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Properties

Label 2-165-33.2-c1-0-6
Degree $2$
Conductor $165$
Sign $0.338 - 0.941i$
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  + (−0.140 + 0.431i)2-s + (0.180 + 1.72i)3-s + (1.45 + 1.05i)4-s + (0.951 − 0.309i)5-s + (−0.768 − 0.163i)6-s + (2.72 − 3.74i)7-s + (−1.39 + 1.01i)8-s + (−2.93 + 0.621i)9-s + 0.453i·10-s + (−2.24 + 2.44i)11-s + (−1.55 + 2.69i)12-s + (−2.34 − 0.762i)13-s + (1.23 + 1.69i)14-s + (0.703 + 1.58i)15-s + (0.867 + 2.67i)16-s + (−0.465 − 1.43i)17-s + ⋯
L(s)  = 1  + (−0.0990 + 0.304i)2-s + (0.104 + 0.994i)3-s + (0.725 + 0.527i)4-s + (0.425 − 0.138i)5-s + (−0.313 − 0.0667i)6-s + (1.02 − 1.41i)7-s + (−0.492 + 0.357i)8-s + (−0.978 + 0.207i)9-s + 0.143i·10-s + (−0.675 + 0.737i)11-s + (−0.448 + 0.776i)12-s + (−0.651 − 0.211i)13-s + (0.329 + 0.454i)14-s + (0.181 + 0.408i)15-s + (0.216 + 0.667i)16-s + (−0.112 − 0.347i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.338 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.338 - 0.941i$
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: \((2,\ 165,\ (\ :1/2),\ 0.338 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10874 + 0.779637i\)
\(L(\frac12)\) \(\approx\) \(1.10874 + 0.779637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.180 - 1.72i)T \)
5 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (2.24 - 2.44i)T \)
good2 \( 1 + (0.140 - 0.431i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-2.72 + 3.74i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (2.34 + 0.762i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.465 + 1.43i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.58 + 2.18i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 5.63iT - 23T^{2} \)
29 \( 1 + (-0.463 - 0.336i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.36 - 7.28i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.396 - 0.287i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.82 + 4.96i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.51iT - 43T^{2} \)
47 \( 1 + (1.96 + 2.71i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-12.6 - 4.11i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.21 - 3.04i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-7.73 + 2.51i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 7.19T + 67T^{2} \)
71 \( 1 + (6.97 - 2.26i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.93 - 5.41i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.68 - 1.19i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.83 + 5.63i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 3.19iT - 89T^{2} \)
97 \( 1 + (0.622 - 1.91i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.07702397934324666428221141678, −11.86050737055650961800108965467, −10.67928557378411710093033042729, −10.36528104204010890820008389727, −8.859202116852452126515964127659, −7.79937334931209427765863767184, −6.92789269508318709272798923819, −5.20281231833260371341969889608, −4.24200210449464525512965598681, −2.51988396817329928573615346826, 1.81788175575433346352436588346, 2.67611436150616270974025837654, 5.51012135425241524308854909387, 5.99916924712336148706517644858, 7.45740392521290042567088014325, 8.448023304057821907770374485225, 9.572960612012538027483929159257, 10.99645776759751724700736739718, 11.62031391067767721061660964553, 12.43315485144716980376815537100

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