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Properties

Label 2-165-33.29-c1-0-0
Degree $2$
Conductor $165$
Sign $0.0505 - 0.998i$
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  + (−1.99 − 1.44i)2-s + (−0.0877 − 1.72i)3-s + (1.25 + 3.86i)4-s + (0.587 + 0.809i)5-s + (−2.32 + 3.57i)6-s + (−4.90 + 1.59i)7-s + (1.57 − 4.84i)8-s + (−2.98 + 0.303i)9-s − 2.46i·10-s + (−2.74 + 1.86i)11-s + (6.58 − 2.51i)12-s + (−0.329 + 0.453i)13-s + (12.0 + 3.92i)14-s + (1.34 − 1.08i)15-s + (−3.56 + 2.59i)16-s + (−0.0267 + 0.0194i)17-s + ⋯
L(s)  = 1  + (−1.40 − 1.02i)2-s + (−0.0506 − 0.998i)3-s + (0.628 + 1.93i)4-s + (0.262 + 0.361i)5-s + (−0.951 + 1.45i)6-s + (−1.85 + 0.602i)7-s + (0.556 − 1.71i)8-s + (−0.994 + 0.101i)9-s − 0.778i·10-s + (−0.827 + 0.561i)11-s + (1.89 − 0.725i)12-s + (−0.0914 + 0.125i)13-s + (3.23 + 1.04i)14-s + (0.348 − 0.280i)15-s + (−0.891 + 0.647i)16-s + (−0.00648 + 0.00471i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0505 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0505 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.0505 - 0.998i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.0505 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0285193 + 0.0271128i\)
\(L(\frac12)\) \(\approx\) \(0.0285193 + 0.0271128i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0877 + 1.72i)T \)
5 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (2.74 - 1.86i)T \)
good2 \( 1 + (1.99 + 1.44i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (4.90 - 1.59i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.329 - 0.453i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.0267 - 0.0194i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.705 + 0.229i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.72iT - 23T^{2} \)
29 \( 1 + (1.76 + 5.44i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.85 + 3.52i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.05 - 3.24i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.702 - 2.16i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.71iT - 43T^{2} \)
47 \( 1 + (5.66 + 1.84i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.24 - 4.46i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.63 - 1.17i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.95 + 4.06i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.72T + 67T^{2} \)
71 \( 1 + (-2.27 - 3.13i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.92 + 1.59i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.03 - 8.30i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.20 + 6.68i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 11.6iT - 89T^{2} \)
97 \( 1 + (-0.610 - 0.443i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.75465229005262311479492110387, −12.07141198223835746649490044053, −10.96167818267295682768170406110, −9.901249798010885843290132844360, −9.289705791504791724404814304554, −8.098961162656231463307719878064, −7.05030411342903994952618803595, −6.03141508255973603402211693039, −3.08202373409870914601732816303, −2.24306836202224010328912776740, 0.05343983610238181131692514642, 3.39105022381709468722749694996, 5.39634034201772424278337975837, 6.30434725264805321778671101768, 7.45763609523354160500405279264, 8.780364382119943673482362215994, 9.423948986368795383721398842499, 10.22291474390381173721520494451, 10.82439443340059798212577037512, 12.70520087957353144469365021480

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