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Properties

Label 2-18e2-108.23-c1-0-11
Degree $2$
Conductor $324$
Sign $0.870 - 0.492i$
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  + (1.33 + 0.462i)2-s + (1.57 + 1.23i)4-s + (3.39 + 0.598i)5-s + (−1.88 − 2.24i)7-s + (1.52 + 2.38i)8-s + (4.25 + 2.37i)10-s + (−0.453 − 2.57i)11-s + (−5.09 + 1.85i)13-s + (−1.47 − 3.87i)14-s + (0.938 + 3.88i)16-s + (−1.15 − 0.667i)17-s + (−0.0790 + 0.0456i)19-s + (4.59 + 5.13i)20-s + (0.585 − 3.65i)22-s + (0.650 + 0.545i)23-s + ⋯
L(s)  = 1  + (0.944 + 0.327i)2-s + (0.785 + 0.618i)4-s + (1.51 + 0.267i)5-s + (−0.711 − 0.848i)7-s + (0.539 + 0.841i)8-s + (1.34 + 0.749i)10-s + (−0.136 − 0.776i)11-s + (−1.41 + 0.514i)13-s + (−0.394 − 1.03i)14-s + (0.234 + 0.972i)16-s + (−0.280 − 0.161i)17-s + (−0.0181 + 0.0104i)19-s + (1.02 + 1.14i)20-s + (0.124 − 0.778i)22-s + (0.135 + 0.113i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.870 - 0.492i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.870 - 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.43791 + 0.641225i\)
\(L(\frac12)\) \(\approx\) \(2.43791 + 0.641225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 - 0.462i)T \)
3 \( 1 \)
good5 \( 1 + (-3.39 - 0.598i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (1.88 + 2.24i)T + (-1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.453 + 2.57i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (5.09 - 1.85i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.15 + 0.667i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0790 - 0.0456i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.650 - 0.545i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-0.171 + 0.470i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (4.28 - 5.10i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (-2.75 + 4.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.976 - 2.68i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (5.59 - 0.986i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-5.06 + 4.24i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 7.37iT - 53T^{2} \)
59 \( 1 + (0.528 - 2.99i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (1.85 - 1.55i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (1.64 + 4.53i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (5.91 - 10.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.83 + 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.576 + 1.58i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-2.02 - 0.738i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-12.5 + 7.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.14 - 6.49i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.86171953416049106523600000318, −10.70706684825127857785531863549, −10.00389895849378419704460344576, −8.988428615508013032559752433393, −7.40059158079464326518781470184, −6.68551944117714986542092683346, −5.81078217154853992220622477716, −4.80466250777227761525100972469, −3.34694798224735136095397689259, −2.17917591920559008201984475575, 2.02960708675828924660291846429, 2.81775385780110127724525830726, 4.67032818931246969501070968018, 5.55144634590761923113893495333, 6.26673050698427244218578335646, 7.38725240758187976525718357510, 9.189279946122172271617673932866, 9.795703266087568813504832669675, 10.47620806152002295024748569055, 11.87653712120406678509876867943

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