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Properties

Label 2-165-11.3-c1-0-0
Degree $2$
Conductor $165$
Sign $-0.872 - 0.488i$
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  + (−1.08 + 0.786i)2-s + (−0.309 − 0.951i)3-s + (−0.0646 + 0.198i)4-s + (−0.809 − 0.587i)5-s + (1.08 + 0.786i)6-s + (−1.16 + 3.59i)7-s + (−0.913 − 2.81i)8-s + (−0.809 + 0.587i)9-s + 1.33·10-s + (−0.569 + 3.26i)11-s + 0.209·12-s + (−3.99 + 2.90i)13-s + (−1.56 − 4.81i)14-s + (−0.309 + 0.951i)15-s + (2.86 + 2.07i)16-s + (−2.63 − 1.91i)17-s + ⋯
L(s)  = 1  + (−0.765 + 0.556i)2-s + (−0.178 − 0.549i)3-s + (−0.0323 + 0.0994i)4-s + (−0.361 − 0.262i)5-s + (0.442 + 0.321i)6-s + (−0.441 + 1.35i)7-s + (−0.322 − 0.994i)8-s + (−0.269 + 0.195i)9-s + 0.423·10-s + (−0.171 + 0.985i)11-s + 0.0603·12-s + (−1.10 + 0.804i)13-s + (−0.418 − 1.28i)14-s + (−0.0797 + 0.245i)15-s + (0.715 + 0.519i)16-s + (−0.639 − 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.872 - 0.488i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0914680 + 0.350603i\)
\(L(\frac12)\) \(\approx\) \(0.0914680 + 0.350603i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (0.569 - 3.26i)T \)
good2 \( 1 + (1.08 - 0.786i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (1.16 - 3.59i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.99 - 2.90i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.63 + 1.91i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.424 - 1.30i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.93T + 23T^{2} \)
29 \( 1 + (0.537 - 1.65i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.79 + 3.48i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.45 + 4.46i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.91 - 8.97i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + (-0.248 - 0.766i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.65 - 6.28i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.75 + 11.5i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.95 + 5.04i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.77T + 67T^{2} \)
71 \( 1 + (-2.21 - 1.61i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.89 - 8.91i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.3 - 7.50i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.76 - 2.00i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 7.85T + 89T^{2} \)
97 \( 1 + (-11.0 + 8.00i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.76379160975401795523955973491, −12.42878773124770449281756178636, −11.53318779521690635533259980359, −9.678462597960770570265218221695, −9.178153475402942729096114134175, −8.009181053946271806732698768876, −7.16926804479036971065189564896, −6.12378561939880861512266593991, −4.56521478749187526353173341970, −2.48496526281175380056055944660, 0.43007829073308725796055039342, 2.98013072016319402439510971490, 4.44375798848912268892855707728, 5.90792387127451822359538407295, 7.38415709700902104384751935677, 8.502617954737514338605330412933, 9.692269465835732583493731261286, 10.51291428351238875879139178088, 10.89869503289635704842741411044, 12.08047820394861048188162339125

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