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Properties

Label 2-165-33.2-c1-0-14
Degree $2$
Conductor $165$
Sign $-0.902 + 0.431i$
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  + (0.664 − 2.04i)2-s + (0.340 − 1.69i)3-s + (−2.11 − 1.53i)4-s + (−0.951 + 0.309i)5-s + (−3.24 − 1.82i)6-s + (−1.16 + 1.60i)7-s + (−1.07 + 0.781i)8-s + (−2.76 − 1.15i)9-s + 2.14i·10-s + (3.11 − 1.14i)11-s + (−3.33 + 3.07i)12-s + (5.55 + 1.80i)13-s + (2.50 + 3.44i)14-s + (0.200 + 1.72i)15-s + (−0.735 − 2.26i)16-s + (0.00961 + 0.0296i)17-s + ⋯
L(s)  = 1  + (0.469 − 1.44i)2-s + (0.196 − 0.980i)3-s + (−1.05 − 0.769i)4-s + (−0.425 + 0.138i)5-s + (−1.32 − 0.744i)6-s + (−0.439 + 0.605i)7-s + (−0.380 + 0.276i)8-s + (−0.922 − 0.386i)9-s + 0.679i·10-s + (0.938 − 0.345i)11-s + (−0.962 + 0.886i)12-s + (1.54 + 0.500i)13-s + (0.668 + 0.920i)14-s + (0.0517 + 0.444i)15-s + (−0.183 − 0.566i)16-s + (0.00233 + 0.00717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.902 + 0.431i$
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: \((2,\ 165,\ (\ :1/2),\ -0.902 + 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.308002 - 1.35699i\)
\(L(\frac12)\) \(\approx\) \(0.308002 - 1.35699i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.340 + 1.69i)T \)
5 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (-3.11 + 1.14i)T \)
good2 \( 1 + (-0.664 + 2.04i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (1.16 - 1.60i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-5.55 - 1.80i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.00961 - 0.0296i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.10 + 4.27i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 8.23iT - 23T^{2} \)
29 \( 1 + (-0.972 - 0.706i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.153 + 0.472i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.98 - 2.16i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.54 - 1.84i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.79iT - 43T^{2} \)
47 \( 1 + (0.700 + 0.963i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-9.74 - 3.16i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.78 - 2.46i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (8.33 - 2.70i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 6.68T + 67T^{2} \)
71 \( 1 + (-0.561 + 0.182i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.63 - 3.62i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (16.4 + 5.33i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.17 + 9.76i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 1.78iT - 89T^{2} \)
97 \( 1 + (5.65 - 17.4i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.22755629301196851682588501169, −11.57644764801507842193168558321, −10.98293454989159351025911863491, −9.350292706853496237941918557019, −8.603469855793213914911691076493, −7.01008710294192262937232273175, −5.90455146141833560516862320920, −3.97489171174760311163038938738, −2.95149421687771835322314314278, −1.41571931930375544339520968956, 3.74760509249298473538604477925, 4.39593358529598380001496155072, 5.86684135278841080238406945428, 6.72826350212067309192814039049, 8.162210626952411654774357121298, 8.756589839173977213679705704994, 10.19100273599326163663970367334, 11.09419461826336187061740790628, 12.59609694758227000960445370312, 13.66978627609063949245799896589

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