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Properties

Label 2-165-33.8-c1-0-6
Degree $2$
Conductor $165$
Sign $0.00247 - 0.999i$
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  + (−1.37 + 0.997i)2-s + (1.42 + 0.979i)3-s + (0.271 − 0.834i)4-s + (0.587 − 0.809i)5-s + (−2.93 + 0.0801i)6-s + (3.82 + 1.24i)7-s + (−0.588 − 1.81i)8-s + (1.08 + 2.79i)9-s + 1.69i·10-s + (−0.328 + 3.30i)11-s + (1.20 − 0.926i)12-s + (−3.28 − 4.51i)13-s + (−6.49 + 2.11i)14-s + (1.63 − 0.579i)15-s + (4.03 + 2.93i)16-s + (−0.766 − 0.556i)17-s + ⋯
L(s)  = 1  + (−0.970 + 0.705i)2-s + (0.824 + 0.565i)3-s + (0.135 − 0.417i)4-s + (0.262 − 0.361i)5-s + (−1.19 + 0.0327i)6-s + (1.44 + 0.470i)7-s + (−0.208 − 0.640i)8-s + (0.360 + 0.932i)9-s + 0.536i·10-s + (−0.0991 + 0.995i)11-s + (0.347 − 0.267i)12-s + (−0.910 − 1.25i)13-s + (−1.73 + 0.564i)14-s + (0.421 − 0.149i)15-s + (1.00 + 0.732i)16-s + (−0.185 − 0.135i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00247 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00247 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.00247 - 0.999i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.00247 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.704027 + 0.702289i\)
\(L(\frac12)\) \(\approx\) \(0.704027 + 0.702289i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.42 - 0.979i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (0.328 - 3.30i)T \)
good2 \( 1 + (1.37 - 0.997i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-3.82 - 1.24i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.28 + 4.51i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.766 + 0.556i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.31 - 1.40i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.88iT - 23T^{2} \)
29 \( 1 + (-1.65 + 5.10i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.46 + 1.06i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.46 + 7.59i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.413 - 1.27i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.82iT - 43T^{2} \)
47 \( 1 + (-7.05 + 2.29i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.57 + 7.67i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.84 - 0.600i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.71 - 10.6i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 1.99T + 67T^{2} \)
71 \( 1 + (1.44 - 1.98i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-9.21 - 2.99i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.25 + 1.72i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.50 + 1.81i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 11.0iT - 89T^{2} \)
97 \( 1 + (6.43 - 4.67i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.12379860165792066014031959598, −12.15846767113110172974704019543, −10.57005651235657864322290402080, −9.777803602898910340425773674503, −8.825783953627044691405737711816, −8.026480128119469815757840054428, −7.42098254580477052680011469817, −5.47231658209144838214255172741, −4.32495979476547415458479444121, −2.20669950212459837514758541792, 1.48776229612223101560607450098, 2.64710322113084138919064933630, 4.61367376732657197113713889863, 6.52660359323955465292038306405, 7.81472271451469230348929364041, 8.563387778389694767666557906298, 9.410328169472685156421700802042, 10.64850828983186801862644366041, 11.29747261392553583162661670642, 12.33554254388577898909009625902

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