Properties

Label 2-825-1.1-c5-0-35
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  + 3.27·2-s − 9·3-s − 21.2·4-s − 29.4·6-s + 115.·7-s − 174.·8-s + 81·9-s − 121·11-s + 191.·12-s + 883.·13-s + 378.·14-s + 110.·16-s − 519.·17-s + 265.·18-s − 1.59e3·19-s − 1.04e3·21-s − 395.·22-s + 1.46e3·23-s + 1.56e3·24-s + 2.88e3·26-s − 729·27-s − 2.46e3·28-s − 7.18e3·29-s + 7.83e3·31-s + 5.94e3·32-s + 1.08e3·33-s − 1.70e3·34-s + ⋯
L(s)  = 1  + 0.578·2-s − 0.577·3-s − 0.665·4-s − 0.333·6-s + 0.891·7-s − 0.963·8-s + 0.333·9-s − 0.301·11-s + 0.384·12-s + 1.44·13-s + 0.515·14-s + 0.108·16-s − 0.436·17-s + 0.192·18-s − 1.01·19-s − 0.514·21-s − 0.174·22-s + 0.576·23-s + 0.556·24-s + 0.838·26-s − 0.192·27-s − 0.593·28-s − 1.58·29-s + 1.46·31-s + 1.02·32-s + 0.174·33-s − 0.252·34-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

\(L(3)\) \(\approx\) \(1.868538163\)
\(L(\frac12)\) \(\approx\) \(1.868538163\)
\(L(\frac{7}{2})\) 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 \( 1 - 3.27T + 32T^{2} \)
7 \( 1 - 115.T + 1.68e4T^{2} \)
13 \( 1 - 883.T + 3.71e5T^{2} \)
17 \( 1 + 519.T + 1.41e6T^{2} \)
19 \( 1 + 1.59e3T + 2.47e6T^{2} \)
23 \( 1 - 1.46e3T + 6.43e6T^{2} \)
29 \( 1 + 7.18e3T + 2.05e7T^{2} \)
31 \( 1 - 7.83e3T + 2.86e7T^{2} \)
37 \( 1 - 1.03e4T + 6.93e7T^{2} \)
41 \( 1 + 3.16e3T + 1.15e8T^{2} \)
43 \( 1 + 1.59e4T + 1.47e8T^{2} \)
47 \( 1 + 1.07e4T + 2.29e8T^{2} \)
53 \( 1 - 2.69e3T + 4.18e8T^{2} \)
59 \( 1 + 4.69e4T + 7.14e8T^{2} \)
61 \( 1 - 2.11e3T + 8.44e8T^{2} \)
67 \( 1 - 5.85e4T + 1.35e9T^{2} \)
71 \( 1 + 5.70e4T + 1.80e9T^{2} \)
73 \( 1 + 2.92e4T + 2.07e9T^{2} \)
79 \( 1 - 7.36e4T + 3.07e9T^{2} \)
83 \( 1 - 3.90e4T + 3.93e9T^{2} \)
89 \( 1 - 1.28e4T + 5.58e9T^{2} \)
97 \( 1 - 2.36e4T + 8.58e9T^{2} \)
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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

−9.424275969692908520181353615506, −8.550328736871000519771721360986, −7.943334665508919372304650734482, −6.53527532776760505596666915905, −5.88151259107278914356609674803, −4.90531659352731061217767111699, −4.32311449854270908893654120298, −3.29380563690085412583703826954, −1.77408696669920202968660601991, −0.60143291256109746847266096642, 0.60143291256109746847266096642, 1.77408696669920202968660601991, 3.29380563690085412583703826954, 4.32311449854270908893654120298, 4.90531659352731061217767111699, 5.88151259107278914356609674803, 6.53527532776760505596666915905, 7.943334665508919372304650734482, 8.550328736871000519771721360986, 9.424275969692908520181353615506

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