Properties

Label 2-825-1.1-c5-0-36
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  + 5.64·2-s + 9·3-s − 0.188·4-s + 50.7·6-s − 145.·7-s − 181.·8-s + 81·9-s − 121·11-s − 1.69·12-s + 69.9·13-s − 817.·14-s − 1.01e3·16-s + 500.·17-s + 456.·18-s − 670.·19-s − 1.30e3·21-s − 682.·22-s − 791.·23-s − 1.63e3·24-s + 394.·26-s + 729·27-s + 27.3·28-s + 1.54e3·29-s + 2.70e3·31-s + 68.2·32-s − 1.08e3·33-s + 2.82e3·34-s + ⋯
L(s)  = 1  + 0.997·2-s + 0.577·3-s − 0.00588·4-s + 0.575·6-s − 1.11·7-s − 1.00·8-s + 0.333·9-s − 0.301·11-s − 0.00339·12-s + 0.114·13-s − 1.11·14-s − 0.994·16-s + 0.420·17-s + 0.332·18-s − 0.426·19-s − 0.645·21-s − 0.300·22-s − 0.311·23-s − 0.579·24-s + 0.114·26-s + 0.192·27-s + 0.00658·28-s + 0.341·29-s + 0.505·31-s + 0.0117·32-s − 0.174·33-s + 0.418·34-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: $\chi_{825} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.877884043\)
\(L(\frac12)\) \(\approx\) \(2.877884043\)
\(L(\frac{7}{2})\) 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 \( 1 - 5.64T + 32T^{2} \)
7 \( 1 + 145.T + 1.68e4T^{2} \)
13 \( 1 - 69.9T + 3.71e5T^{2} \)
17 \( 1 - 500.T + 1.41e6T^{2} \)
19 \( 1 + 670.T + 2.47e6T^{2} \)
23 \( 1 + 791.T + 6.43e6T^{2} \)
29 \( 1 - 1.54e3T + 2.05e7T^{2} \)
31 \( 1 - 2.70e3T + 2.86e7T^{2} \)
37 \( 1 - 2.66e3T + 6.93e7T^{2} \)
41 \( 1 - 9.62e3T + 1.15e8T^{2} \)
43 \( 1 - 6.68e3T + 1.47e8T^{2} \)
47 \( 1 - 1.16e3T + 2.29e8T^{2} \)
53 \( 1 + 2.88e4T + 4.18e8T^{2} \)
59 \( 1 - 2.35e4T + 7.14e8T^{2} \)
61 \( 1 - 1.76e4T + 8.44e8T^{2} \)
67 \( 1 + 1.65e4T + 1.35e9T^{2} \)
71 \( 1 - 7.20e4T + 1.80e9T^{2} \)
73 \( 1 - 4.54e4T + 2.07e9T^{2} \)
79 \( 1 + 2.16e4T + 3.07e9T^{2} \)
83 \( 1 + 6.93e3T + 3.93e9T^{2} \)
89 \( 1 - 4.27e4T + 5.58e9T^{2} \)
97 \( 1 - 2.09e4T + 8.58e9T^{2} \)
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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

−9.509466977606516008424136571652, −8.694128891843557596694585803002, −7.76528065066203192817561623947, −6.59681478688753380727396214219, −5.97113568836800681539075944419, −4.89775317683149519071407058270, −3.94079489210083763665820737705, −3.21085468562664707665454223682, −2.36846396987066372542909394937, −0.62003610740880892615285872975, 0.62003610740880892615285872975, 2.36846396987066372542909394937, 3.21085468562664707665454223682, 3.94079489210083763665820737705, 4.89775317683149519071407058270, 5.97113568836800681539075944419, 6.59681478688753380727396214219, 7.76528065066203192817561623947, 8.694128891843557596694585803002, 9.509466977606516008424136571652

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