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Properties

Label 2-165-55.4-c1-0-8
Degree $2$
Conductor $165$
Sign $0.724 + 0.689i$
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  + (0.460 − 0.634i)2-s + (−0.951 − 0.309i)3-s + (0.428 + 1.31i)4-s + (0.741 − 2.10i)5-s + (−0.634 + 0.460i)6-s + (1.44 − 0.469i)7-s + (2.52 + 0.820i)8-s + (0.809 + 0.587i)9-s + (−0.996 − 1.44i)10-s + (0.438 − 3.28i)11-s − 1.38i·12-s + (0.274 − 0.377i)13-s + (0.367 − 1.13i)14-s + (−1.35 + 1.77i)15-s + (−0.557 + 0.404i)16-s + (−0.997 − 1.37i)17-s + ⋯
L(s)  = 1  + (0.325 − 0.448i)2-s + (−0.549 − 0.178i)3-s + (0.214 + 0.658i)4-s + (0.331 − 0.943i)5-s + (−0.258 + 0.188i)6-s + (0.545 − 0.177i)7-s + (0.892 + 0.290i)8-s + (0.269 + 0.195i)9-s + (−0.315 − 0.456i)10-s + (0.132 − 0.991i)11-s − 0.399i·12-s + (0.0760 − 0.104i)13-s + (0.0983 − 0.302i)14-s + (−0.350 + 0.458i)15-s + (−0.139 + 0.101i)16-s + (−0.241 − 0.332i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.724 + 0.689i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.724 + 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23284 - 0.493102i\)
\(L(\frac12)\) \(\approx\) \(1.23284 - 0.493102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (-0.741 + 2.10i)T \)
11 \( 1 + (-0.438 + 3.28i)T \)
good2 \( 1 + (-0.460 + 0.634i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + (-1.44 + 0.469i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.274 + 0.377i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.997 + 1.37i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.73 - 5.33i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 7.68iT - 23T^{2} \)
29 \( 1 + (0.822 + 2.53i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.20 + 2.32i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.90 - 0.945i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.417 - 1.28i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 12.0iT - 43T^{2} \)
47 \( 1 + (2.43 + 0.790i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.71 + 6.48i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.33 + 7.17i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.99 + 6.53i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.89iT - 67T^{2} \)
71 \( 1 + (11.8 - 8.57i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.15 + 0.700i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-9.85 - 7.15i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.92 + 9.53i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.47T + 89T^{2} \)
97 \( 1 + (-8.34 + 11.4i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.73087633294727233395197325862, −11.63571207005427387154330633466, −11.20041569044268229540096222692, −9.815840324750533271657400111118, −8.436723887489275002148173519063, −7.67029897327671532146793601232, −6.06659023496831213801315056283, −4.93977544082021170900659948028, −3.67504927995126003392606007484, −1.65870073312474134155774340914, 2.11629171911999000082593927720, 4.38831411813799113314267132103, 5.43517158742059354128354543880, 6.62360668025870524935221775371, 7.17696593286640918584765133840, 8.988249069267406116647517277433, 10.35570687313991224163468630740, 10.71429204133614933811837737352, 11.82386065762537803830064571936, 13.08248060755030808855363196146

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