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Properties

Label 2-165-11.3-c1-0-1
Degree $2$
Conductor $165$
Sign $-0.932 - 0.361i$
Analytic cond. $1.31753$
Root an. cond. $1.14783$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.24 + 1.62i)2-s + (0.309 + 0.951i)3-s + (1.75 − 5.40i)4-s + (0.809 + 0.587i)5-s + (−2.24 − 1.62i)6-s + (−0.703 + 2.16i)7-s + (3.15 + 9.71i)8-s + (−0.809 + 0.587i)9-s − 2.77·10-s + (−0.105 + 3.31i)11-s + 5.68·12-s + (0.352 − 0.256i)13-s + (−1.95 − 6.00i)14-s + (−0.309 + 0.951i)15-s + (−13.7 − 9.96i)16-s + (−4.04 − 2.93i)17-s + ⋯
L(s)  = 1  + (−1.58 + 1.15i)2-s + (0.178 + 0.549i)3-s + (0.878 − 2.70i)4-s + (0.361 + 0.262i)5-s + (−0.915 − 0.665i)6-s + (−0.266 + 0.818i)7-s + (1.11 + 3.43i)8-s + (−0.269 + 0.195i)9-s − 0.876·10-s + (−0.0317 + 0.999i)11-s + 1.64·12-s + (0.0977 − 0.0710i)13-s + (−0.521 − 1.60i)14-s + (−0.0797 + 0.245i)15-s + (−3.42 − 2.49i)16-s + (−0.981 − 0.712i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.932 - 0.361i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -0.932 - 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0979272 + 0.523778i\)
\(L(\frac12)\) \(\approx\) \(0.0979272 + 0.523778i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.105 - 3.31i)T \)
good2 \( 1 + (2.24 - 1.62i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.703 - 2.16i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.352 + 0.256i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.04 + 2.93i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.45 - 4.46i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.845T + 23T^{2} \)
29 \( 1 + (0.821 - 2.52i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.77 + 2.74i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.73 - 8.42i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.32 + 4.08i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.00T + 43T^{2} \)
47 \( 1 + (-0.144 - 0.445i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.76 + 6.37i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.21 + 3.74i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.39 + 1.74i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.47T + 67T^{2} \)
71 \( 1 + (-9.15 - 6.65i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.60 + 8.01i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.79 + 6.38i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.78 - 3.47i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 5.89T + 89T^{2} \)
97 \( 1 + (-6.99 + 5.08i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.74813255529706748226194519448, −11.91612964005691303372341225200, −10.68822101661992780126595660904, −9.833075683948293297088460240954, −9.238326912381604711445034960645, −8.284632058054414737701448503395, −7.13420338911303083409731327497, −6.11967595849196432161909211041, −5.03911894547221110663831418261, −2.20666047878598324043916916328, 0.827252526430782166359971287934, 2.44595016318926577257107142346, 3.85795702243843454333148259591, 6.51047727975939996899784930649, 7.54536629916821936620996707519, 8.603294255099647895724311855253, 9.256542944942617526823851032902, 10.47656006993316064903600055072, 11.09188293451182925654106702669, 12.14217409931930010657306826706

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