Properties

Label 2-825-1.1-c5-0-80
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  + 5.09·2-s + 9·3-s − 6.03·4-s + 45.8·6-s + 170.·7-s − 193.·8-s + 81·9-s + 121·11-s − 54.3·12-s + 364.·13-s + 867.·14-s − 794.·16-s + 1.30e3·17-s + 412.·18-s + 619.·19-s + 1.53e3·21-s + 616.·22-s − 922.·23-s − 1.74e3·24-s + 1.85e3·26-s + 729·27-s − 1.02e3·28-s − 3.43e3·29-s + 1.47e3·31-s + 2.15e3·32-s + 1.08e3·33-s + 6.66e3·34-s + ⋯
L(s)  = 1  + 0.900·2-s + 0.577·3-s − 0.188·4-s + 0.520·6-s + 1.31·7-s − 1.07·8-s + 0.333·9-s + 0.301·11-s − 0.108·12-s + 0.597·13-s + 1.18·14-s − 0.775·16-s + 1.09·17-s + 0.300·18-s + 0.393·19-s + 0.758·21-s + 0.271·22-s − 0.363·23-s − 0.618·24-s + 0.538·26-s + 0.192·27-s − 0.247·28-s − 0.758·29-s + 0.276·31-s + 0.371·32-s + 0.174·33-s + 0.988·34-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

\(L(3)\) \(\approx\) \(5.102981694\)
\(L(\frac12)\) \(\approx\) \(5.102981694\)
\(L(\frac{7}{2})\) 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 \( 1 - 5.09T + 32T^{2} \)
7 \( 1 - 170.T + 1.68e4T^{2} \)
13 \( 1 - 364.T + 3.71e5T^{2} \)
17 \( 1 - 1.30e3T + 1.41e6T^{2} \)
19 \( 1 - 619.T + 2.47e6T^{2} \)
23 \( 1 + 922.T + 6.43e6T^{2} \)
29 \( 1 + 3.43e3T + 2.05e7T^{2} \)
31 \( 1 - 1.47e3T + 2.86e7T^{2} \)
37 \( 1 - 5.28e3T + 6.93e7T^{2} \)
41 \( 1 + 7.66e3T + 1.15e8T^{2} \)
43 \( 1 + 1.73e4T + 1.47e8T^{2} \)
47 \( 1 + 6.94e3T + 2.29e8T^{2} \)
53 \( 1 - 1.73e4T + 4.18e8T^{2} \)
59 \( 1 - 3.40e4T + 7.14e8T^{2} \)
61 \( 1 - 4.85e4T + 8.44e8T^{2} \)
67 \( 1 - 3.43e4T + 1.35e9T^{2} \)
71 \( 1 - 4.84e4T + 1.80e9T^{2} \)
73 \( 1 + 5.94e4T + 2.07e9T^{2} \)
79 \( 1 - 3.59e3T + 3.07e9T^{2} \)
83 \( 1 - 6.43e4T + 3.93e9T^{2} \)
89 \( 1 - 2.69e4T + 5.58e9T^{2} \)
97 \( 1 - 1.65e4T + 8.58e9T^{2} \)
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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

−9.418932184579612606149260878670, −8.435893519344903113980492060469, −8.002358008382458492064515803366, −6.81320575503578654512730829420, −5.63562368410957857468299386936, −5.00134047337201393363177699558, −4.00650696574819263876082553467, −3.32978701290571390849319690401, −2.02552505856185953283847959589, −0.918966766439278514611563003644, 0.918966766439278514611563003644, 2.02552505856185953283847959589, 3.32978701290571390849319690401, 4.00650696574819263876082553467, 5.00134047337201393363177699558, 5.63562368410957857468299386936, 6.81320575503578654512730829420, 8.002358008382458492064515803366, 8.435893519344903113980492060469, 9.418932184579612606149260878670

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