Properties

Label 2-165-1.1-c3-0-3
Degree $2$
Conductor $165$
Sign $1$
Analytic cond. $9.73531$
Root an. cond. $3.12014$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.793·2-s − 3·3-s − 7.37·4-s + 5·5-s − 2.38·6-s − 2.90·7-s − 12.1·8-s + 9·9-s + 3.96·10-s + 11·11-s + 22.1·12-s + 68.4·13-s − 2.30·14-s − 15·15-s + 49.2·16-s − 31.0·17-s + 7.14·18-s + 54.9·19-s − 36.8·20-s + 8.72·21-s + 8.72·22-s + 180.·23-s + 36.5·24-s + 25·25-s + 54.3·26-s − 27·27-s + 21.4·28-s + ⋯
L(s)  = 1  + 0.280·2-s − 0.577·3-s − 0.921·4-s + 0.447·5-s − 0.161·6-s − 0.157·7-s − 0.539·8-s + 0.333·9-s + 0.125·10-s + 0.301·11-s + 0.531·12-s + 1.46·13-s − 0.0440·14-s − 0.258·15-s + 0.770·16-s − 0.443·17-s + 0.0935·18-s + 0.663·19-s − 0.412·20-s + 0.0906·21-s + 0.0845·22-s + 1.63·23-s + 0.311·24-s + 0.200·25-s + 0.409·26-s − 0.192·27-s + 0.144·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(9.73531\)
Root analytic conductor: \(3.12014\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.434502568\)
\(L(\frac12)\) \(\approx\) \(1.434502568\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
5 \( 1 - 5T \)
11 \( 1 - 11T \)
good2 \( 1 - 0.793T + 8T^{2} \)
7 \( 1 + 2.90T + 343T^{2} \)
13 \( 1 - 68.4T + 2.19e3T^{2} \)
17 \( 1 + 31.0T + 4.91e3T^{2} \)
19 \( 1 - 54.9T + 6.85e3T^{2} \)
23 \( 1 - 180.T + 1.21e4T^{2} \)
29 \( 1 - 67.3T + 2.43e4T^{2} \)
31 \( 1 - 153.T + 2.97e4T^{2} \)
37 \( 1 + 324.T + 5.06e4T^{2} \)
41 \( 1 + 25.4T + 6.89e4T^{2} \)
43 \( 1 - 133.T + 7.95e4T^{2} \)
47 \( 1 - 113.T + 1.03e5T^{2} \)
53 \( 1 - 91.6T + 1.48e5T^{2} \)
59 \( 1 - 434.T + 2.05e5T^{2} \)
61 \( 1 + 60.2T + 2.26e5T^{2} \)
67 \( 1 + 439.T + 3.00e5T^{2} \)
71 \( 1 - 436.T + 3.57e5T^{2} \)
73 \( 1 + 91.5T + 3.89e5T^{2} \)
79 \( 1 - 947.T + 4.93e5T^{2} \)
83 \( 1 + 944.T + 5.71e5T^{2} \)
89 \( 1 - 413.T + 7.04e5T^{2} \)
97 \( 1 + 1.46e3T + 9.12e5T^{2} \)
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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

−12.53514142548475685761190233839, −11.40250628816904776583673730796, −10.38926805398512018628733245544, −9.282261317456584000562431960811, −8.487024843523878192796846856601, −6.79471294058941706190637239887, −5.76476596052275293442869129411, −4.73248873960455429108959037937, −3.40080168087615840203684801713, −1.01696290428988740337621192414, 1.01696290428988740337621192414, 3.40080168087615840203684801713, 4.73248873960455429108959037937, 5.76476596052275293442869129411, 6.79471294058941706190637239887, 8.487024843523878192796846856601, 9.282261317456584000562431960811, 10.38926805398512018628733245544, 11.40250628816904776583673730796, 12.53514142548475685761190233839

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