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Properties

Label 2-165-5.4-c3-0-0
Degree $2$
Conductor $165$
Sign $0.753 + 0.657i$
Analytic cond. $9.73531$
Root an. cond. $3.12014$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.52i·2-s − 3i·3-s − 22.5·4-s + (−7.34 + 8.42i)5-s + 16.5·6-s + 22.7i·7-s − 80.3i·8-s − 9·9-s + (−46.5 − 40.6i)10-s − 11·11-s + 67.6i·12-s − 25.8i·13-s − 125.·14-s + (25.2 + 22.0i)15-s + 263.·16-s − 103. i·17-s + ⋯
L(s)  = 1  + 1.95i·2-s − 0.577i·3-s − 2.81·4-s + (−0.657 + 0.753i)5-s + 1.12·6-s + 1.22i·7-s − 3.55i·8-s − 0.333·9-s + (−1.47 − 1.28i)10-s − 0.301·11-s + 1.62i·12-s − 0.552i·13-s − 2.39·14-s + (0.435 + 0.379i)15-s + 4.12·16-s − 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.753 + 0.657i$
Analytic conductor: \(9.73531\)
Root analytic conductor: \(3.12014\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :3/2),\ 0.753 + 0.657i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0432452 - 0.0162060i\)
\(L(\frac12)\) \(\approx\) \(0.0432452 - 0.0162060i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 + (7.34 - 8.42i)T \)
11 \( 1 + 11T \)
good2 \( 1 - 5.52iT - 8T^{2} \)
7 \( 1 - 22.7iT - 343T^{2} \)
13 \( 1 + 25.8iT - 2.19e3T^{2} \)
17 \( 1 + 103. iT - 4.91e3T^{2} \)
19 \( 1 - 91.5T + 6.85e3T^{2} \)
23 \( 1 - 78.7iT - 1.21e4T^{2} \)
29 \( 1 + 243.T + 2.43e4T^{2} \)
31 \( 1 + 177.T + 2.97e4T^{2} \)
37 \( 1 + 71.7iT - 5.06e4T^{2} \)
41 \( 1 + 321.T + 6.89e4T^{2} \)
43 \( 1 - 64.4iT - 7.95e4T^{2} \)
47 \( 1 + 76.7iT - 1.03e5T^{2} \)
53 \( 1 - 181. iT - 1.48e5T^{2} \)
59 \( 1 + 623.T + 2.05e5T^{2} \)
61 \( 1 + 86.9T + 2.26e5T^{2} \)
67 \( 1 + 162. iT - 3.00e5T^{2} \)
71 \( 1 - 326.T + 3.57e5T^{2} \)
73 \( 1 - 728. iT - 3.89e5T^{2} \)
79 \( 1 + 261.T + 4.93e5T^{2} \)
83 \( 1 + 1.30e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.22e3T + 7.04e5T^{2} \)
97 \( 1 + 1.30e3iT - 9.12e5T^{2} \)
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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

−13.60456680875182825341161277614, −12.56728808921385378100235682724, −11.48685582186248846618533611834, −9.671784995205323272201547349506, −8.772375391867676903491398076170, −7.60697226639922729734715724623, −7.23496511047987896218546310051, −5.89325478915909098996551654043, −5.16709014958145745166716068572, −3.27811647990656732106634727619, 0.02306275864800523768713098576, 1.50235188176998245923392535532, 3.54502537872088334310966291932, 4.13766175019699073566810618339, 5.20738395710121369758774213182, 7.79762362210686996467510762002, 8.841660321453230875623603630022, 9.762347983919035408710553191100, 10.68105009977891070125099520221, 11.30017607313936820809885302568

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