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Properties

Label 2-18e2-36.11-c1-0-8
Degree $2$
Conductor $324$
Sign $0.995 - 0.0962i$
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  + (−1.40 + 0.190i)2-s + (1.92 − 0.535i)4-s + (1.93 − 1.11i)5-s + (3.35 + 1.93i)7-s + (−2.59 + 1.11i)8-s + (−2.5 + 1.93i)10-s + (−0.866 + 1.5i)11-s + (−1 − 1.73i)13-s + (−5.06 − 2.07i)14-s + (3.42 − 2.06i)16-s − 4.47i·17-s + (3.13 − 3.19i)20-s + (0.927 − 2.26i)22-s + (3.46 + 6i)23-s + (1.73 + 2.23i)26-s + ⋯
L(s)  = 1  + (−0.990 + 0.135i)2-s + (0.963 − 0.267i)4-s + (0.866 − 0.499i)5-s + (1.26 + 0.731i)7-s + (−0.918 + 0.395i)8-s + (−0.790 + 0.612i)10-s + (−0.261 + 0.452i)11-s + (−0.277 − 0.480i)13-s + (−1.35 − 0.554i)14-s + (0.856 − 0.515i)16-s − 1.08i·17-s + (0.700 − 0.713i)20-s + (0.197 − 0.483i)22-s + (0.722 + 1.25i)23-s + (0.339 + 0.438i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.995 - 0.0962i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.995 - 0.0962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10575 + 0.0533308i\)
\(L(\frac12)\) \(\approx\) \(1.10575 + 0.0533308i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.190i)T \)
3 \( 1 \)
good5 \( 1 + (-1.93 + 1.11i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.35 - 1.93i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.866 - 1.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.47iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-3.46 - 6i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.87 - 2.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.35 + 1.93i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-7.74 + 4.47i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.70 + 3.87i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.73 + 3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.23iT - 53T^{2} \)
59 \( 1 + (1.73 + 3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.70 - 3.87i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 + (6.70 + 3.87i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.06 - 10.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.47iT - 89T^{2} \)
97 \( 1 + (5.5 - 9.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.54761201699629475346491404918, −10.54151278929335827392795752895, −9.570660055889714249652041393862, −8.921831529453415512885186792570, −7.976257456922119444324851449238, −7.07440560228400812055969570133, −5.57606406405927563237021289273, −5.07647154848516363188659078678, −2.61971195312539969638161056619, −1.47403551636966006944133902116, 1.43637849241101149154827117920, 2.70486903593500102404765870027, 4.47699873582168140844378123716, 6.01956404187274510027677794049, 6.89580452998342742546920928258, 8.003771072720594578654795950462, 8.682524335379201426750753880568, 9.954054764709998227892498781047, 10.59278645535129955830429227845, 11.18478550033118574269978362008

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