Properties

Label 2-165-1.1-c5-0-11
Degree $2$
Conductor $165$
Sign $1$
Analytic cond. $26.4633$
Root an. cond. $5.14425$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.55·2-s − 9·3-s + 11.0·4-s + 25·5-s − 59.0·6-s + 146.·7-s − 137.·8-s + 81·9-s + 163.·10-s − 121·11-s − 99.1·12-s + 170.·13-s + 960.·14-s − 225·15-s − 1.25e3·16-s + 1.56e3·17-s + 531.·18-s + 569.·19-s + 275.·20-s − 1.31e3·21-s − 793.·22-s + 3.15e3·23-s + 1.23e3·24-s + 625·25-s + 1.11e3·26-s − 729·27-s + 1.61e3·28-s + ⋯
L(s)  = 1  + 1.15·2-s − 0.577·3-s + 0.344·4-s + 0.447·5-s − 0.669·6-s + 1.12·7-s − 0.760·8-s + 0.333·9-s + 0.518·10-s − 0.301·11-s − 0.198·12-s + 0.279·13-s + 1.31·14-s − 0.258·15-s − 1.22·16-s + 1.31·17-s + 0.386·18-s + 0.362·19-s + 0.153·20-s − 0.652·21-s − 0.349·22-s + 1.24·23-s + 0.438·24-s + 0.200·25-s + 0.323·26-s − 0.192·27-s + 0.389·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(26.4633\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.401646108\)
\(L(\frac12)\) \(\approx\) \(3.401646108\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
5 \( 1 - 25T \)
11 \( 1 + 121T \)
good2 \( 1 - 6.55T + 32T^{2} \)
7 \( 1 - 146.T + 1.68e4T^{2} \)
13 \( 1 - 170.T + 3.71e5T^{2} \)
17 \( 1 - 1.56e3T + 1.41e6T^{2} \)
19 \( 1 - 569.T + 2.47e6T^{2} \)
23 \( 1 - 3.15e3T + 6.43e6T^{2} \)
29 \( 1 - 3.98e3T + 2.05e7T^{2} \)
31 \( 1 - 2.99e3T + 2.86e7T^{2} \)
37 \( 1 - 7.85e3T + 6.93e7T^{2} \)
41 \( 1 + 5.20e3T + 1.15e8T^{2} \)
43 \( 1 - 1.38e4T + 1.47e8T^{2} \)
47 \( 1 - 6.85e3T + 2.29e8T^{2} \)
53 \( 1 + 3.83e3T + 4.18e8T^{2} \)
59 \( 1 - 9.64e3T + 7.14e8T^{2} \)
61 \( 1 + 2.11e4T + 8.44e8T^{2} \)
67 \( 1 + 4.34e4T + 1.35e9T^{2} \)
71 \( 1 + 5.26e4T + 1.80e9T^{2} \)
73 \( 1 + 6.43e4T + 2.07e9T^{2} \)
79 \( 1 - 2.89e4T + 3.07e9T^{2} \)
83 \( 1 + 4.64e3T + 3.93e9T^{2} \)
89 \( 1 + 1.03e5T + 5.58e9T^{2} \)
97 \( 1 - 7.58e4T + 8.58e9T^{2} \)
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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

−12.04557954328586215228065548202, −11.25925037755360635044320686379, −10.16302585159666504076736976046, −8.851811704478158946884983000373, −7.54031646514276314576006592346, −6.11097717284170366577635046256, −5.26840326735560715748487049569, −4.48073631061681286237257232700, −2.93058766300651227263226916171, −1.13250124010902434281094106825, 1.13250124010902434281094106825, 2.93058766300651227263226916171, 4.48073631061681286237257232700, 5.26840326735560715748487049569, 6.11097717284170366577635046256, 7.54031646514276314576006592346, 8.851811704478158946884983000373, 10.16302585159666504076736976046, 11.25925037755360635044320686379, 12.04557954328586215228065548202

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