Properties

Label 2-18-9.2-c2-0-0
Degree $2$
Conductor $18$
Sign $0.870 + 0.491i$
Analytic cond. $0.490464$
Root an. cond. $0.700331$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (2.44 − 1.73i)3-s + (0.999 + 1.73i)4-s + (−4.5 + 2.59i)5-s + (−4.22 + 0.389i)6-s + (−3.17 + 5.49i)7-s − 2.82i·8-s + (2.99 − 8.48i)9-s + 7.34·10-s + (8.17 + 4.71i)11-s + (5.44 + 2.51i)12-s + (−9.84 − 17.0i)13-s + (7.77 − 4.48i)14-s + (−6.52 + 14.1i)15-s + (−2.00 + 3.46i)16-s − 1.90i·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.816 − 0.577i)3-s + (0.249 + 0.433i)4-s + (−0.900 + 0.519i)5-s + (−0.704 + 0.0648i)6-s + (−0.453 + 0.785i)7-s − 0.353i·8-s + (0.333 − 0.942i)9-s + 0.734·10-s + (0.743 + 0.429i)11-s + (0.454 + 0.209i)12-s + (−0.757 − 1.31i)13-s + (0.555 − 0.320i)14-s + (−0.434 + 0.943i)15-s + (−0.125 + 0.216i)16-s − 0.112i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $0.870 + 0.491i$
Analytic conductor: \(0.490464\)
Root analytic conductor: \(0.700331\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{18} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 18,\ (\ :1),\ 0.870 + 0.491i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.695591 - 0.182947i\)
\(L(\frac12)\) \(\approx\) \(0.695591 - 0.182947i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (-2.44 + 1.73i)T \)
good5 \( 1 + (4.5 - 2.59i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (3.17 - 5.49i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-8.17 - 4.71i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (9.84 + 17.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 1.90iT - 289T^{2} \)
19 \( 1 - 4.69T + 361T^{2} \)
23 \( 1 + (-8.17 + 4.71i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (2.84 + 1.64i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-20.5 - 35.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 17.3T + 1.36e3T^{2} \)
41 \( 1 + (53.5 - 30.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (0.477 - 0.826i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (12.2 + 7.05i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 9.53iT - 2.80e3T^{2} \)
59 \( 1 + (-79.2 + 45.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-37.5 + 65.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (15.4 + 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 85.9iT - 5.04e3T^{2} \)
73 \( 1 + 96.0T + 5.32e3T^{2} \)
79 \( 1 + (14.8 - 25.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (76.1 + 43.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 41.3iT - 7.92e3T^{2} \)
97 \( 1 + (47.9 - 83.0i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−18.73470949071888080555396036025, −17.57547673562482447420443834407, −15.62508536121149116438871026816, −14.73764241489512448276768414001, −12.77057074571954182052727572579, −11.77847013109460800075356973662, −9.812433465428032710850995173630, −8.333607289168685714296216901136, −7.01625851001993022562047922659, −3.08266487875230505300547482247, 4.16831666093679399606816968694, 7.22373635247377566575366078629, 8.678128793745155372043192588380, 9.880974393067798906745917632475, 11.62936365107692201197538088819, 13.68761001830931743560191895933, 14.94911498001339251343967744904, 16.26555310766820021965715127173, 16.87382936778583479900571953794, 19.13739471986139193363604939785

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