Properties

Label 2-165-33.17-c1-0-9
Degree $2$
Conductor $165$
Sign $0.882 - 0.470i$
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  + (0.491 + 1.51i)2-s + (0.957 − 1.44i)3-s + (−0.425 + 0.309i)4-s + (0.951 + 0.309i)5-s + (2.65 + 0.737i)6-s + (−1.78 − 2.46i)7-s + (1.89 + 1.37i)8-s + (−1.16 − 2.76i)9-s + 1.58i·10-s + (−0.668 + 3.24i)11-s + (0.0390 + 0.910i)12-s + (−1.51 + 0.491i)13-s + (2.84 − 3.91i)14-s + (1.35 − 1.07i)15-s + (−1.47 + 4.54i)16-s + (−0.898 + 2.76i)17-s + ⋯
L(s)  = 1  + (0.347 + 1.06i)2-s + (0.552 − 0.833i)3-s + (−0.212 + 0.154i)4-s + (0.425 + 0.138i)5-s + (1.08 + 0.301i)6-s + (−0.676 − 0.930i)7-s + (0.670 + 0.486i)8-s + (−0.389 − 0.921i)9-s + 0.502i·10-s + (−0.201 + 0.979i)11-s + (0.0112 + 0.262i)12-s + (−0.419 + 0.136i)13-s + (0.759 − 1.04i)14-s + (0.350 − 0.278i)15-s + (−0.368 + 1.13i)16-s + (−0.217 + 0.670i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.882 - 0.470i$
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: \((2,\ 165,\ (\ :1/2),\ 0.882 - 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59101 + 0.397883i\)
\(L(\frac12)\) \(\approx\) \(1.59101 + 0.397883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.957 + 1.44i)T \)
5 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (0.668 - 3.24i)T \)
good2 \( 1 + (-0.491 - 1.51i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (1.78 + 2.46i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.51 - 0.491i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.898 - 2.76i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.39 - 1.91i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.51iT - 23T^{2} \)
29 \( 1 + (4.81 - 3.49i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.0658 - 0.202i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.92 + 2.85i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.89 - 1.37i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 11.8iT - 43T^{2} \)
47 \( 1 + (-7.63 + 10.5i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (11.2 - 3.65i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.196 + 0.271i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-11.0 - 3.59i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 8.55T + 67T^{2} \)
71 \( 1 + (-8.16 - 2.65i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.13 - 8.44i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (8.85 - 2.87i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.49 - 7.69i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 9.73iT - 89T^{2} \)
97 \( 1 + (3.23 + 9.96i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.12989661133930968777017500385, −12.45027656279046941264864986108, −10.77660004610282566344555419733, −9.790478555994996042126267842314, −8.432468343315371670032533931565, −7.21879602185987282383506906030, −6.87409574654196569275601040333, −5.70947522387440982904239490169, −4.05962747579718588145356000510, −2.09160110100981263594499413293, 2.45739212119443344772299854225, 3.25464309232416784598367294845, 4.71878019377759827517307938504, 5.99632641390460631747406499154, 7.80337788090523016906698334237, 9.212480113307875747066169089294, 9.682696838614219939417125970901, 10.89294369072503956368467360271, 11.59031304587187491210849931054, 12.84955401966841791127402078751

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