Properties

Label 2-825-1.1-c5-0-29
Degree $2$
Conductor $825$
Sign $1$
Analytic cond. $132.316$
Root an. cond. $11.5028$
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  − 7.39·2-s + 9·3-s + 22.7·4-s − 66.5·6-s + 150.·7-s + 68.5·8-s + 81·9-s + 121·11-s + 204.·12-s − 868.·13-s − 1.11e3·14-s − 1.23e3·16-s − 2.31e3·17-s − 599.·18-s − 2.65e3·19-s + 1.35e3·21-s − 895.·22-s − 2.53e3·23-s + 616.·24-s + 6.42e3·26-s + 729·27-s + 3.43e3·28-s − 819.·29-s + 7.30e3·31-s + 6.94e3·32-s + 1.08e3·33-s + 1.71e4·34-s + ⋯
L(s)  = 1  − 1.30·2-s + 0.577·3-s + 0.710·4-s − 0.755·6-s + 1.16·7-s + 0.378·8-s + 0.333·9-s + 0.301·11-s + 0.410·12-s − 1.42·13-s − 1.52·14-s − 1.20·16-s − 1.94·17-s − 0.435·18-s − 1.68·19-s + 0.671·21-s − 0.394·22-s − 1.00·23-s + 0.218·24-s + 1.86·26-s + 0.192·27-s + 0.826·28-s − 0.181·29-s + 1.36·31-s + 1.19·32-s + 0.174·33-s + 2.54·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(132.316\)
Root analytic conductor: \(11.5028\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9775242633\)
\(L(\frac12)\) \(\approx\) \(0.9775242633\)
\(L(\frac{7}{2})\) 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 \( 1 \)
11 \( 1 - 121T \)
good2 \( 1 + 7.39T + 32T^{2} \)
7 \( 1 - 150.T + 1.68e4T^{2} \)
13 \( 1 + 868.T + 3.71e5T^{2} \)
17 \( 1 + 2.31e3T + 1.41e6T^{2} \)
19 \( 1 + 2.65e3T + 2.47e6T^{2} \)
23 \( 1 + 2.53e3T + 6.43e6T^{2} \)
29 \( 1 + 819.T + 2.05e7T^{2} \)
31 \( 1 - 7.30e3T + 2.86e7T^{2} \)
37 \( 1 - 2.99e3T + 6.93e7T^{2} \)
41 \( 1 + 4.56e3T + 1.15e8T^{2} \)
43 \( 1 + 1.02e3T + 1.47e8T^{2} \)
47 \( 1 - 2.44e4T + 2.29e8T^{2} \)
53 \( 1 - 1.33e4T + 4.18e8T^{2} \)
59 \( 1 + 2.97e4T + 7.14e8T^{2} \)
61 \( 1 + 1.72e4T + 8.44e8T^{2} \)
67 \( 1 - 1.86e4T + 1.35e9T^{2} \)
71 \( 1 - 4.77e4T + 1.80e9T^{2} \)
73 \( 1 - 1.91e4T + 2.07e9T^{2} \)
79 \( 1 + 3.71e3T + 3.07e9T^{2} \)
83 \( 1 - 5.19e4T + 3.93e9T^{2} \)
89 \( 1 + 5.29e4T + 5.58e9T^{2} \)
97 \( 1 - 1.14e5T + 8.58e9T^{2} \)
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

−9.281842098741222845614049760954, −8.607069897911245586021193655006, −8.076817850133117976293077227557, −7.26267727523093545735401907835, −6.40244731193612492542702551977, −4.66762620399445473526331901606, −4.32627488086398726242331700658, −2.26418626274941352210762176696, −1.98683395979185323760915482646, −0.51588955211323301310764933248, 0.51588955211323301310764933248, 1.98683395979185323760915482646, 2.26418626274941352210762176696, 4.32627488086398726242331700658, 4.66762620399445473526331901606, 6.40244731193612492542702551977, 7.26267727523093545735401907835, 8.076817850133117976293077227557, 8.607069897911245586021193655006, 9.281842098741222845614049760954

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