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Properties

Label 2-18-9.5-c4-0-3
Degree $2$
Conductor $18$
Sign $0.522 + 0.852i$
Analytic cond. $1.86065$
Root an. cond. $1.36405$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.44 − 1.41i)2-s + (−1.67 − 8.84i)3-s + (3.99 − 6.92i)4-s + (6.41 + 3.70i)5-s + (−16.6 − 19.2i)6-s + (30.1 + 52.2i)7-s − 22.6i·8-s + (−75.3 + 29.7i)9-s + 20.9·10-s + (55.2 − 31.8i)11-s + (−67.9 − 23.7i)12-s + (−144. + 251. i)13-s + (147. + 85.3i)14-s + (21.9 − 62.9i)15-s + (−32.0 − 55.4i)16-s − 460. i·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.186 − 0.982i)3-s + (0.249 − 0.433i)4-s + (0.256 + 0.148i)5-s + (−0.461 − 0.535i)6-s + (0.616 + 1.06i)7-s − 0.353i·8-s + (−0.930 + 0.366i)9-s + 0.209·10-s + (0.456 − 0.263i)11-s + (−0.472 − 0.164i)12-s + (−0.857 + 1.48i)13-s + (0.754 + 0.435i)14-s + (0.0976 − 0.279i)15-s + (−0.125 − 0.216i)16-s − 1.59i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(1.86065\)
Root analytic conductor: \(1.36405\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{18} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 18,\ (\ :2),\ 0.522 + 0.852i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.39965 - 0.783661i\)
\(L(\frac12)\) \(\approx\) \(1.39965 - 0.783661i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.44 + 1.41i)T \)
3 \( 1 + (1.67 + 8.84i)T \)
good5 \( 1 + (-6.41 - 3.70i)T + (312.5 + 541. i)T^{2} \)
7 \( 1 + (-30.1 - 52.2i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-55.2 + 31.8i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (144. - 251. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 460. iT - 8.35e4T^{2} \)
19 \( 1 - 187.T + 1.30e5T^{2} \)
23 \( 1 + (35.8 + 20.7i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (1.21e3 - 699. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (318. - 551. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 847.T + 1.87e6T^{2} \)
41 \( 1 + (438. + 253. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (702. + 1.21e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-2.14e3 + 1.23e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 773. iT - 7.89e6T^{2} \)
59 \( 1 + (-3.89e3 - 2.24e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-1.30e3 - 2.25e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-1.69e3 + 2.92e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 2.77e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.52e3T + 2.83e7T^{2} \)
79 \( 1 + (2.91e3 + 5.04e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (4.43e3 - 2.56e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 1.22e3iT - 6.27e7T^{2} \)
97 \( 1 + (-7.36e3 - 1.27e4i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−18.10762432180741677667091385787, −16.48741817343912861084128337361, −14.59990816109401825704119445111, −13.77847998313919257376596523736, −12.08653959156300982303442217086, −11.52426004223574450207932896268, −9.143854154715248480620945706870, −7.02472971564279392719221851465, −5.34350836507070368719130875235, −2.18552047483969715815150323590, 3.98279971393923136417242305414, 5.57339504985227318995248366895, 7.81510011025190175749563368848, 9.911498780051949482223532652971, 11.21057486391577487503924363923, 12.96988003478996458713398693831, 14.50950401562852958552056091828, 15.30500580832450508151855404656, 17.03462821531786141297448984789, 17.34527156485260331305140593160

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