Properties

Label 2-165-1.1-c1-0-6
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.414·2-s − 3-s − 1.82·4-s − 5-s − 0.414·6-s − 4.82·7-s − 1.58·8-s + 9-s − 0.414·10-s − 11-s + 1.82·12-s + 5.65·13-s − 1.99·14-s + 15-s + 3·16-s − 6.82·17-s + 0.414·18-s − 1.17·19-s + 1.82·20-s + 4.82·21-s − 0.414·22-s − 4·23-s + 1.58·24-s + 25-s + 2.34·26-s − 27-s + 8.82·28-s + ⋯
L(s)  = 1  + 0.292·2-s − 0.577·3-s − 0.914·4-s − 0.447·5-s − 0.169·6-s − 1.82·7-s − 0.560·8-s + 0.333·9-s − 0.130·10-s − 0.301·11-s + 0.527·12-s + 1.56·13-s − 0.534·14-s + 0.258·15-s + 0.750·16-s − 1.65·17-s + 0.0976·18-s − 0.268·19-s + 0.408·20-s + 1.05·21-s − 0.0883·22-s − 0.834·23-s + 0.323·24-s + 0.200·25-s + 0.459·26-s − 0.192·27-s + 1.66·28-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: \(1\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.414T + 2T^{2} \)
7 \( 1 + 4.82T + 7T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 + 6.82T + 17T^{2} \)
19 \( 1 + 1.17T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 0.343T + 37T^{2} \)
41 \( 1 + 0.828T + 41T^{2} \)
43 \( 1 + 3.17T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 0.343T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 + 10T + 83T^{2} \)
89 \( 1 + 7.65T + 89T^{2} \)
97 \( 1 - 0.343T + 97T^{2} \)
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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

−12.71356479536445702086775698581, −11.40122504862480372315304683882, −10.31600728048091969811308172255, −9.307790568978610657685973125159, −8.380360614730459104025703531425, −6.64325833235590056459070713525, −5.95319055097928357696185137670, −4.37694419147338609265442814529, −3.38533677853718646521597180185, 0, 3.38533677853718646521597180185, 4.37694419147338609265442814529, 5.95319055097928357696185137670, 6.64325833235590056459070713525, 8.380360614730459104025703531425, 9.307790568978610657685973125159, 10.31600728048091969811308172255, 11.40122504862480372315304683882, 12.71356479536445702086775698581

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