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Properties

Label 2-165-165.29-c1-0-15
Degree $2$
Conductor $165$
Sign $0.380 + 0.924i$
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  + (0.303 − 0.417i)2-s + (0.725 − 1.57i)3-s + (0.535 + 1.64i)4-s + (−0.723 − 2.11i)5-s + (−0.436 − 0.779i)6-s + (−0.781 − 2.40i)7-s + (1.83 + 0.595i)8-s + (−1.94 − 2.28i)9-s + (−1.10 − 0.339i)10-s + (0.813 + 3.21i)11-s + (2.98 + 0.353i)12-s + (4.13 + 3.00i)13-s + (−1.24 − 0.403i)14-s + (−3.85 − 0.396i)15-s + (−2.00 + 1.45i)16-s + (−2.17 − 3.00i)17-s + ⋯
L(s)  = 1  + (0.214 − 0.295i)2-s + (0.418 − 0.908i)3-s + (0.267 + 0.824i)4-s + (−0.323 − 0.946i)5-s + (−0.178 − 0.318i)6-s + (−0.295 − 0.909i)7-s + (0.647 + 0.210i)8-s + (−0.648 − 0.760i)9-s + (−0.348 − 0.107i)10-s + (0.245 + 0.969i)11-s + (0.860 + 0.102i)12-s + (1.14 + 0.833i)13-s + (−0.331 − 0.107i)14-s + (−0.994 − 0.102i)15-s + (−0.500 + 0.363i)16-s + (−0.528 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.380 + 0.924i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.380 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18471 - 0.794002i\)
\(L(\frac12)\) \(\approx\) \(1.18471 - 0.794002i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.725 + 1.57i)T \)
5 \( 1 + (0.723 + 2.11i)T \)
11 \( 1 + (-0.813 - 3.21i)T \)
good2 \( 1 + (-0.303 + 0.417i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.781 + 2.40i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.13 - 3.00i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.17 + 3.00i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.97 - 0.642i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.64T + 23T^{2} \)
29 \( 1 + (-1.79 - 5.53i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.64 - 3.37i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (8.16 - 2.65i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.856 + 2.63i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.05T + 43T^{2} \)
47 \( 1 + (1.10 - 3.38i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.04 + 3.66i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.98 - 1.29i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.38 + 6.02i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 11.2iT - 67T^{2} \)
71 \( 1 + (0.927 + 1.27i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.17 + 6.69i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.759 - 1.04i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.83 - 9.41i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 6.15iT - 89T^{2} \)
97 \( 1 + (-1.61 + 2.22i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.48311207304164579091721278825, −12.08183099566902096261849178267, −10.99094910461074089506912819531, −9.329277950890826855041289742019, −8.419428781153385011169137433688, −7.40360199567080769652323273731, −6.64468670803613138011822392500, −4.53796484915304081916937396620, −3.46572399274216233193219396480, −1.60622831983221852908910366941, 2.70168225419824591570418203146, 3.97459047496944374057160333776, 5.71304187241195987937143256031, 6.24285869187666481066500281886, 7.962378720996683941845655054644, 8.985512234964016265917256865443, 10.19244644416625089777448801351, 10.85436077810552994215578696032, 11.66947554494121369374919761226, 13.46660657420158016770618742212

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