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Properties

Label 2-18-9.4-c3-0-2
Degree $2$
Conductor $18$
Sign $0.173 + 0.984i$
Analytic cond. $1.06203$
Root an. cond. $1.03055$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s − 5.19i·3-s + (−1.99 + 3.46i)4-s + (4.5 − 7.79i)5-s + (−9 + 5.19i)6-s + (15.5 + 26.8i)7-s + 7.99·8-s − 27·9-s − 18·10-s + (7.5 + 12.9i)11-s + (18 + 10.3i)12-s + (18.5 − 32.0i)13-s + (31 − 53.6i)14-s + (−40.5 − 23.3i)15-s + (−8 − 13.8i)16-s − 42·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s − 0.999i·3-s + (−0.249 + 0.433i)4-s + (0.402 − 0.697i)5-s + (−0.612 + 0.353i)6-s + (0.836 + 1.44i)7-s + 0.353·8-s − 9-s − 0.569·10-s + (0.205 + 0.356i)11-s + (0.433 + 0.249i)12-s + (0.394 − 0.683i)13-s + (0.591 − 1.02i)14-s + (−0.697 − 0.402i)15-s + (−0.125 − 0.216i)16-s − 0.599·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(1.06203\)
Root analytic conductor: \(1.03055\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{18} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 18,\ (\ :3/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.721181 - 0.605143i\)
\(L(\frac12)\) \(\approx\) \(0.721181 - 0.605143i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 + 5.19iT \)
good5 \( 1 + (-4.5 + 7.79i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-15.5 - 26.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-7.5 - 12.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-18.5 + 32.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 42T + 4.91e3T^{2} \)
19 \( 1 + 28T + 6.85e3T^{2} \)
23 \( 1 + (97.5 - 168. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (55.5 + 96.1i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-102.5 + 177. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 166T + 5.06e4T^{2} \)
41 \( 1 + (-130.5 + 226. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-21.5 - 37.2i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (88.5 + 153. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 114T + 1.48e5T^{2} \)
59 \( 1 + (79.5 - 137. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (95.5 + 165. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-210.5 + 364. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 156T + 3.57e5T^{2} \)
73 \( 1 - 182T + 3.89e5T^{2} \)
79 \( 1 + (566.5 + 981. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-541.5 - 937. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + (-450.5 - 780. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−17.98010892956469024606134947500, −17.32899427275944951131865274870, −15.29020789404722998881645073590, −13.54289523139430969864649595321, −12.40302537557065707422760620618, −11.42938516121310386509632186967, −9.168554738670215362145146658577, −8.062690670972476804372231461517, −5.60789669161822730948025485554, −1.94503011243440762792153454089, 4.38699503553042529087882060291, 6.60179753450900366957381205288, 8.509455018666476487755178860952, 10.25633008352779349364677513132, 11.03224542513244268239564416890, 14.00263829233119909279857834873, 14.48447618557902344058107544074, 16.13157874639066754573935098721, 17.07765926648066177146356404859, 18.14228405724161650303447802579

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