Properties

Label 2-165-55.14-c1-0-4
Degree $2$
Conductor $165$
Sign $-0.244 - 0.969i$
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  + (1.60 + 2.20i)2-s + (0.951 − 0.309i)3-s + (−1.67 + 5.16i)4-s + (−0.762 − 2.10i)5-s + (2.20 + 1.60i)6-s + (−0.462 − 0.150i)7-s + (−8.89 + 2.88i)8-s + (0.809 − 0.587i)9-s + (3.41 − 5.04i)10-s + (3.06 + 1.26i)11-s + 5.43i·12-s + (−2.12 − 2.92i)13-s + (−0.409 − 1.26i)14-s + (−1.37 − 1.76i)15-s + (−11.8 − 8.59i)16-s + (2.14 − 2.94i)17-s + ⋯
L(s)  = 1  + (1.13 + 1.55i)2-s + (0.549 − 0.178i)3-s + (−0.839 + 2.58i)4-s + (−0.341 − 0.940i)5-s + (0.900 + 0.654i)6-s + (−0.174 − 0.0568i)7-s + (−3.14 + 1.02i)8-s + (0.269 − 0.195i)9-s + (1.07 − 1.59i)10-s + (0.924 + 0.381i)11-s + 1.56i·12-s + (−0.588 − 0.810i)13-s + (−0.109 − 0.336i)14-s + (−0.354 − 0.455i)15-s + (−2.95 − 2.14i)16-s + (0.519 − 0.714i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.244 - 0.969i$
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: \((2,\ 165,\ (\ :1/2),\ -0.244 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22811 + 1.57610i\)
\(L(\frac12)\) \(\approx\) \(1.22811 + 1.57610i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.762 + 2.10i)T \)
11 \( 1 + (-3.06 - 1.26i)T \)
good2 \( 1 + (-1.60 - 2.20i)T + (-0.618 + 1.90i)T^{2} \)
7 \( 1 + (0.462 + 0.150i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.12 + 2.92i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.14 + 2.94i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.504 - 1.55i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.54iT - 23T^{2} \)
29 \( 1 + (0.326 - 1.00i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.93 - 3.58i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (8.17 + 2.65i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.39 - 10.4i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.41iT - 43T^{2} \)
47 \( 1 + (-4.95 + 1.61i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.875 - 1.20i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.46 + 7.58i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-8.78 - 6.38i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 0.432iT - 67T^{2} \)
71 \( 1 + (4.86 + 3.53i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.10 + 0.359i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.57 - 2.60i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.86 + 2.57i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.38T + 89T^{2} \)
97 \( 1 + (0.822 + 1.13i)T + (-29.9 + 92.2i)T^{2} \)
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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

−13.19634248429635139931166432430, −12.57014188896524006736149244815, −11.88500912792167472728604026226, −9.547909840061267252545814588397, −8.579760109119942542010342948374, −7.70837374817674534252915092342, −6.84525544689411057898959682661, −5.48683240060210063336659794016, −4.50286339375698091193174879131, −3.33983226179419139129071108080, 2.05522392491991835466342358524, 3.40964131545026268939047833067, 4.09181770334505156662329224096, 5.69262984928133567832015637065, 7.01729444284119238770534626987, 8.980610141757804005662008115172, 9.876758080697200926150516178174, 10.79295362222426604519025644896, 11.64891823977335240698773869715, 12.36230301355041163228267620856

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