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Properties

Label 2-165-55.14-c1-0-3
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: $\chi_{165} (124, \cdot )$
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

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

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