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Properties

Label 2-18e2-12.11-c1-0-6
Degree $2$
Conductor $324$
Sign $0.805 + 0.593i$
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  + (−0.637 − 1.26i)2-s + (−1.18 + 1.61i)4-s − 0.792i·5-s + 2.71i·7-s + (2.78 + 0.469i)8-s + (−1 + 0.505i)10-s + 3.42·11-s + 3.37·13-s + (3.42 − 1.73i)14-s + (−1.18 − 3.82i)16-s − 2.52i·17-s + 2.20i·19-s + (1.27 + 0.939i)20-s + (−2.18 − 4.32i)22-s + 2.15·23-s + ⋯
L(s)  = 1  + (−0.451 − 0.892i)2-s + (−0.593 + 0.805i)4-s − 0.354i·5-s + 1.02i·7-s + (0.986 + 0.166i)8-s + (−0.316 + 0.159i)10-s + 1.03·11-s + 0.935·13-s + (0.915 − 0.462i)14-s + (−0.296 − 0.955i)16-s − 0.612i·17-s + 0.506i·19-s + (0.285 + 0.210i)20-s + (−0.466 − 0.922i)22-s + 0.448·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.805 + 0.593i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.805 + 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02915 - 0.338120i\)
\(L(\frac12)\) \(\approx\) \(1.02915 - 0.338120i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.637 + 1.26i)T \)
3 \( 1 \)
good5 \( 1 + 0.792iT - 5T^{2} \)
7 \( 1 - 2.71iT - 7T^{2} \)
11 \( 1 - 3.42T + 11T^{2} \)
13 \( 1 - 3.37T + 13T^{2} \)
17 \( 1 + 2.52iT - 17T^{2} \)
19 \( 1 - 2.20iT - 19T^{2} \)
23 \( 1 - 2.15T + 23T^{2} \)
29 \( 1 + 0.792iT - 29T^{2} \)
31 \( 1 - 1.70iT - 31T^{2} \)
37 \( 1 - 4.74T + 37T^{2} \)
41 \( 1 + 0.147iT - 41T^{2} \)
43 \( 1 + 6.94iT - 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 - 8.51iT - 53T^{2} \)
59 \( 1 + 5.17T + 59T^{2} \)
61 \( 1 - 3.37T + 61T^{2} \)
67 \( 1 - 6.94iT - 67T^{2} \)
71 \( 1 + 1.75T + 71T^{2} \)
73 \( 1 + 2.37T + 73T^{2} \)
79 \( 1 - 10.1iT - 79T^{2} \)
83 \( 1 - 7.25T + 83T^{2} \)
89 \( 1 + 5.34iT - 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−11.59522891283963803807503526610, −10.70487966405183441072621148821, −9.476364670979115587848707836010, −8.929190944861431774951882512622, −8.160327729148297603294338702376, −6.73852319565927459050930207420, −5.41280772224833807551011349461, −4.14367620601738205677447143481, −2.89631727969944702863485228412, −1.37286885085961951782937017533, 1.20363598546546291373557786315, 3.65806377405000152538572059586, 4.73248921665591412137331898598, 6.25663943934427353448200428378, 6.78983342913710477881781560057, 7.84096323716550304143911089435, 8.803779146996555404542716101005, 9.713932128564381846238372855184, 10.70969238377702755835100653241, 11.32159459757821840127653294577

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