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Properties

Label 2-18-3.2-c12-0-0
Degree $2$
Conductor $18$
Sign $-0.816 + 0.577i$
Analytic cond. $16.4518$
Root an. cond. $4.05609$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 45.2i·2-s − 2.04e3·4-s + 2.05e4i·5-s + 3.35e4·7-s − 9.26e4i·8-s − 9.32e5·10-s + 1.71e6i·11-s − 5.19e6·13-s + 1.51e6i·14-s + 4.19e6·16-s − 3.13e7i·17-s − 6.60e7·19-s − 4.21e7i·20-s − 7.75e7·22-s − 1.72e8i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s + 1.31i·5-s + 0.285·7-s − 0.353i·8-s − 0.932·10-s + 0.967i·11-s − 1.07·13-s + 0.201i·14-s + 0.250·16-s − 1.29i·17-s − 1.40·19-s − 0.659i·20-s − 0.684·22-s − 1.16i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $-0.816 + 0.577i$
Analytic conductor: \(16.4518\)
Root analytic conductor: \(4.05609\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{18} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 18,\ (\ :6),\ -0.816 + 0.577i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.220889 - 0.694977i\)
\(L(\frac12)\) \(\approx\) \(0.220889 - 0.694977i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 45.2iT \)
3 \( 1 \)
good5 \( 1 - 2.05e4iT - 2.44e8T^{2} \)
7 \( 1 - 3.35e4T + 1.38e10T^{2} \)
11 \( 1 - 1.71e6iT - 3.13e12T^{2} \)
13 \( 1 + 5.19e6T + 2.32e13T^{2} \)
17 \( 1 + 3.13e7iT - 5.82e14T^{2} \)
19 \( 1 + 6.60e7T + 2.21e15T^{2} \)
23 \( 1 + 1.72e8iT - 2.19e16T^{2} \)
29 \( 1 - 4.04e8iT - 3.53e17T^{2} \)
31 \( 1 + 1.73e9T + 7.87e17T^{2} \)
37 \( 1 - 4.29e9T + 6.58e18T^{2} \)
41 \( 1 - 8.42e9iT - 2.25e19T^{2} \)
43 \( 1 - 2.71e9T + 3.99e19T^{2} \)
47 \( 1 - 8.00e9iT - 1.16e20T^{2} \)
53 \( 1 - 1.16e10iT - 4.91e20T^{2} \)
59 \( 1 + 1.07e10iT - 1.77e21T^{2} \)
61 \( 1 + 3.74e10T + 2.65e21T^{2} \)
67 \( 1 - 7.46e10T + 8.18e21T^{2} \)
71 \( 1 - 7.21e10iT - 1.64e22T^{2} \)
73 \( 1 + 7.22e10T + 2.29e22T^{2} \)
79 \( 1 - 3.17e11T + 5.90e22T^{2} \)
83 \( 1 + 1.22e11iT - 1.06e23T^{2} \)
89 \( 1 - 2.36e11iT - 2.46e23T^{2} \)
97 \( 1 + 1.34e11T + 6.93e23T^{2} \)
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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

−16.51921311414961822109780103279, −14.79587223845900709831801292028, −14.58672182690319023663664124672, −12.68639789252526026843061598844, −10.94482426352989567945839697814, −9.562971217129862458655237462806, −7.58748045305022106002296336561, −6.60545183937517885039668823130, −4.65397331821090732381970650134, −2.53436428829489754739677558188, 0.27309252011008683145190181076, 1.83820940461222875089441113121, 4.04732245275144891177602329047, 5.52738222820608218839524591322, 8.143873876351027827017920751480, 9.279779321535803293214060340964, 10.92304159676229824796348545054, 12.35599652170272402860594893378, 13.23799105504029971697980062431, 14.84229774903433363192870780566

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