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Properties

Label 2-165-33.29-c1-0-2
Degree $2$
Conductor $165$
Sign $0.854 + 0.520i$
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  + (−1.45 − 1.05i)2-s + (−1.62 + 0.600i)3-s + (0.383 + 1.17i)4-s + (0.587 + 0.809i)5-s + (3.00 + 0.843i)6-s + (−0.145 + 0.0472i)7-s + (−0.422 + 1.30i)8-s + (2.27 − 1.95i)9-s − 1.80i·10-s + (3.24 + 0.670i)11-s + (−1.33 − 1.68i)12-s + (1.93 − 2.66i)13-s + (0.261 + 0.0850i)14-s + (−1.44 − 0.960i)15-s + (3.99 − 2.90i)16-s + (4.27 − 3.10i)17-s + ⋯
L(s)  = 1  + (−1.02 − 0.748i)2-s + (−0.937 + 0.346i)3-s + (0.191 + 0.589i)4-s + (0.262 + 0.361i)5-s + (1.22 + 0.344i)6-s + (−0.0549 + 0.0178i)7-s + (−0.149 + 0.459i)8-s + (0.759 − 0.650i)9-s − 0.569i·10-s + (0.979 + 0.202i)11-s + (−0.384 − 0.486i)12-s + (0.536 − 0.738i)13-s + (0.0699 + 0.0227i)14-s + (−0.372 − 0.248i)15-s + (0.999 − 0.726i)16-s + (1.03 − 0.754i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.854 + 0.520i$
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: \((2,\ 165,\ (\ :1/2),\ 0.854 + 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.543383 - 0.152464i\)
\(L(\frac12)\) \(\approx\) \(0.543383 - 0.152464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.62 - 0.600i)T \)
5 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (-3.24 - 0.670i)T \)
good2 \( 1 + (1.45 + 1.05i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (0.145 - 0.0472i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.93 + 2.66i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.27 + 3.10i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.650 - 0.211i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.40iT - 23T^{2} \)
29 \( 1 + (0.713 + 2.19i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-6.14 - 4.46i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.02 - 6.22i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.545 - 1.67i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.03iT - 43T^{2} \)
47 \( 1 + (-3.53 - 1.14i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.80 + 10.7i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (8.84 - 2.87i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.87 + 3.95i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + (1.79 + 2.47i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-8.83 + 2.87i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.524 - 0.721i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.29 + 6.02i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 4.99iT - 89T^{2} \)
97 \( 1 + (-6.77 - 4.92i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.07195648340648499670504621007, −11.64418363954520421707231815811, −10.57642819411819977688401524283, −9.911330868741377885186156186970, −9.139036670485148114479021897051, −7.69352217080847132044062730114, −6.30423181655683073489853690077, −5.19706709924409155456777937990, −3.33928338231322447369075281495, −1.20817586576420834056495233590, 1.18051907841372281236199724953, 4.18149626310075462970411829481, 5.92734545458257296885851034271, 6.54714941848899117949041095839, 7.70804231049442093761865079894, 8.761513064113872031830581562285, 9.719335001955597430368914196690, 10.74169849188196887323700646398, 11.95518502558513003010163752459, 12.70349174514211076257861817261

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