Properties

Label 2-165-33.17-c1-0-14
Degree $2$
Conductor $165$
Sign $-0.255 + 0.966i$
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  + (−0.0403 − 0.124i)2-s + (−0.644 − 1.60i)3-s + (1.60 − 1.16i)4-s + (−0.951 − 0.309i)5-s + (−0.173 + 0.144i)6-s + (−1.50 − 2.06i)7-s + (−0.420 − 0.305i)8-s + (−2.17 + 2.07i)9-s + 0.130i·10-s + (−1.12 + 3.11i)11-s + (−2.90 − 1.82i)12-s + (4.12 − 1.34i)13-s + (−0.196 + 0.270i)14-s + (0.115 + 1.72i)15-s + (1.20 − 3.70i)16-s + (1.85 − 5.72i)17-s + ⋯
L(s)  = 1  + (−0.0285 − 0.0877i)2-s + (−0.371 − 0.928i)3-s + (0.802 − 0.582i)4-s + (−0.425 − 0.138i)5-s + (−0.0708 + 0.0591i)6-s + (−0.568 − 0.782i)7-s + (−0.148 − 0.108i)8-s + (−0.723 + 0.690i)9-s + 0.0412i·10-s + (−0.339 + 0.940i)11-s + (−0.839 − 0.527i)12-s + (1.14 − 0.372i)13-s + (−0.0524 + 0.0721i)14-s + (0.0298 + 0.446i)15-s + (0.301 − 0.926i)16-s + (0.450 − 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.255 + 0.966i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -0.255 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.626823 - 0.813631i\)
\(L(\frac12)\) \(\approx\) \(0.626823 - 0.813631i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.644 + 1.60i)T \)
5 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (1.12 - 3.11i)T \)
good2 \( 1 + (0.0403 + 0.124i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (1.50 + 2.06i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-4.12 + 1.34i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.85 + 5.72i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.284 - 0.391i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.22iT - 23T^{2} \)
29 \( 1 + (-8.15 + 5.92i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.89 - 8.89i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.66 - 3.38i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.66 - 2.65i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.35iT - 43T^{2} \)
47 \( 1 + (-0.504 + 0.694i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (8.89 - 2.89i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (4.80 + 6.60i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.13 - 1.34i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 4.84T + 67T^{2} \)
71 \( 1 + (-1.92 - 0.624i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.412 + 0.568i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (9.67 - 3.14i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.211 + 0.651i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 5.26iT - 89T^{2} \)
97 \( 1 + (-2.30 - 7.09i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.38411436994345792639489815859, −11.63609838614101884345018146374, −10.68008411034192773495541790336, −9.784860400254029562634438214296, −8.056092509865324151125889148080, −7.12778701407635021852245883283, −6.38531616194728641151087598608, −5.03831737265339184513317633155, −2.99472458683999466010496882677, −1.11883348560213773681356480926, 2.96101732578448492284793514134, 3.95302861415973517734110980165, 5.84002164302305650068706625956, 6.46231282663930129636774132632, 8.243749512311644451308329093845, 8.843081065030335313972715331585, 10.44599425652963757381900371510, 11.04958701514118590580325051369, 12.00321862407076621642221074568, 12.79217586003656048103877589400

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