Properties

Label 2-165-33.2-c1-0-13
Degree $2$
Conductor $165$
Sign $-0.420 + 0.907i$
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  + (0.796 − 2.45i)2-s + (1.45 + 0.945i)3-s + (−3.76 − 2.73i)4-s + (−0.951 + 0.309i)5-s + (3.47 − 2.80i)6-s + (2.05 − 2.82i)7-s + (−5.52 + 4.01i)8-s + (1.21 + 2.74i)9-s + 2.57i·10-s + (−3.28 − 0.490i)11-s + (−2.87 − 7.52i)12-s + (1.33 + 0.433i)13-s + (−5.29 − 7.29i)14-s + (−1.67 − 0.451i)15-s + (2.57 + 7.91i)16-s + (2.35 + 7.24i)17-s + ⋯
L(s)  = 1  + (0.563 − 1.73i)2-s + (0.837 + 0.546i)3-s + (−1.88 − 1.36i)4-s + (−0.425 + 0.138i)5-s + (1.41 − 1.14i)6-s + (0.776 − 1.06i)7-s + (−1.95 + 1.41i)8-s + (0.403 + 0.914i)9-s + 0.815i·10-s + (−0.989 − 0.147i)11-s + (−0.829 − 2.17i)12-s + (0.370 + 0.120i)13-s + (−1.41 − 1.94i)14-s + (−0.431 − 0.116i)15-s + (0.642 + 1.97i)16-s + (0.570 + 1.75i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.420 + 0.907i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -0.420 + 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.892701 - 1.39737i\)
\(L(\frac12)\) \(\approx\) \(0.892701 - 1.39737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.45 - 0.945i)T \)
5 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (3.28 + 0.490i)T \)
good2 \( 1 + (-0.796 + 2.45i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-2.05 + 2.82i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.33 - 0.433i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.35 - 7.24i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.49 - 2.05i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 0.0193iT - 23T^{2} \)
29 \( 1 + (1.69 + 1.23i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.712 - 2.19i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.92 + 4.30i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.92 + 1.39i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.48iT - 43T^{2} \)
47 \( 1 + (4.14 + 5.70i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.30 + 1.39i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.94 + 8.17i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.32 + 0.756i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.98T + 67T^{2} \)
71 \( 1 + (-5.33 + 1.73i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.35 - 7.37i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.17 + 0.705i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.89 + 5.84i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 12.7iT - 89T^{2} \)
97 \( 1 + (4.05 - 12.4i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.59165538816821834222758358879, −11.26852677998563384984491263365, −10.54635028699818952898337860884, −10.10445748887254519721454307232, −8.620748751356656900795615872425, −7.73242150663878723671625613445, −5.26816871482188203618548319801, −4.08765750663272578021374613687, −3.42828801395989834842869558572, −1.76766159959968236382623179419, 2.99155262470807654100179859505, 4.73689356063384727636881309688, 5.63043272634899945633773075012, 7.08094749393097511352860657588, 7.81463505411510745192498990202, 8.535685035937065890335842527827, 9.426736900983338373816720909431, 11.63916811794081124238586922012, 12.59794008678411470075410831556, 13.49454531551592652510690449128

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