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Properties

Label 2-165-55.4-c1-0-11
Degree $2$
Conductor $165$
Sign $-0.784 + 0.619i$
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  + (1.38 − 1.91i)2-s + (−0.951 − 0.309i)3-s + (−1.10 − 3.41i)4-s + (−1.97 − 1.05i)5-s + (−1.91 + 1.38i)6-s + (1.63 − 0.529i)7-s + (−3.56 − 1.15i)8-s + (0.809 + 0.587i)9-s + (−4.75 + 2.31i)10-s + (−0.406 + 3.29i)11-s + 3.58i·12-s + (2.07 − 2.85i)13-s + (1.25 − 3.85i)14-s + (1.55 + 1.60i)15-s + (−1.37 + 0.996i)16-s + (0.219 + 0.302i)17-s + ⋯
L(s)  = 1  + (0.982 − 1.35i)2-s + (−0.549 − 0.178i)3-s + (−0.554 − 1.70i)4-s + (−0.882 − 0.469i)5-s + (−0.780 + 0.567i)6-s + (0.616 − 0.200i)7-s + (−1.26 − 0.409i)8-s + (0.269 + 0.195i)9-s + (−1.50 + 0.731i)10-s + (−0.122 + 0.992i)11-s + 1.03i·12-s + (0.576 − 0.792i)13-s + (0.334 − 1.02i)14-s + (0.400 + 0.415i)15-s + (−0.342 + 0.249i)16-s + (0.0532 + 0.0732i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.784 + 0.619i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -0.784 + 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.468714 - 1.35015i\)
\(L(\frac12)\) \(\approx\) \(0.468714 - 1.35015i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (1.97 + 1.05i)T \)
11 \( 1 + (0.406 - 3.29i)T \)
good2 \( 1 + (-1.38 + 1.91i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + (-1.63 + 0.529i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.07 + 2.85i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.219 - 0.302i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.78 + 5.48i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.80iT - 23T^{2} \)
29 \( 1 + (-1.08 - 3.34i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-8.16 - 5.93i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (10.9 - 3.56i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.747 - 2.30i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.224iT - 43T^{2} \)
47 \( 1 + (4.44 + 1.44i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.43 + 1.97i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.65 + 5.09i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.75 + 2.00i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9.08iT - 67T^{2} \)
71 \( 1 + (-7.09 + 5.15i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (14.2 - 4.62i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.11 - 5.16i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.37 - 6.02i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 + (-8.61 + 11.8i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.32855001879961526394278433554, −11.59925435530210926159147349843, −10.90435938491640088153453030202, −9.938920706166474608776319009175, −8.360465502998344884325402114357, −7.06621489772671121740120191150, −5.18680860547769293082393635050, −4.64714352045491912050042402627, −3.26500644525415542160931094227, −1.31313711757966032634140415655, 3.59671288796755792180807898432, 4.57646638114460284287985882889, 5.81450566880712771354196506783, 6.63073796898061446624166063987, 7.82658703645907851348917297770, 8.547113827595787415114608404285, 10.47461619016622507576033156015, 11.58099559378460273682454592298, 12.20929177425922387457901139369, 13.61112301535289979099972120503

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