↑

Properties

Label 2-165-55.4-c1-0-2
Degree $2$
Conductor $165$
Sign $0.613 - 0.789i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.169 − 0.233i)2-s + (−0.951 − 0.309i)3-s + (0.592 + 1.82i)4-s + (1.62 + 1.53i)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 + (−1.06 − 1.96i)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 + (0.725 + 0.688i)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 + (−0.275 − 0.507i)15-s + (−0.709 + 0.515i)16-s + (0.547 + 0.753i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.613 - 0.789i$
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: \((2,\ 165,\ (\ :1/2),\ 0.613 - 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00635 + 0.492156i\)
\(L(\frac12)\) \(\approx\) \(1.00635 + 0.492156i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 1.53i)T \)
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 \( 1 + (8.09 - 11.1i)T + (-29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.99299693782560064339209892896, −12.12447861637994668737350909099, −11.03483926327353572158528737281, −10.10080965724412242674967376681, −9.058461516696484322083580452305, −7.47306062922984032080140197847, −6.61233624696023948831629743658, −5.66124892459321789675487641511, −3.68791690732029380637266927511, −2.48432469987682740004163223222, 1.25809606213844087090326323780, 3.72612366967210507765559764800, 5.40134409861443635924541015689, 6.08558109254303548707887514705, 6.98043635159794616704471405203, 8.991417218003657260700328795413, 9.720420778311586723724387802729, 10.54541075017487212278711132366, 11.64908686561557862289841930395, 12.75283334173764053023175082151

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