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Properties

Label 2-18-9.4-c3-0-0
Degree $2$
Conductor $18$
Sign $0.512 - 0.858i$
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 + (−1.81 + 4.87i)3-s + (−1.99 + 3.46i)4-s + (9.93 − 17.2i)5-s + (−10.2 + 1.73i)6-s + (2.93 + 5.08i)7-s − 7.99·8-s + (−20.4 − 17.6i)9-s + 39.7·10-s + (−9.37 − 16.2i)11-s + (−13.2 − 16.0i)12-s + (−22.9 + 39.7i)13-s + (−5.87 + 10.1i)14-s + (65.8 + 79.5i)15-s + (−8 − 13.8i)16-s + 16.8·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.348 + 0.937i)3-s + (−0.249 + 0.433i)4-s + (0.888 − 1.53i)5-s + (−0.697 + 0.117i)6-s + (0.158 + 0.274i)7-s − 0.353·8-s + (−0.756 − 0.653i)9-s + 1.25·10-s + (−0.256 − 0.444i)11-s + (−0.318 − 0.385i)12-s + (−0.489 + 0.847i)13-s + (−0.112 + 0.194i)14-s + (1.13 + 1.36i)15-s + (−0.125 − 0.216i)16-s + 0.240·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $0.512 - 0.858i$
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: \((2,\ 18,\ (\ :3/2),\ 0.512 - 0.858i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.01665 + 0.577400i\)
\(L(\frac12)\) \(\approx\) \(1.01665 + 0.577400i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.81 - 4.87i)T \)
good5 \( 1 + (-9.93 + 17.2i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-2.93 - 5.08i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (9.37 + 16.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (22.9 - 39.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 16.8T + 4.91e3T^{2} \)
19 \( 1 + 10.3T + 6.85e3T^{2} \)
23 \( 1 + (24.9 - 43.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (5.45 + 9.44i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (75.8 - 131. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 346.T + 5.06e4T^{2} \)
41 \( 1 + (132. - 229. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-205. - 356. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (236. + 408. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 290.T + 1.48e5T^{2} \)
59 \( 1 + (26.6 - 46.1i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-146. - 254. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-199. + 345. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 647.T + 3.57e5T^{2} \)
73 \( 1 + 478.T + 3.89e5T^{2} \)
79 \( 1 + (187. + 324. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (466. + 808. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 368.T + 7.04e5T^{2} \)
97 \( 1 + (137. + 237. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−17.87140107662534098706355907882, −16.71630551852798337783752917682, −16.22742338922348381242242736622, −14.63683679109220782921056131412, −13.25250709963598213379394897253, −11.84939578941188341887547656442, −9.703925142424272870655477537562, −8.632252995053353520206374177106, −5.81611137140904375916041931556, −4.67515235124625793810476954388, 2.49126116116257301076434845504, 5.85119623212478453380424068911, 7.35951559821777938586083022979, 10.11058409929588419318488500924, 11.12470557929548630851373349131, 12.71408902175183919861051798423, 13.89662624150097310414775033587, 14.86360813757324261361322118031, 17.34368284064117606481042047552, 18.17631068870080158431366727726

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