Properties

Label 2-18-3.2-c6-0-0
Degree $2$
Conductor $18$
Sign $-0.816 - 0.577i$
Analytic cond. $4.14097$
Root an. cond. $2.03493$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.65i·2-s − 32.0·4-s + 173. i·5-s − 484·7-s − 181. i·8-s − 984.·10-s + 1.34e3i·11-s + 3.36e3·13-s − 2.73e3i·14-s + 1.02e3·16-s + 12.7i·17-s + 5.74e3·19-s − 5.56e3i·20-s − 7.58e3·22-s − 3.37e3i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s + 1.39i·5-s − 1.41·7-s − 0.353i·8-s − 0.984·10-s + 1.00i·11-s + 1.53·13-s − 0.997i·14-s + 0.250·16-s + 0.00259i·17-s + 0.837·19-s − 0.695i·20-s − 0.712·22-s − 0.277i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(4.14097\)
Root analytic conductor: \(2.03493\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{18} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 18,\ (\ :3),\ -0.816 - 0.577i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.329918 + 1.03801i\)
\(L(\frac12)\) \(\approx\) \(0.329918 + 1.03801i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 5.65iT \)
3 \( 1 \)
good5 \( 1 - 173. iT - 1.56e4T^{2} \)
7 \( 1 + 484T + 1.17e5T^{2} \)
11 \( 1 - 1.34e3iT - 1.77e6T^{2} \)
13 \( 1 - 3.36e3T + 4.82e6T^{2} \)
17 \( 1 - 12.7iT - 2.41e7T^{2} \)
19 \( 1 - 5.74e3T + 4.70e7T^{2} \)
23 \( 1 + 3.37e3iT - 1.48e8T^{2} \)
29 \( 1 - 2.93e4iT - 5.94e8T^{2} \)
31 \( 1 + 3.97e4T + 8.87e8T^{2} \)
37 \( 1 - 5.25e4T + 2.56e9T^{2} \)
41 \( 1 + 3.70e4iT - 4.75e9T^{2} \)
43 \( 1 - 3.80e3T + 6.32e9T^{2} \)
47 \( 1 - 7.67e4iT - 1.07e10T^{2} \)
53 \( 1 - 2.38e5iT - 2.21e10T^{2} \)
59 \( 1 + 2.49e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.32e4T + 5.15e10T^{2} \)
67 \( 1 - 1.68e5T + 9.04e10T^{2} \)
71 \( 1 + 5.31e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.36e5T + 1.51e11T^{2} \)
79 \( 1 + 3.51e4T + 2.43e11T^{2} \)
83 \( 1 + 1.09e4iT - 3.26e11T^{2} \)
89 \( 1 + 1.29e5iT - 4.96e11T^{2} \)
97 \( 1 + 3.21e5T + 8.32e11T^{2} \)
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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

−18.06910386729150301377131706949, −16.32436931584549506497081861955, −15.37463139302735411915003575388, −14.12812289850175296566168762207, −12.79943948634922932546375385271, −10.75716920754941246673460170820, −9.404076889061244728090908165569, −7.23424034812212747387496025669, −6.19948274389871141938995126915, −3.40488705269129499046290458171, 0.77636111964462645662059400391, 3.63132619809715011724446302242, 5.80918410576232217533082322387, 8.549264004115012358299792564715, 9.630622809509253975325652294077, 11.43224405226010532435699155407, 12.91917231506903117264513196133, 13.51344595741453586559357503664, 15.96786541734827632151247851394, 16.62029485204699025951331168978

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