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Properties

Label 2-165-55.9-c1-0-9
Degree $2$
Conductor $165$
Sign $0.880 - 0.474i$
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  + (2.46 + 0.800i)2-s + (−0.587 − 0.809i)3-s + (3.81 + 2.77i)4-s + (−2.15 + 0.599i)5-s + (−0.800 − 2.46i)6-s + (2.00 − 2.76i)7-s + (4.14 + 5.70i)8-s + (−0.309 + 0.951i)9-s + (−5.79 − 0.247i)10-s + (−2.29 + 2.39i)11-s − 4.71i·12-s + (−4.35 − 1.41i)13-s + (7.16 − 5.20i)14-s + (1.75 + 1.39i)15-s + (2.72 + 8.39i)16-s + (−1.88 + 0.611i)17-s + ⋯
L(s)  = 1  + (1.74 + 0.566i)2-s + (−0.339 − 0.467i)3-s + (1.90 + 1.38i)4-s + (−0.963 + 0.268i)5-s + (−0.326 − 1.00i)6-s + (0.759 − 1.04i)7-s + (1.46 + 2.01i)8-s + (−0.103 + 0.317i)9-s + (−1.83 − 0.0781i)10-s + (−0.693 + 0.720i)11-s − 1.36i·12-s + (−1.20 − 0.392i)13-s + (1.91 − 1.39i)14-s + (0.452 + 0.358i)15-s + (0.681 + 2.09i)16-s + (−0.456 + 0.148i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.880 - 0.474i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.880 - 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21983 + 0.559590i\)
\(L(\frac12)\) \(\approx\) \(2.21983 + 0.559590i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 - 0.599i)T \)
11 \( 1 + (2.29 - 2.39i)T \)
good2 \( 1 + (-2.46 - 0.800i)T + (1.61 + 1.17i)T^{2} \)
7 \( 1 + (-2.00 + 2.76i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (4.35 + 1.41i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.88 - 0.611i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.14 + 0.833i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5.03iT - 23T^{2} \)
29 \( 1 + (-3.51 - 2.55i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.0844 - 0.259i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.144 - 0.198i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.08 - 0.784i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 11.7iT - 43T^{2} \)
47 \( 1 + (-7.61 - 10.4i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.924 + 0.300i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.93 - 3.58i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.41 + 7.41i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 2.14iT - 67T^{2} \)
71 \( 1 + (2.80 + 8.64i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.44 + 10.2i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.237 - 0.732i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (9.26 - 3.00i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + (-12.9 - 4.21i)T + (78.4 + 57.0i)T^{2} \)
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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

−12.80643617222286101452830066319, −12.33437850097581130356788324655, −11.29602649935202750136802930472, −10.51787784847662479090008325989, −7.894606739070546720558298072911, −7.45368432121874223663302126020, −6.56331952508713572959394397668, −4.90184574335910013973233009969, −4.42877029072438304100937434500, −2.76069906938941212011333337500, 2.49302932686116740164346288431, 3.89320422556511745886776272593, 5.03124083292882589490026044496, 5.55689507283810273428014686054, 7.20047084733768214213840255631, 8.660088514035558852311062184964, 10.25712886533672578334889487095, 11.43749702029039469900436302872, 11.74987041571585041227071619828, 12.50722081086592467043173607302

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