Properties

Label 2-165-33.29-c1-0-6
Degree $2$
Conductor $165$
Sign $0.203 - 0.978i$
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  + (2.09 + 1.52i)2-s + (−1.46 − 0.921i)3-s + (1.44 + 4.45i)4-s + (0.587 + 0.809i)5-s + (−1.66 − 4.15i)6-s + (0.326 − 0.106i)7-s + (−2.14 + 6.61i)8-s + (1.30 + 2.70i)9-s + 2.58i·10-s + (−2.88 − 1.63i)11-s + (1.98 − 7.87i)12-s + (3.76 − 5.17i)13-s + (0.845 + 0.274i)14-s + (−0.116 − 1.72i)15-s + (−6.96 + 5.05i)16-s + (0.839 − 0.610i)17-s + ⋯
L(s)  = 1  + (1.47 + 1.07i)2-s + (−0.846 − 0.532i)3-s + (0.724 + 2.22i)4-s + (0.262 + 0.361i)5-s + (−0.680 − 1.69i)6-s + (0.123 − 0.0401i)7-s + (−0.759 + 2.33i)8-s + (0.433 + 0.901i)9-s + 0.817i·10-s + (−0.869 − 0.494i)11-s + (0.573 − 2.27i)12-s + (1.04 − 1.43i)13-s + (0.225 + 0.0733i)14-s + (−0.0299 − 0.446i)15-s + (−1.74 + 1.26i)16-s + (0.203 − 0.147i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.203 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.203 - 0.978i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.203 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48973 + 1.21134i\)
\(L(\frac12)\) \(\approx\) \(1.48973 + 1.21134i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.46 + 0.921i)T \)
5 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (2.88 + 1.63i)T \)
good2 \( 1 + (-2.09 - 1.52i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (-0.326 + 0.106i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.76 + 5.17i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.839 + 0.610i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (5.97 + 1.94i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.43iT - 23T^{2} \)
29 \( 1 + (1.36 + 4.20i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.974 - 0.708i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.0967 - 0.297i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.390 - 1.20i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 + (-8.00 - 2.60i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.614 - 0.846i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (7.13 - 2.31i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.01 - 2.77i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.66T + 67T^{2} \)
71 \( 1 + (5.34 + 7.36i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.99 - 1.29i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.88 - 3.96i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.7 + 7.77i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.84iT - 89T^{2} \)
97 \( 1 + (8.10 + 5.88i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.18988669947504915205238729775, −12.56867281651281983494693091895, −11.29685931035446856697772683632, −10.55200843999677248983979793529, −8.234246603920811643224828128568, −7.50690752182186731216639760921, −6.19166956346441320395681494274, −5.79128136682384871454544781687, −4.59469965393680031270854773001, −2.94295102760184084693258585261, 1.86701098500572039145329513086, 3.83230132057226628434983823241, 4.67189705771430828887058452492, 5.68662791682505454758204990576, 6.63008253502389407698460454953, 8.954710054599300923824239530581, 10.25875030730816088867488462353, 10.80317806792880802591100083197, 11.77846985239259332140302398389, 12.52633604297344832297345237285

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