Properties

Label 2-165-1.1-c3-0-5
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  − 4.20·2-s + 3·3-s + 9.65·4-s + 5·5-s − 12.6·6-s + 15.3·7-s − 6.96·8-s + 9·9-s − 21.0·10-s − 11·11-s + 28.9·12-s + 24.4·13-s − 64.6·14-s + 15·15-s − 48.0·16-s − 54.9·17-s − 37.8·18-s + 119.·19-s + 48.2·20-s + 46.1·21-s + 46.2·22-s − 191.·23-s − 20.8·24-s + 25·25-s − 102.·26-s + 27·27-s + 148.·28-s + ⋯
L(s)  = 1  − 1.48·2-s + 0.577·3-s + 1.20·4-s + 0.447·5-s − 0.857·6-s + 0.830·7-s − 0.307·8-s + 0.333·9-s − 0.664·10-s − 0.301·11-s + 0.696·12-s + 0.522·13-s − 1.23·14-s + 0.258·15-s − 0.750·16-s − 0.783·17-s − 0.495·18-s + 1.44·19-s + 0.539·20-s + 0.479·21-s + 0.447·22-s − 1.73·23-s − 0.177·24-s + 0.200·25-s − 0.776·26-s + 0.192·27-s + 1.00·28-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

\(L(2)\) \(\approx\) \(1.185051886\)
\(L(\frac12)\) \(\approx\) \(1.185051886\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.20T + 8T^{2} \)
7 \( 1 - 15.3T + 343T^{2} \)
13 \( 1 - 24.4T + 2.19e3T^{2} \)
17 \( 1 + 54.9T + 4.91e3T^{2} \)
19 \( 1 - 119.T + 6.85e3T^{2} \)
23 \( 1 + 191.T + 1.21e4T^{2} \)
29 \( 1 - 225.T + 2.43e4T^{2} \)
31 \( 1 - 303.T + 2.97e4T^{2} \)
37 \( 1 - 109.T + 5.06e4T^{2} \)
41 \( 1 - 348.T + 6.89e4T^{2} \)
43 \( 1 - 92.7T + 7.95e4T^{2} \)
47 \( 1 - 306.T + 1.03e5T^{2} \)
53 \( 1 + 216.T + 1.48e5T^{2} \)
59 \( 1 + 692.T + 2.05e5T^{2} \)
61 \( 1 + 152.T + 2.26e5T^{2} \)
67 \( 1 + 62.9T + 3.00e5T^{2} \)
71 \( 1 - 554.T + 3.57e5T^{2} \)
73 \( 1 - 122.T + 3.89e5T^{2} \)
79 \( 1 - 476.T + 4.93e5T^{2} \)
83 \( 1 + 913.T + 5.71e5T^{2} \)
89 \( 1 - 1.60e3T + 7.04e5T^{2} \)
97 \( 1 + 498.T + 9.12e5T^{2} \)
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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

−12.05979539214435049964005441293, −10.99415673521210488480663481696, −10.09059433867984295385550895603, −9.278285300415975880965978643356, −8.258936056653913063605766391868, −7.70820782069187426116160127553, −6.29839544205149230210849197289, −4.55622592142425963746678797951, −2.47907653558667954946509723646, −1.15355991645704823257212866536, 1.15355991645704823257212866536, 2.47907653558667954946509723646, 4.55622592142425963746678797951, 6.29839544205149230210849197289, 7.70820782069187426116160127553, 8.258936056653913063605766391868, 9.278285300415975880965978643356, 10.09059433867984295385550895603, 10.99415673521210488480663481696, 12.05979539214435049964005441293

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