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Properties

Label 2-18e2-12.11-c3-0-58
Degree $2$
Conductor $324$
Sign $-0.748 + 0.663i$
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  + (−1.16 + 2.57i)2-s + (−5.30 − 5.98i)4-s − 16.7i·5-s − 19.3i·7-s + (21.5 − 6.74i)8-s + (43.1 + 19.4i)10-s + 4.88·11-s − 12.0·13-s + (49.7 + 22.3i)14-s + (−7.64 + 63.5i)16-s − 71.2i·17-s + 68.3i·19-s + (−100. + 88.8i)20-s + (−5.66 + 12.5i)22-s − 136.·23-s + ⋯
L(s)  = 1  + (−0.410 + 0.912i)2-s + (−0.663 − 0.748i)4-s − 1.49i·5-s − 1.04i·7-s + (0.954 − 0.298i)8-s + (1.36 + 0.613i)10-s + 0.133·11-s − 0.257·13-s + (0.950 + 0.427i)14-s + (−0.119 + 0.992i)16-s − 1.01i·17-s + 0.824i·19-s + (−1.11 + 0.993i)20-s + (−0.0548 + 0.122i)22-s − 1.23·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.748 + 0.663i$
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: \((2,\ 324,\ (\ :3/2),\ -0.748 + 0.663i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6577778346\)
\(L(\frac12)\) \(\approx\) \(0.6577778346\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 - 2.57i)T \)
3 \( 1 \)
good5 \( 1 + 16.7iT - 125T^{2} \)
7 \( 1 + 19.3iT - 343T^{2} \)
11 \( 1 - 4.88T + 1.33e3T^{2} \)
13 \( 1 + 12.0T + 2.19e3T^{2} \)
17 \( 1 + 71.2iT - 4.91e3T^{2} \)
19 \( 1 - 68.3iT - 6.85e3T^{2} \)
23 \( 1 + 136.T + 1.21e4T^{2} \)
29 \( 1 + 219. iT - 2.43e4T^{2} \)
31 \( 1 - 329. iT - 2.97e4T^{2} \)
37 \( 1 + 133.T + 5.06e4T^{2} \)
41 \( 1 + 34.1iT - 6.89e4T^{2} \)
43 \( 1 + 0.644iT - 7.95e4T^{2} \)
47 \( 1 - 186.T + 1.03e5T^{2} \)
53 \( 1 - 266. iT - 1.48e5T^{2} \)
59 \( 1 + 208.T + 2.05e5T^{2} \)
61 \( 1 + 1.60T + 2.26e5T^{2} \)
67 \( 1 + 428. iT - 3.00e5T^{2} \)
71 \( 1 - 386.T + 3.57e5T^{2} \)
73 \( 1 + 776.T + 3.89e5T^{2} \)
79 \( 1 - 79.0iT - 4.93e5T^{2} \)
83 \( 1 + 925.T + 5.71e5T^{2} \)
89 \( 1 + 1.04e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.46e3T + 9.12e5T^{2} \)
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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

−10.43644253800890620761607369014, −9.717506734470841048290801381549, −8.806796559182442091314962181700, −7.974558139786201327621260637860, −7.14487291770483215834862962579, −5.87378345510078107085734515757, −4.85725354343246649276400926887, −4.05578831114491492230288323089, −1.41109806174719063318411683142, −0.28334696291131448097662798211, 2.03046824370504365271004673295, 2.88697607401749957624032833673, 4.02858724164728668098877212300, 5.65830255414914788636885483257, 6.82612237517206373430577298858, 7.905998859901341343295683633037, 8.908303078522385229070772117355, 9.874601331514615132911934009167, 10.66388452744388845754044039886, 11.41592052945416710864426006306

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