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Properties

Label 2-165-55.4-c1-0-7
Degree $2$
Conductor $165$
Sign $0.855 - 0.517i$
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  + (−0.169 + 0.233i)2-s + (0.951 + 0.309i)3-s + (0.592 + 1.82i)4-s + (−0.962 − 2.01i)5-s + (−0.233 + 0.169i)6-s + (3.48 − 1.13i)7-s + (−1.07 − 0.349i)8-s + (0.809 + 0.587i)9-s + (0.635 + 0.117i)10-s + (1.25 + 3.06i)11-s + 1.91i·12-s + (−2.83 + 3.90i)13-s + (−0.327 + 1.00i)14-s + (−0.292 − 2.21i)15-s + (−2.83 + 2.06i)16-s + (−2.25 − 3.10i)17-s + ⋯
L(s)  = 1  + (−0.120 + 0.165i)2-s + (0.549 + 0.178i)3-s + (0.296 + 0.911i)4-s + (−0.430 − 0.902i)5-s + (−0.0953 + 0.0693i)6-s + (1.31 − 0.428i)7-s + (−0.380 − 0.123i)8-s + (0.269 + 0.195i)9-s + (0.200 + 0.0371i)10-s + (0.378 + 0.925i)11-s + 0.553i·12-s + (−0.786 + 1.08i)13-s + (−0.0874 + 0.269i)14-s + (−0.0754 − 0.572i)15-s + (−0.709 + 0.515i)16-s + (−0.547 − 0.753i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.855 - 0.517i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.855 - 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29591 + 0.361085i\)
\(L(\frac12)\) \(\approx\) \(1.29591 + 0.361085i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 + (0.962 + 2.01i)T \)
11 \( 1 + (-1.25 - 3.06i)T \)
good2 \( 1 + (0.169 - 0.233i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + (-3.48 + 1.13i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.83 - 3.90i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.25 + 3.10i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.15 + 6.62i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.20iT - 23T^{2} \)
29 \( 1 + (1.52 + 4.70i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.96 + 4.33i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.27 + 0.737i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.33 + 4.09i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.772iT - 43T^{2} \)
47 \( 1 + (6.15 + 1.99i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.32 - 8.70i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-4.48 - 13.7i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.48 + 1.07i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 6.10iT - 67T^{2} \)
71 \( 1 + (-2.66 + 1.93i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.182 + 0.0591i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.55 + 1.85i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.899 - 1.23i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 3.04T + 89T^{2} \)
97 \( 1 + (-8.09 + 11.1i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.91826840672881618063882770700, −11.72804675924484365553594973161, −11.37999339469271298782782386205, −9.431436347652979957417929760581, −8.851863455150925525364593976942, −7.58004055385228400747267766789, −7.22408689001111449293885767100, −4.77381046359106992866767932484, −4.18296330391606595039639451164, −2.15819143424376968579479815639, 1.82470647666623010943140674139, 3.33180007793804837614546807719, 5.19622102306968192573398884983, 6.35455581458560836641914984017, 7.71427324683897348419143457596, 8.487270511528219015434896486407, 9.894048383182643777318712695081, 10.85711574567968492916790995421, 11.45887027838623966747151574550, 12.62410015396538372074512207230

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