Properties

Label 2-825-1.1-c5-0-16
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.9·2-s − 9·3-s + 87.0·4-s + 98.1·6-s − 180.·7-s − 600.·8-s + 81·9-s + 121·11-s − 783.·12-s − 61.0·13-s + 1.97e3·14-s + 3.76e3·16-s + 133.·17-s − 883.·18-s + 1.99e3·19-s + 1.62e3·21-s − 1.32e3·22-s + 445.·23-s + 5.40e3·24-s + 666.·26-s − 729·27-s − 1.57e4·28-s − 6.71e3·29-s − 8.73e3·31-s − 2.18e4·32-s − 1.08e3·33-s − 1.45e3·34-s + ⋯
L(s)  = 1  − 1.92·2-s − 0.577·3-s + 2.71·4-s + 1.11·6-s − 1.39·7-s − 3.31·8-s + 0.333·9-s + 0.301·11-s − 1.56·12-s − 0.100·13-s + 2.68·14-s + 3.67·16-s + 0.111·17-s − 0.642·18-s + 1.26·19-s + 0.804·21-s − 0.581·22-s + 0.175·23-s + 1.91·24-s + 0.193·26-s − 0.192·27-s − 3.78·28-s − 1.48·29-s − 1.63·31-s − 3.77·32-s − 0.174·33-s − 0.215·34-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

\(L(3)\) \(\approx\) \(0.3546793447\)
\(L(\frac12)\) \(\approx\) \(0.3546793447\)
\(L(\frac{7}{2})\) 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 \( 1 + 10.9T + 32T^{2} \)
7 \( 1 + 180.T + 1.68e4T^{2} \)
13 \( 1 + 61.0T + 3.71e5T^{2} \)
17 \( 1 - 133.T + 1.41e6T^{2} \)
19 \( 1 - 1.99e3T + 2.47e6T^{2} \)
23 \( 1 - 445.T + 6.43e6T^{2} \)
29 \( 1 + 6.71e3T + 2.05e7T^{2} \)
31 \( 1 + 8.73e3T + 2.86e7T^{2} \)
37 \( 1 - 1.49e4T + 6.93e7T^{2} \)
41 \( 1 - 2.98e3T + 1.15e8T^{2} \)
43 \( 1 - 6.57e3T + 1.47e8T^{2} \)
47 \( 1 - 1.91e4T + 2.29e8T^{2} \)
53 \( 1 + 2.00e4T + 4.18e8T^{2} \)
59 \( 1 + 1.06e4T + 7.14e8T^{2} \)
61 \( 1 - 8.34e3T + 8.44e8T^{2} \)
67 \( 1 + 4.41e4T + 1.35e9T^{2} \)
71 \( 1 - 4.78e4T + 1.80e9T^{2} \)
73 \( 1 + 2.79e4T + 2.07e9T^{2} \)
79 \( 1 - 2.13e4T + 3.07e9T^{2} \)
83 \( 1 + 9.30e4T + 3.93e9T^{2} \)
89 \( 1 + 1.30e5T + 5.58e9T^{2} \)
97 \( 1 - 1.68e5T + 8.58e9T^{2} \)
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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

−9.434312417697342521076101141520, −9.040675710715031379748582113692, −7.60239947188951337672814212810, −7.25453050776198457455515806197, −6.25038527125330992484175365294, −5.65254537854282210002724126772, −3.65015341551642388709726777097, −2.63940337959941905672016699196, −1.36491137688305928849167539966, −0.39814640627144420110330139099, 0.39814640627144420110330139099, 1.36491137688305928849167539966, 2.63940337959941905672016699196, 3.65015341551642388709726777097, 5.65254537854282210002724126772, 6.25038527125330992484175365294, 7.25453050776198457455515806197, 7.60239947188951337672814212810, 9.040675710715031379748582113692, 9.434312417697342521076101141520

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