 53T^{2} \)
59 \( 1 + (0.718 + 4.07i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-0.421 - 0.353i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-3.46 + 9.51i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (0.616 + 1.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.18 + 2.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.33 - 9.16i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (16.8 - 6.15i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-4.81 - 2.77i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.83 + 10.4i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.30812217437609931831847486827, −10.76540023700808921939942323032, −9.495783376562967250150657993444, −8.332258012648658695280709512686, −7.71474245404779579786112092321, −7.08260754258458803052180562139, −5.72616519497107892569119682166, −4.46404663429133373725642313580, −3.58909958548639959025933174903, −0.65401335713787066966228177463, 1.60612340081470280522048844812, 3.34168474685931655745778221870, 4.29564964098774836396083486744, 5.36020119715507430344706160816, 7.31262302279161795892654035455, 8.182225929910094741400782982052, 8.706137297544990540939537618840, 9.959687801395102820745924334645, 10.90794659149539811864237458093, 11.78310860042581226056746804627

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