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Properties

Label 2-165-165.29-c1-0-17
Degree $2$
Conductor $165$
Sign $-0.189 + 0.981i$
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  + (1.26 − 1.74i)2-s + (1.42 − 0.990i)3-s + (−0.813 − 2.50i)4-s + (−2.23 + 0.0487i)5-s + (0.0725 − 3.72i)6-s + (1.15 + 3.55i)7-s + (−1.29 − 0.420i)8-s + (1.03 − 2.81i)9-s + (−2.74 + 3.95i)10-s + (−2.90 + 1.60i)11-s + (−3.63 − 2.75i)12-s + (−2.36 − 1.71i)13-s + (7.65 + 2.48i)14-s + (−3.12 + 2.28i)15-s + (1.88 − 1.37i)16-s + (3.60 + 4.95i)17-s + ⋯
L(s)  = 1  + (0.894 − 1.23i)2-s + (0.820 − 0.571i)3-s + (−0.406 − 1.25i)4-s + (−0.999 + 0.0218i)5-s + (0.0296 − 1.52i)6-s + (0.436 + 1.34i)7-s + (−0.458 − 0.148i)8-s + (0.345 − 0.938i)9-s + (−0.867 + 1.25i)10-s + (−0.875 + 0.483i)11-s + (−1.04 − 0.794i)12-s + (−0.655 − 0.476i)13-s + (2.04 + 0.665i)14-s + (−0.807 + 0.589i)15-s + (0.472 − 0.342i)16-s + (0.873 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.189 + 0.981i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -0.189 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22346 - 1.48272i\)
\(L(\frac12)\) \(\approx\) \(1.22346 - 1.48272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.42 + 0.990i)T \)
5 \( 1 + (2.23 - 0.0487i)T \)
11 \( 1 + (2.90 - 1.60i)T \)
good2 \( 1 + (-1.26 + 1.74i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + (-1.15 - 3.55i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.36 + 1.71i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.60 - 4.95i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.69 + 0.550i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.68T + 23T^{2} \)
29 \( 1 + (1.73 + 5.34i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.15 + 0.840i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.51 - 1.14i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.0785 + 0.241i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.64T + 43T^{2} \)
47 \( 1 + (2.93 - 9.02i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.45 + 3.23i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.0480 + 0.0156i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.70 - 2.34i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.45iT - 67T^{2} \)
71 \( 1 + (0.490 + 0.674i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.79 + 5.52i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.17 - 7.12i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.718 + 0.988i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.49iT - 89T^{2} \)
97 \( 1 + (4.47 - 6.16i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.38995245753423965895843032687, −12.07696213603951259495406138260, −10.88078686896668373869290247353, −9.720972831671004891521238313277, −8.278244278682410384569769876420, −7.67391978032256746694251233912, −5.70165305834691101435788377948, −4.36577016078431580435529076830, −3.07565868607490683327628449255, −2.07399223785916146571245880449, 3.39485231875129862409538887543, 4.37409810628212292774262921859, 5.19207999172128579764434754324, 7.24816003775984078555641828321, 7.52841462207008344754498627376, 8.526040510071635379061297215276, 10.12389490971274284424299285096, 11.05823773383930914999839766294, 12.52141316889386292438730435054, 13.62708065256300288217353821701

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