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Properties

Label 2-18e2-12.11-c1-0-13
Degree $2$
Conductor $324$
Sign $-0.537 + 0.843i$
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  + (−1.35 − 0.396i)2-s + (1.68 + 1.07i)4-s − 2.52i·5-s − 1.27i·7-s + (−1.86 − 2.12i)8-s + (−1 + 3.42i)10-s − 0.505·11-s − 2.37·13-s + (−0.505 + 1.73i)14-s + (1.68 + 3.62i)16-s − 0.792i·17-s − 4.70i·19-s + (2.71 − 4.25i)20-s + (0.686 + 0.200i)22-s − 3.22·23-s + ⋯
L(s)  = 1  + (−0.959 − 0.280i)2-s + (0.843 + 0.537i)4-s − 1.12i·5-s − 0.482i·7-s + (−0.658 − 0.752i)8-s + (−0.316 + 1.08i)10-s − 0.152·11-s − 0.657·13-s + (−0.135 + 0.462i)14-s + (0.421 + 0.906i)16-s − 0.192i·17-s − 1.07i·19-s + (0.607 − 0.951i)20-s + (0.146 + 0.0426i)22-s − 0.671·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.537 + 0.843i$
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: \((2,\ 324,\ (\ :1/2),\ -0.537 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327490 - 0.597359i\)
\(L(\frac12)\) \(\approx\) \(0.327490 - 0.597359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.396i)T \)
3 \( 1 \)
good5 \( 1 + 2.52iT - 5T^{2} \)
7 \( 1 + 1.27iT - 7T^{2} \)
11 \( 1 + 0.505T + 11T^{2} \)
13 \( 1 + 2.37T + 13T^{2} \)
17 \( 1 + 0.792iT - 17T^{2} \)
19 \( 1 + 4.70iT - 19T^{2} \)
23 \( 1 + 3.22T + 23T^{2} \)
29 \( 1 + 2.52iT - 29T^{2} \)
31 \( 1 + 8.12iT - 31T^{2} \)
37 \( 1 + 6.74T + 37T^{2} \)
41 \( 1 - 6.78iT - 41T^{2} \)
43 \( 1 + 7.72iT - 43T^{2} \)
47 \( 1 + 1.19T + 47T^{2} \)
53 \( 1 + 1.87iT - 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 + 2.37T + 61T^{2} \)
67 \( 1 - 7.72iT - 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 - 3.37T + 73T^{2} \)
79 \( 1 - 9.88iT - 79T^{2} \)
83 \( 1 - 7.64T + 83T^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−11.27345256695804721300024652422, −10.16803048411685386482145627661, −9.437720340488286407399316331097, −8.572985635494802218219238127114, −7.71262576741327165962897209431, −6.73678865664180480434445847523, −5.27564601599521067790283106462, −4.01669742209851357022854295658, −2.32008891818036613307317260837, −0.64195917548714101357827461841, 2.04904739439121086414643269009, 3.28600803525524826809115520290, 5.31553205847738379837647907785, 6.37976295625940337249709873086, 7.19471384984110588888212956379, 8.122686931172929760105544411214, 9.121085878950089940043935091249, 10.23937667136628453972205446470, 10.60913395114682657812018206801, 11.76427096784468702796573020640

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