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Properties

Label 2-18-9.4-c3-0-1
Degree $2$
Conductor $18$
Sign $0.934 - 0.355i$
Analytic cond. $1.06203$
Root an. cond. $1.03055$
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 + 1.73i)2-s + (3.31 − 4.00i)3-s + (−1.99 + 3.46i)4-s + (−5.43 + 9.41i)5-s + (10.2 + 1.73i)6-s + (−12.4 − 21.5i)7-s − 7.99·8-s + (−5.06 − 26.5i)9-s − 21.7·10-s + (21.3 + 37.0i)11-s + (7.24 + 19.4i)12-s + (−7.56 + 13.1i)13-s + (24.8 − 43.0i)14-s + (19.6 + 52.9i)15-s + (−8 − 13.8i)16-s − 13.8·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.637 − 0.770i)3-s + (−0.249 + 0.433i)4-s + (−0.486 + 0.842i)5-s + (0.697 + 0.117i)6-s + (−0.671 − 1.16i)7-s − 0.353·8-s + (−0.187 − 0.982i)9-s − 0.687·10-s + (0.585 + 1.01i)11-s + (0.174 + 0.468i)12-s + (−0.161 + 0.279i)13-s + (0.474 − 0.822i)14-s + (0.339 + 0.911i)15-s + (−0.125 − 0.216i)16-s − 0.197·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $0.934 - 0.355i$
Analytic conductor: \(1.06203\)
Root analytic conductor: \(1.03055\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{18} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 18,\ (\ :3/2),\ 0.934 - 0.355i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.25371 + 0.230233i\)
\(L(\frac12)\) \(\approx\) \(1.25371 + 0.230233i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 + (-3.31 + 4.00i)T \)
good5 \( 1 + (5.43 - 9.41i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (12.4 + 21.5i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-21.3 - 37.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (7.56 - 13.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 13.8T + 4.91e3T^{2} \)
19 \( 1 - 143.T + 6.85e3T^{2} \)
23 \( 1 + (9.56 - 16.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (113. + 195. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (29.6 - 51.4i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 84.1T + 5.06e4T^{2} \)
41 \( 1 + (101. - 176. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (162. + 282. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (5.47 + 9.47i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 140.T + 1.48e5T^{2} \)
59 \( 1 + (57.3 - 99.3i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-377. - 653. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-383. + 664. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 335.T + 3.57e5T^{2} \)
73 \( 1 - 167.T + 3.89e5T^{2} \)
79 \( 1 + (-12.6 - 21.9i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (143. + 249. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 860.T + 7.04e5T^{2} \)
97 \( 1 + (-201. - 348. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−18.30021021962872889399323045360, −17.04411175449106600726258323842, −15.43062941847074119135500388072, −14.30441924306445031095314370466, −13.34376940576790711248211056619, −11.83011894895909056258898003511, −9.659404539468818438178579664217, −7.49280612897997457221569944911, −6.83613776844462763888020529733, −3.62768242447689383205795706978, 3.33354277687313836026402320475, 5.32425418152774817954800103619, 8.586624919062136920710889624436, 9.547834098055063183043822650011, 11.43944045384362872394399344443, 12.72052001372066200764273099936, 14.16854064168078325209429092594, 15.60504420665671828690061860086, 16.41142751801919002512807144296, 18.64231613099320194426544512282

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