Properties

Label 2-825-1.1-c5-0-138
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 + 77.9·4-s − 94.3·6-s + 152.·7-s − 481.·8-s + 81·9-s + 121·11-s + 701.·12-s + 260.·13-s − 1.60e3·14-s + 2.55e3·16-s − 586.·17-s − 849.·18-s − 614.·19-s + 1.37e3·21-s − 1.26e3·22-s + 1.37e3·23-s − 4.33e3·24-s − 2.72e3·26-s + 729·27-s + 1.19e4·28-s − 4.06e3·29-s − 2.68e3·31-s − 1.13e4·32-s + 1.08e3·33-s + 6.15e3·34-s + ⋯
L(s)  = 1  − 1.85·2-s + 0.577·3-s + 2.43·4-s − 1.07·6-s + 1.17·7-s − 2.66·8-s + 0.333·9-s + 0.301·11-s + 1.40·12-s + 0.426·13-s − 2.18·14-s + 2.49·16-s − 0.492·17-s − 0.617·18-s − 0.390·19-s + 0.681·21-s − 0.558·22-s + 0.542·23-s − 1.53·24-s − 0.791·26-s + 0.192·27-s + 2.87·28-s − 0.896·29-s − 0.502·31-s − 1.96·32-s + 0.174·33-s + 0.912·34-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 \( 1 - 152.T + 1.68e4T^{2} \)
13 \( 1 - 260.T + 3.71e5T^{2} \)
17 \( 1 + 586.T + 1.41e6T^{2} \)
19 \( 1 + 614.T + 2.47e6T^{2} \)
23 \( 1 - 1.37e3T + 6.43e6T^{2} \)
29 \( 1 + 4.06e3T + 2.05e7T^{2} \)
31 \( 1 + 2.68e3T + 2.86e7T^{2} \)
37 \( 1 + 5.30e3T + 6.93e7T^{2} \)
41 \( 1 + 3.30e3T + 1.15e8T^{2} \)
43 \( 1 - 1.95e4T + 1.47e8T^{2} \)
47 \( 1 + 1.54e4T + 2.29e8T^{2} \)
53 \( 1 + 1.73e4T + 4.18e8T^{2} \)
59 \( 1 + 5.15e4T + 7.14e8T^{2} \)
61 \( 1 + 3.00e4T + 8.44e8T^{2} \)
67 \( 1 - 3.35e4T + 1.35e9T^{2} \)
71 \( 1 + 7.41e4T + 1.80e9T^{2} \)
73 \( 1 + 3.94e4T + 2.07e9T^{2} \)
79 \( 1 + 2.63e4T + 3.07e9T^{2} \)
83 \( 1 + 5.54e4T + 3.93e9T^{2} \)
89 \( 1 + 1.19e4T + 5.58e9T^{2} \)
97 \( 1 + 1.23e5T + 8.58e9T^{2} \)
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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

−8.970779690624769721229100637775, −8.370260135548802570355111721126, −7.65671564902208825555995612861, −6.96600054434470613955627628373, −5.89364318032810331877057677762, −4.46308109862332177505595571408, −3.06080289600461421282673557320, −1.87827805639411217032559007564, −1.36082101776373264380946692242, 0, 1.36082101776373264380946692242, 1.87827805639411217032559007564, 3.06080289600461421282673557320, 4.46308109862332177505595571408, 5.89364318032810331877057677762, 6.96600054434470613955627628373, 7.65671564902208825555995612861, 8.370260135548802570355111721126, 8.970779690624769721229100637775

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