Properties

Label 2-825-1.1-c5-0-154
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  + 10.4·2-s − 9·3-s + 76.7·4-s − 93.8·6-s − 21.9·7-s + 467.·8-s + 81·9-s + 121·11-s − 691.·12-s − 460.·13-s − 228.·14-s + 2.41e3·16-s − 2.31e3·17-s + 844.·18-s + 800.·19-s + 197.·21-s + 1.26e3·22-s − 4.78e3·23-s − 4.20e3·24-s − 4.80e3·26-s − 729·27-s − 1.68e3·28-s + 911.·29-s − 944.·31-s + 1.02e4·32-s − 1.08e3·33-s − 2.41e4·34-s + ⋯
L(s)  = 1  + 1.84·2-s − 0.577·3-s + 2.39·4-s − 1.06·6-s − 0.168·7-s + 2.58·8-s + 0.333·9-s + 0.301·11-s − 1.38·12-s − 0.756·13-s − 0.311·14-s + 2.35·16-s − 1.94·17-s + 0.614·18-s + 0.508·19-s + 0.0975·21-s + 0.555·22-s − 1.88·23-s − 1.49·24-s − 1.39·26-s − 0.192·27-s − 0.405·28-s + 0.201·29-s − 0.176·31-s + 1.76·32-s − 0.174·33-s − 3.58·34-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 \( 1 - 10.4T + 32T^{2} \)
7 \( 1 + 21.9T + 1.68e4T^{2} \)
13 \( 1 + 460.T + 3.71e5T^{2} \)
17 \( 1 + 2.31e3T + 1.41e6T^{2} \)
19 \( 1 - 800.T + 2.47e6T^{2} \)
23 \( 1 + 4.78e3T + 6.43e6T^{2} \)
29 \( 1 - 911.T + 2.05e7T^{2} \)
31 \( 1 + 944.T + 2.86e7T^{2} \)
37 \( 1 + 1.28e4T + 6.93e7T^{2} \)
41 \( 1 - 8.71e3T + 1.15e8T^{2} \)
43 \( 1 - 2.05e4T + 1.47e8T^{2} \)
47 \( 1 - 9.28e3T + 2.29e8T^{2} \)
53 \( 1 + 2.32e4T + 4.18e8T^{2} \)
59 \( 1 - 2.07e4T + 7.14e8T^{2} \)
61 \( 1 + 6.62e3T + 8.44e8T^{2} \)
67 \( 1 + 5.79e4T + 1.35e9T^{2} \)
71 \( 1 - 3.83e4T + 1.80e9T^{2} \)
73 \( 1 + 8.43e3T + 2.07e9T^{2} \)
79 \( 1 + 1.68e3T + 3.07e9T^{2} \)
83 \( 1 + 7.79e3T + 3.93e9T^{2} \)
89 \( 1 + 8.30e4T + 5.58e9T^{2} \)
97 \( 1 + 1.46e4T + 8.58e9T^{2} \)
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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

−9.174930036440988043294846531742, −7.73048462784012717249366248619, −6.85693084488443854533495817244, −6.23071225635600083334163959042, −5.42064608160843308867344955042, −4.49823426600331224326826515965, −3.93665609793883041220292203020, −2.65453609877619904798305741251, −1.78747175982882703012740809442, 0, 1.78747175982882703012740809442, 2.65453609877619904798305741251, 3.93665609793883041220292203020, 4.49823426600331224326826515965, 5.42064608160843308867344955042, 6.23071225635600083334163959042, 6.85693084488443854533495817244, 7.73048462784012717249366248619, 9.174930036440988043294846531742

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