Properties

Label 2-165-1.1-c5-0-5
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.64·2-s − 9·3-s − 0.188·4-s + 25·5-s + 50.7·6-s + 145.·7-s + 181.·8-s + 81·9-s − 141.·10-s − 121·11-s + 1.69·12-s − 69.9·13-s − 817.·14-s − 225·15-s − 1.01e3·16-s − 500.·17-s − 456.·18-s − 670.·19-s − 4.70·20-s − 1.30e3·21-s + 682.·22-s + 791.·23-s − 1.63e3·24-s + 625·25-s + 394.·26-s − 729·27-s − 27.3·28-s + ⋯
L(s)  = 1  − 0.997·2-s − 0.577·3-s − 0.00588·4-s + 0.447·5-s + 0.575·6-s + 1.11·7-s + 1.00·8-s + 0.333·9-s − 0.445·10-s − 0.301·11-s + 0.00339·12-s − 0.114·13-s − 1.11·14-s − 0.258·15-s − 0.994·16-s − 0.420·17-s − 0.332·18-s − 0.426·19-s − 0.00263·20-s − 0.645·21-s + 0.300·22-s + 0.311·23-s − 0.579·24-s + 0.200·25-s + 0.114·26-s − 0.192·27-s − 0.00658·28-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

\(L(3)\) \(\approx\) \(0.9596889671\)
\(L(\frac12)\) \(\approx\) \(0.9596889671\)
\(L(\frac{7}{2})\) 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 \( 1 + 5.64T + 32T^{2} \)
7 \( 1 - 145.T + 1.68e4T^{2} \)
13 \( 1 + 69.9T + 3.71e5T^{2} \)
17 \( 1 + 500.T + 1.41e6T^{2} \)
19 \( 1 + 670.T + 2.47e6T^{2} \)
23 \( 1 - 791.T + 6.43e6T^{2} \)
29 \( 1 - 1.54e3T + 2.05e7T^{2} \)
31 \( 1 - 2.70e3T + 2.86e7T^{2} \)
37 \( 1 + 2.66e3T + 6.93e7T^{2} \)
41 \( 1 - 9.62e3T + 1.15e8T^{2} \)
43 \( 1 + 6.68e3T + 1.47e8T^{2} \)
47 \( 1 + 1.16e3T + 2.29e8T^{2} \)
53 \( 1 - 2.88e4T + 4.18e8T^{2} \)
59 \( 1 - 2.35e4T + 7.14e8T^{2} \)
61 \( 1 - 1.76e4T + 8.44e8T^{2} \)
67 \( 1 - 1.65e4T + 1.35e9T^{2} \)
71 \( 1 - 7.20e4T + 1.80e9T^{2} \)
73 \( 1 + 4.54e4T + 2.07e9T^{2} \)
79 \( 1 + 2.16e4T + 3.07e9T^{2} \)
83 \( 1 - 6.93e3T + 3.93e9T^{2} \)
89 \( 1 - 4.27e4T + 5.58e9T^{2} \)
97 \( 1 + 2.09e4T + 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.57312022533148106213853824767, −10.74842132224563290986945716651, −9.972486242375844595378535073880, −8.830046302966854482740088656169, −7.994437793409911235673916520874, −6.85529341403178437414155954229, −5.35296769742148392019876554355, −4.41737165682017092840881371958, −2.03549059882497724123099092593, −0.77937066771802246596478905562, 0.77937066771802246596478905562, 2.03549059882497724123099092593, 4.41737165682017092840881371958, 5.35296769742148392019876554355, 6.85529341403178437414155954229, 7.994437793409911235673916520874, 8.830046302966854482740088656169, 9.972486242375844595378535073880, 10.74842132224563290986945716651, 11.57312022533148106213853824767

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