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Properties

Label 2-18e2-12.11-c3-0-27
Degree $2$
Conductor $324$
Sign $-0.434 - 0.900i$
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  + (−1.50 + 2.39i)2-s + (−3.47 − 7.20i)4-s + 12.5i·5-s + 7.52i·7-s + (22.4 + 2.50i)8-s + (−30.1 − 18.9i)10-s + 67.4·11-s + 29.7·13-s + (−18.0 − 11.3i)14-s + (−39.8 + 50.1i)16-s − 66.7i·17-s − 59.4i·19-s + (90.6 − 43.7i)20-s + (−101. + 161. i)22-s + 58.7·23-s + ⋯
L(s)  = 1  + (−0.531 + 0.846i)2-s + (−0.434 − 0.900i)4-s + 1.12i·5-s + 0.406i·7-s + (0.993 + 0.110i)8-s + (−0.953 − 0.598i)10-s + 1.84·11-s + 0.634·13-s + (−0.343 − 0.215i)14-s + (−0.622 + 0.782i)16-s − 0.952i·17-s − 0.717i·19-s + (1.01 − 0.489i)20-s + (−0.983 + 1.56i)22-s + 0.532·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.434 - 0.900i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ -0.434 - 0.900i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.503194543\)
\(L(\frac12)\) \(\approx\) \(1.503194543\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.50 - 2.39i)T \)
3 \( 1 \)
good5 \( 1 - 12.5iT - 125T^{2} \)
7 \( 1 - 7.52iT - 343T^{2} \)
11 \( 1 - 67.4T + 1.33e3T^{2} \)
13 \( 1 - 29.7T + 2.19e3T^{2} \)
17 \( 1 + 66.7iT - 4.91e3T^{2} \)
19 \( 1 + 59.4iT - 6.85e3T^{2} \)
23 \( 1 - 58.7T + 1.21e4T^{2} \)
29 \( 1 - 198. iT - 2.43e4T^{2} \)
31 \( 1 - 123. iT - 2.97e4T^{2} \)
37 \( 1 + 159.T + 5.06e4T^{2} \)
41 \( 1 - 4.44iT - 6.89e4T^{2} \)
43 \( 1 - 508. iT - 7.95e4T^{2} \)
47 \( 1 - 625.T + 1.03e5T^{2} \)
53 \( 1 - 513. iT - 1.48e5T^{2} \)
59 \( 1 + 417.T + 2.05e5T^{2} \)
61 \( 1 + 860.T + 2.26e5T^{2} \)
67 \( 1 + 722. iT - 3.00e5T^{2} \)
71 \( 1 - 168.T + 3.57e5T^{2} \)
73 \( 1 + 882.T + 3.89e5T^{2} \)
79 \( 1 + 301. iT - 4.93e5T^{2} \)
83 \( 1 + 270.T + 5.71e5T^{2} \)
89 \( 1 + 16.7iT - 7.04e5T^{2} \)
97 \( 1 + 185.T + 9.12e5T^{2} \)
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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

−11.18299590754133478172745849465, −10.56921821583971009987228542829, −9.151223593854855594338433156069, −8.983236505858405915586537702751, −7.37174188312252990593583830144, −6.76116957275775078431990613010, −5.98483408143151901132933792130, −4.58423811314138503636608938252, −3.10426650834187377406181604220, −1.26907348207025556486238109153, 0.809892225056669140032155420598, 1.72051693152620401777759192928, 3.75952658316215625941579281899, 4.29407459704337166696021415734, 5.91532578026902669196606135284, 7.26245507099109957168666508485, 8.532270168913284158723022527459, 8.926535698625134680915160122459, 9.911039629421608292642417727243, 10.88711493248714107219633354855

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