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Properties

Label 2-18-9.5-c4-0-1
Degree $2$
Conductor $18$
Sign $0.998 + 0.0448i$
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 + (5.62 + 7.02i)3-s + (3.99 − 6.92i)4-s + (−1.91 − 1.10i)5-s + (23.7 + 9.24i)6-s + (−21.9 − 38.0i)7-s − 22.6i·8-s + (−17.6 + 79.0i)9-s − 6.25·10-s + (−183. + 106. i)11-s + (71.1 − 10.9i)12-s + (88.7 − 153. i)13-s + (−107. − 62.2i)14-s + (−3.01 − 19.6i)15-s + (−32.0 − 55.4i)16-s + 43.3i·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.625 + 0.780i)3-s + (0.249 − 0.433i)4-s + (−0.0765 − 0.0441i)5-s + (0.658 + 0.256i)6-s + (−0.448 − 0.777i)7-s − 0.353i·8-s + (−0.217 + 0.976i)9-s − 0.0625·10-s + (−1.51 + 0.876i)11-s + (0.494 − 0.0757i)12-s + (0.524 − 0.909i)13-s + (−0.549 − 0.317i)14-s + (−0.0133 − 0.0873i)15-s + (−0.125 − 0.216i)16-s + 0.150i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0448i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.998 + 0.0448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $0.998 + 0.0448i$
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: \((2,\ 18,\ (\ :2),\ 0.998 + 0.0448i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.77608 - 0.0398238i\)
\(L(\frac12)\) \(\approx\) \(1.77608 - 0.0398238i\)
\(L(3)\) 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 \( 1 + (-5.62 - 7.02i)T \)
good5 \( 1 + (1.91 + 1.10i)T + (312.5 + 541. i)T^{2} \)
7 \( 1 + (21.9 + 38.0i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (183. - 106. i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-88.7 + 153. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 43.3iT - 8.35e4T^{2} \)
19 \( 1 - 528.T + 1.30e5T^{2} \)
23 \( 1 + (-517. - 298. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-380. + 219. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (756. - 1.31e3i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 702.T + 1.87e6T^{2} \)
41 \( 1 + (-517. - 298. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-46.1 - 79.8i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.49e3 - 860. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 1.31e3iT - 7.89e6T^{2} \)
59 \( 1 + (3.41e3 + 1.97e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (2.11e3 + 3.65e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (322. - 558. i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 6.39e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.19e3T + 2.83e7T^{2} \)
79 \( 1 + (-4.78e3 - 8.28e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (3.07e3 - 1.77e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 502. iT - 6.27e7T^{2} \)
97 \( 1 + (-3.71e3 - 6.43e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−18.03354238777314867551601575610, −16.11906354419166925321343171757, −15.35199319076354101451217221949, −13.85707981248762826971788753824, −12.86182398509986029557546278956, −10.77848574187079351204045126078, −9.836628985114543156199026892568, −7.69791458295153657662761402014, −5.05410374698400068331021020299, −3.20693269571621856355343984246, 2.95719226246400729747482457102, 5.81798189516294971838275586069, 7.52917689240083198541965448688, 9.015973749489160047282688416547, 11.50529636784761245336533749207, 12.94900031385885486675941285860, 13.78817632927448843693064853889, 15.24595350805086809090703566400, 16.31995745173276146448380591929, 18.29518690434008981800781789011

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