Properties

Label 2-495-1.1-c5-0-68
Degree $2$
Conductor $495$
Sign $-1$
Analytic cond. $79.3899$
Root an. cond. $8.91010$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9.91·2-s + 66.3·4-s + 25·5-s + 92.6·7-s − 340.·8-s − 247.·10-s − 121·11-s + 800.·13-s − 918.·14-s + 1.25e3·16-s + 117.·17-s + 831.·19-s + 1.65e3·20-s + 1.20e3·22-s − 2.95e3·23-s + 625·25-s − 7.93e3·26-s + 6.14e3·28-s − 5.76e3·29-s − 61.7·31-s − 1.56e3·32-s − 1.16e3·34-s + 2.31e3·35-s − 1.02e4·37-s − 8.24e3·38-s − 8.52e3·40-s − 9.59e3·41-s + ⋯
L(s)  = 1  − 1.75·2-s + 2.07·4-s + 0.447·5-s + 0.714·7-s − 1.88·8-s − 0.784·10-s − 0.301·11-s + 1.31·13-s − 1.25·14-s + 1.22·16-s + 0.0988·17-s + 0.528·19-s + 0.927·20-s + 0.528·22-s − 1.16·23-s + 0.200·25-s − 2.30·26-s + 1.48·28-s − 1.27·29-s − 0.0115·31-s − 0.270·32-s − 0.173·34-s + 0.319·35-s − 1.22·37-s − 0.926·38-s − 0.842·40-s − 0.891·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-1$
Analytic conductor: \(79.3899\)
Root analytic conductor: \(8.91010\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 495,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - 25T \)
11 \( 1 + 121T \)
good2 \( 1 + 9.91T + 32T^{2} \)
7 \( 1 - 92.6T + 1.68e4T^{2} \)
13 \( 1 - 800.T + 3.71e5T^{2} \)
17 \( 1 - 117.T + 1.41e6T^{2} \)
19 \( 1 - 831.T + 2.47e6T^{2} \)
23 \( 1 + 2.95e3T + 6.43e6T^{2} \)
29 \( 1 + 5.76e3T + 2.05e7T^{2} \)
31 \( 1 + 61.7T + 2.86e7T^{2} \)
37 \( 1 + 1.02e4T + 6.93e7T^{2} \)
41 \( 1 + 9.59e3T + 1.15e8T^{2} \)
43 \( 1 + 1.74e4T + 1.47e8T^{2} \)
47 \( 1 + 1.87e4T + 2.29e8T^{2} \)
53 \( 1 - 9.70e3T + 4.18e8T^{2} \)
59 \( 1 - 2.44e4T + 7.14e8T^{2} \)
61 \( 1 - 3.39e4T + 8.44e8T^{2} \)
67 \( 1 + 4.98e3T + 1.35e9T^{2} \)
71 \( 1 - 6.29e4T + 1.80e9T^{2} \)
73 \( 1 + 5.42e4T + 2.07e9T^{2} \)
79 \( 1 + 5.69e4T + 3.07e9T^{2} \)
83 \( 1 + 4.96e4T + 3.93e9T^{2} \)
89 \( 1 - 8.79e4T + 5.58e9T^{2} \)
97 \( 1 + 4.48e4T + 8.58e9T^{2} \)
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

−9.763410270314011626924187920621, −8.643733848685720394461864018645, −8.254842168384662534379301259615, −7.26899869782249388239385801971, −6.30711961394165983635936202470, −5.25924599944913516133628342267, −3.52658661723187206186345970158, −2.00218550281837574719358827836, −1.34779579441483380094084436390, 0, 1.34779579441483380094084436390, 2.00218550281837574719358827836, 3.52658661723187206186345970158, 5.25924599944913516133628342267, 6.30711961394165983635936202470, 7.26899869782249388239385801971, 8.254842168384662534379301259615, 8.643733848685720394461864018645, 9.763410270314011626924187920621

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