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Properties

Label 2-165-55.49-c1-0-8
Degree $2$
Conductor $165$
Sign $0.862 + 0.506i$
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  + (1.20 − 0.392i)2-s + (0.587 − 0.809i)3-s + (−0.311 + 0.226i)4-s + (2.20 − 0.347i)5-s + (0.392 − 1.20i)6-s + (0.284 + 0.390i)7-s + (−1.78 + 2.45i)8-s + (−0.309 − 0.951i)9-s + (2.53 − 1.28i)10-s + (−0.982 − 3.16i)11-s + 0.384i·12-s + (−2.99 + 0.971i)13-s + (0.496 + 0.361i)14-s + (1.01 − 1.99i)15-s + (−0.952 + 2.93i)16-s + (2.38 + 0.775i)17-s + ⋯
L(s)  = 1  + (0.854 − 0.277i)2-s + (0.339 − 0.467i)3-s + (−0.155 + 0.113i)4-s + (0.987 − 0.155i)5-s + (0.160 − 0.493i)6-s + (0.107 + 0.147i)7-s + (−0.629 + 0.866i)8-s + (−0.103 − 0.317i)9-s + (0.801 − 0.407i)10-s + (−0.296 − 0.955i)11-s + 0.111i·12-s + (−0.829 + 0.269i)13-s + (0.132 + 0.0964i)14-s + (0.262 − 0.514i)15-s + (−0.238 + 0.732i)16-s + (0.578 + 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.862 + 0.506i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.862 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77280 - 0.482430i\)
\(L(\frac12)\) \(\approx\) \(1.77280 - 0.482430i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 + (-2.20 + 0.347i)T \)
11 \( 1 + (0.982 + 3.16i)T \)
good2 \( 1 + (-1.20 + 0.392i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + (-0.284 - 0.390i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (2.99 - 0.971i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.38 - 0.775i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.00 + 2.18i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 8.53iT - 23T^{2} \)
29 \( 1 + (8.07 - 5.86i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.78 + 5.50i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.88 - 2.59i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.36 - 4.62i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.97iT - 43T^{2} \)
47 \( 1 + (-3.60 + 4.96i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.50 - 0.488i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.305 - 0.222i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.929 + 2.86i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 5.98iT - 67T^{2} \)
71 \( 1 + (-2.57 + 7.93i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.0889 - 0.122i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.31 + 7.11i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.85 - 1.25i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + (3.43 - 1.11i)T + (78.4 - 57.0i)T^{2} \)
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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

−13.11432546599905032493235261229, −12.03888152354842862850116883294, −11.03930182850568264567517138452, −9.545065440140695101452677655714, −8.762750044395759701540680549004, −7.52321060453299195350086082542, −5.94585995609712958968178969863, −5.17199283375304676792772555975, −3.51431199444325696542524046904, −2.19082816778627284283401651155, 2.50639840344489962575764484970, 4.21621922630975287698924916985, 5.17158922116826587369006378405, 6.21470236602015518003337855942, 7.53151547995700074824615135039, 9.120358371497676540673972431674, 9.896195962313923137892157919139, 10.63201071844294724386662420882, 12.49356504798892177954432662601, 12.91463821538964082199572104158

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