Properties

Label 2-825-1.1-c5-0-111
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.870·2-s − 9·3-s − 31.2·4-s − 7.83·6-s + 236.·7-s − 55.0·8-s + 81·9-s − 121·11-s + 281.·12-s + 261.·13-s + 205.·14-s + 951.·16-s − 1.58e3·17-s + 70.5·18-s − 711.·19-s − 2.12e3·21-s − 105.·22-s − 2.96e3·23-s + 495.·24-s + 227.·26-s − 729·27-s − 7.39e3·28-s + 4.42e3·29-s − 4.44e3·31-s + 2.59e3·32-s + 1.08e3·33-s − 1.38e3·34-s + ⋯
L(s)  = 1  + 0.153·2-s − 0.577·3-s − 0.976·4-s − 0.0888·6-s + 1.82·7-s − 0.304·8-s + 0.333·9-s − 0.301·11-s + 0.563·12-s + 0.429·13-s + 0.280·14-s + 0.929·16-s − 1.33·17-s + 0.0512·18-s − 0.452·19-s − 1.05·21-s − 0.0463·22-s − 1.16·23-s + 0.175·24-s + 0.0661·26-s − 0.192·27-s − 1.78·28-s + 0.976·29-s − 0.830·31-s + 0.447·32-s + 0.174·33-s − 0.204·34-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: \(1\)
Selberg data: \((2,\ 825,\ (\ :5/2),\ -1)\)

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 \( 1 + 9T \)
5 \( 1 \)
11 \( 1 + 121T \)
good2 \( 1 - 0.870T + 32T^{2} \)
7 \( 1 - 236.T + 1.68e4T^{2} \)
13 \( 1 - 261.T + 3.71e5T^{2} \)
17 \( 1 + 1.58e3T + 1.41e6T^{2} \)
19 \( 1 + 711.T + 2.47e6T^{2} \)
23 \( 1 + 2.96e3T + 6.43e6T^{2} \)
29 \( 1 - 4.42e3T + 2.05e7T^{2} \)
31 \( 1 + 4.44e3T + 2.86e7T^{2} \)
37 \( 1 - 1.16e4T + 6.93e7T^{2} \)
41 \( 1 - 4.68e3T + 1.15e8T^{2} \)
43 \( 1 + 2.01e4T + 1.47e8T^{2} \)
47 \( 1 - 8.28e3T + 2.29e8T^{2} \)
53 \( 1 - 2.44e4T + 4.18e8T^{2} \)
59 \( 1 - 4.44e4T + 7.14e8T^{2} \)
61 \( 1 + 1.72e4T + 8.44e8T^{2} \)
67 \( 1 + 5.15e4T + 1.35e9T^{2} \)
71 \( 1 - 1.67e4T + 1.80e9T^{2} \)
73 \( 1 + 4.90e4T + 2.07e9T^{2} \)
79 \( 1 - 1.15e4T + 3.07e9T^{2} \)
83 \( 1 + 2.07e4T + 3.93e9T^{2} \)
89 \( 1 - 2.84e4T + 5.58e9T^{2} \)
97 \( 1 - 5.03e4T + 8.58e9T^{2} \)
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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

−8.800246693540748713388777378019, −8.358198067060178760688752975487, −7.48539753110778961871197749160, −6.20417049558566020543114449205, −5.33450337407057342253206061062, −4.55965843889216518274870451233, −4.03902410577290157575518014833, −2.24992140439565892788638824351, −1.16289169987838982676389286958, 0, 1.16289169987838982676389286958, 2.24992140439565892788638824351, 4.03902410577290157575518014833, 4.55965843889216518274870451233, 5.33450337407057342253206061062, 6.20417049558566020543114449205, 7.48539753110778961871197749160, 8.358198067060178760688752975487, 8.800246693540748713388777378019

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