Properties

Label 2-165-33.2-c1-0-10
Degree $2$
Conductor $165$
Sign $0.980 - 0.196i$
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  + (−0.212 + 0.654i)2-s + (1.00 − 1.40i)3-s + (1.23 + 0.897i)4-s + (0.951 − 0.309i)5-s + (0.708 + 0.958i)6-s + (−0.891 + 1.22i)7-s + (−1.96 + 1.42i)8-s + (−0.974 − 2.83i)9-s + 0.687i·10-s + (2.86 − 1.66i)11-s + (2.50 − 0.837i)12-s + (−3.46 − 1.12i)13-s + (−0.613 − 0.844i)14-s + (0.521 − 1.65i)15-s + (0.427 + 1.31i)16-s + (1.34 + 4.15i)17-s + ⋯
L(s)  = 1  + (−0.150 + 0.462i)2-s + (0.581 − 0.813i)3-s + (0.617 + 0.448i)4-s + (0.425 − 0.138i)5-s + (0.289 + 0.391i)6-s + (−0.336 + 0.463i)7-s + (−0.693 + 0.504i)8-s + (−0.324 − 0.945i)9-s + 0.217i·10-s + (0.865 − 0.501i)11-s + (0.724 − 0.241i)12-s + (−0.960 − 0.312i)13-s + (−0.163 − 0.225i)14-s + (0.134 − 0.426i)15-s + (0.106 + 0.329i)16-s + (0.327 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.980 - 0.196i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.980 - 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41271 + 0.139894i\)
\(L(\frac12)\) \(\approx\) \(1.41271 + 0.139894i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.00 + 1.40i)T \)
5 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (-2.86 + 1.66i)T \)
good2 \( 1 + (0.212 - 0.654i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (0.891 - 1.22i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (3.46 + 1.12i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.34 - 4.15i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.51 + 2.08i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.708iT - 23T^{2} \)
29 \( 1 + (4.12 + 2.99i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.57 - 4.83i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.50 + 3.99i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.19 + 4.50i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.31iT - 43T^{2} \)
47 \( 1 + (3.44 + 4.74i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (6.75 + 2.19i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.19 - 9.89i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.99 - 1.94i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 10.8T + 67T^{2} \)
71 \( 1 + (-10.9 + 3.54i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.87 + 5.33i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-12.6 - 4.12i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.58 - 14.0i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 5.44iT - 89T^{2} \)
97 \( 1 + (-3.77 + 11.6i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.59650850779281873308839535193, −12.30089514521703226068758076111, −11.02325608762523721730397513746, −9.419511692828296308009849694169, −8.633740969110221429090109700579, −7.61482890342039070971093970020, −6.61117177797630031905783186074, −5.77500056906206352741611092412, −3.42567892717516004873194872289, −2.10883129497378092432865178169, 2.10324624729315663489927349475, 3.47143562647910579784420920262, 4.96664730972654703726647912722, 6.46938234619618965860192533440, 7.55861391342339111280668966364, 9.396172186054753972488915860274, 9.642679249032462343529861083421, 10.63142689949092812179776358249, 11.58090937767853298733225593943, 12.68017946192724051988035675159

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