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Properties

Label 2-18e2-12.11-c3-0-20
Degree $2$
Conductor $324$
Sign $0.988 - 0.149i$
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  + (−2.14 − 1.84i)2-s + (1.19 + 7.91i)4-s + 4.97i·5-s − 3.54i·7-s + (12.0 − 19.1i)8-s + (9.16 − 10.6i)10-s + 44.3·11-s − 61.3·13-s + (−6.54 + 7.60i)14-s + (−61.1 + 18.9i)16-s − 99.9i·17-s + 85.6i·19-s + (−39.3 + 5.94i)20-s + (−95.1 − 81.8i)22-s + 82.6·23-s + ⋯
L(s)  = 1  + (−0.758 − 0.652i)2-s + (0.149 + 0.988i)4-s + 0.444i·5-s − 0.191i·7-s + (0.531 − 0.846i)8-s + (0.289 − 0.337i)10-s + 1.21·11-s − 1.30·13-s + (−0.124 + 0.145i)14-s + (−0.955 + 0.295i)16-s − 1.42i·17-s + 1.03i·19-s + (−0.439 + 0.0664i)20-s + (−0.922 − 0.793i)22-s + 0.748·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.988 - 0.149i$
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: \((2,\ 324,\ (\ :3/2),\ 0.988 - 0.149i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.196316894\)
\(L(\frac12)\) \(\approx\) \(1.196316894\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.14 + 1.84i)T \)
3 \( 1 \)
good5 \( 1 - 4.97iT - 125T^{2} \)
7 \( 1 + 3.54iT - 343T^{2} \)
11 \( 1 - 44.3T + 1.33e3T^{2} \)
13 \( 1 + 61.3T + 2.19e3T^{2} \)
17 \( 1 + 99.9iT - 4.91e3T^{2} \)
19 \( 1 - 85.6iT - 6.85e3T^{2} \)
23 \( 1 - 82.6T + 1.21e4T^{2} \)
29 \( 1 - 176. iT - 2.43e4T^{2} \)
31 \( 1 - 197. iT - 2.97e4T^{2} \)
37 \( 1 + 172.T + 5.06e4T^{2} \)
41 \( 1 - 44.1iT - 6.89e4T^{2} \)
43 \( 1 + 78.1iT - 7.95e4T^{2} \)
47 \( 1 - 459.T + 1.03e5T^{2} \)
53 \( 1 - 290. iT - 1.48e5T^{2} \)
59 \( 1 - 295.T + 2.05e5T^{2} \)
61 \( 1 - 494.T + 2.26e5T^{2} \)
67 \( 1 + 22.5iT - 3.00e5T^{2} \)
71 \( 1 - 304.T + 3.57e5T^{2} \)
73 \( 1 - 1.16e3T + 3.89e5T^{2} \)
79 \( 1 - 1.11e3iT - 4.93e5T^{2} \)
83 \( 1 + 801.T + 5.71e5T^{2} \)
89 \( 1 + 346. iT - 7.04e5T^{2} \)
97 \( 1 + 582.T + 9.12e5T^{2} \)
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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

−11.10034006105583778764187817296, −10.26045529432286070324327080328, −9.411828304534067740374743458245, −8.656527001266287354004330335605, −7.20930545294135869871710093150, −6.92544163239535558273930536684, −5.03005709342268742274151105894, −3.67215892854532028277944946055, −2.57039751085476218893725896644, −1.04566610856476535811471047002, 0.71042749634116029817908209040, 2.20402529869590825287878367818, 4.25945961091428734625621436193, 5.36113658938423077477765498495, 6.47969691314295704014505240345, 7.31298998228351930529105632220, 8.466258430278539211308215025824, 9.158861504226011622110926016751, 9.946072755221323271183363973534, 11.02637835160358736945641960799

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