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Properties

Label 2-165-55.14-c1-0-8
Degree $2$
Conductor $165$
Sign $0.388 + 0.921i$
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 + (2.23 + 0.0529i)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 + (2.14 − 0.640i)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 + (0.999 + 0.0236i)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 + (0.553 − 0.165i)15-s + (−0.139 − 0.101i)16-s + (0.241 − 0.332i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.388 + 0.921i$
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: \((2,\ 165,\ (\ :1/2),\ 0.388 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06101 - 0.703796i\)
\(L(\frac12)\) \(\approx\) \(1.06101 - 0.703796i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.0529i)T \)
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

−12.79620834336701703998569834923, −11.52150379550333391288134284633, −10.38137888423307244922202452508, −9.572003260476750002010087859558, −9.095898055038949846804474177857, −7.29869244417575322988715789591, −6.33098019926684836371674082469, −5.03497728175245110321192514798, −2.95122039322562884942638140900, −1.66640248454065349936614397650, 2.49769725787335341054600188033, 3.79335157854029511334633783471, 5.82631538329772523294190820066, 6.63849663980596490254642506194, 8.067204108119645702625966354156, 8.824901996406899760090162751739, 9.738421717086263311366202294367, 10.81099853154687021935751391078, 12.30968368242640403727158934632, 12.99509472286470345369396612421

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