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Properties

Label 2-165-33.8-c1-0-13
Degree $2$
Conductor $165$
Sign $0.703 + 0.710i$
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 + (1.08 + 1.34i)3-s + (1.25 − 3.86i)4-s + (−0.587 + 0.809i)5-s + (4.11 + 1.11i)6-s + (−4.90 − 1.59i)7-s + (−1.57 − 4.84i)8-s + (−0.633 + 2.93i)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 + (−1.72 + 0.0877i)15-s + (−3.56 − 2.59i)16-s + (0.0267 + 0.0194i)17-s + ⋯
L(s)  = 1  + (1.40 − 1.02i)2-s + (0.628 + 0.778i)3-s + (0.628 − 1.93i)4-s + (−0.262 + 0.361i)5-s + (1.68 + 0.453i)6-s + (−1.85 − 0.602i)7-s + (−0.556 − 1.71i)8-s + (−0.211 + 0.977i)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 + (−0.446 + 0.0226i)15-s + (−0.891 − 0.647i)16-s + (0.00648 + 0.00471i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.703 + 0.710i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11049 - 0.880527i\)
\(L(\frac12)\) \(\approx\) \(2.11049 - 0.880527i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.08 - 1.34i)T \)
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

−12.81138971331097191373950843305, −11.92330046197861361374516750761, −10.62141985260584543790887507395, −10.10705280336367162384679968125, −9.150012588539056880668054204170, −7.12269329617474491721038905719, −5.95636647689209056466482186705, −4.30799473066123410159966798856, −3.65421977598315608284314720530, −2.63526362505756864938729980863, 3.00968985745930732743753622395, 3.85893650415568677961492554404, 5.72106173363366965728832411036, 6.48556042197184889480302344187, 7.27684598348420994345546659489, 8.615942416876399870832877132897, 9.475801056362402254747769144707, 11.73081618747836933478503188930, 12.52195851159937722801315336088, 13.04981762100872669993289620362

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