Properties

Label 2-165-1.1-c1-0-4
Degree $2$
Conductor $165$
Sign $1$
Analytic cond. $1.31753$
Root an. cond. $1.14783$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s + 3-s + 0.999·4-s − 5-s + 1.73·6-s + 2·7-s − 1.73·8-s + 9-s − 1.73·10-s − 11-s + 0.999·12-s − 1.46·13-s + 3.46·14-s − 15-s − 5·16-s + 1.73·18-s − 1.46·19-s − 0.999·20-s + 2·21-s − 1.73·22-s − 6.92·23-s − 1.73·24-s + 25-s − 2.53·26-s + 27-s + 1.99·28-s + 3.46·29-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.577·3-s + 0.499·4-s − 0.447·5-s + 0.707·6-s + 0.755·7-s − 0.612·8-s + 0.333·9-s − 0.547·10-s − 0.301·11-s + 0.288·12-s − 0.406·13-s + 0.925·14-s − 0.258·15-s − 1.25·16-s + 0.408·18-s − 0.335·19-s − 0.223·20-s + 0.436·21-s − 0.369·22-s − 1.44·23-s − 0.353·24-s + 0.200·25-s − 0.497·26-s + 0.192·27-s + 0.377·28-s + 0.643·29-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.107355552\)
\(L(\frac12)\) \(\approx\) \(2.107355552\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 + T \)
good2 \( 1 - 1.73T + 2T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
13 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 1.46T + 19T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 - 3.46T + 29T^{2} \)
31 \( 1 - 2.92T + 31T^{2} \)
37 \( 1 - 8.92T + 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 - 8.92T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−12.89752814607709570945352996584, −12.12442844884684652899375597193, −11.19000972668998539849998229463, −9.835971336260779695431693116780, −8.538147174097694339979543049872, −7.64462765613727710558979979432, −6.18401801978408377537006310704, −4.84301865879241464751162615703, −3.99060451599707408140172652399, −2.54117248937871306304050278597, 2.54117248937871306304050278597, 3.99060451599707408140172652399, 4.84301865879241464751162615703, 6.18401801978408377537006310704, 7.64462765613727710558979979432, 8.538147174097694339979543049872, 9.835971336260779695431693116780, 11.19000972668998539849998229463, 12.12442844884684652899375597193, 12.89752814607709570945352996584

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