Properties

Label 2-825-1.1-c5-0-30
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  − 5.77·2-s − 9·3-s + 1.37·4-s + 51.9·6-s − 150.·7-s + 176.·8-s + 81·9-s − 121·11-s − 12.3·12-s + 1.14e3·13-s + 867.·14-s − 1.06e3·16-s − 1.57e3·17-s − 467.·18-s + 731.·19-s + 1.35e3·21-s + 698.·22-s + 2.18e3·23-s − 1.59e3·24-s − 6.62e3·26-s − 729·27-s − 205.·28-s + 8.60e3·29-s + 6.70e3·31-s + 495.·32-s + 1.08e3·33-s + 9.07e3·34-s + ⋯
L(s)  = 1  − 1.02·2-s − 0.577·3-s + 0.0428·4-s + 0.589·6-s − 1.15·7-s + 0.977·8-s + 0.333·9-s − 0.301·11-s − 0.0247·12-s + 1.88·13-s + 1.18·14-s − 1.04·16-s − 1.31·17-s − 0.340·18-s + 0.464·19-s + 0.668·21-s + 0.307·22-s + 0.860·23-s − 0.564·24-s − 1.92·26-s − 0.192·27-s − 0.0496·28-s + 1.89·29-s + 1.25·31-s + 0.0856·32-s + 0.174·33-s + 1.34·34-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

\(L(3)\) \(\approx\) \(0.6976089963\)
\(L(\frac12)\) \(\approx\) \(0.6976089963\)
\(L(\frac{7}{2})\) 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 \( 1 + 5.77T + 32T^{2} \)
7 \( 1 + 150.T + 1.68e4T^{2} \)
13 \( 1 - 1.14e3T + 3.71e5T^{2} \)
17 \( 1 + 1.57e3T + 1.41e6T^{2} \)
19 \( 1 - 731.T + 2.47e6T^{2} \)
23 \( 1 - 2.18e3T + 6.43e6T^{2} \)
29 \( 1 - 8.60e3T + 2.05e7T^{2} \)
31 \( 1 - 6.70e3T + 2.86e7T^{2} \)
37 \( 1 + 3.05e3T + 6.93e7T^{2} \)
41 \( 1 + 440.T + 1.15e8T^{2} \)
43 \( 1 + 1.35e4T + 1.47e8T^{2} \)
47 \( 1 - 1.53e4T + 2.29e8T^{2} \)
53 \( 1 + 2.54e4T + 4.18e8T^{2} \)
59 \( 1 + 8.79e3T + 7.14e8T^{2} \)
61 \( 1 - 1.52e4T + 8.44e8T^{2} \)
67 \( 1 + 4.62e4T + 1.35e9T^{2} \)
71 \( 1 - 6.80e4T + 1.80e9T^{2} \)
73 \( 1 + 1.28e4T + 2.07e9T^{2} \)
79 \( 1 - 1.09e4T + 3.07e9T^{2} \)
83 \( 1 - 6.68e4T + 3.93e9T^{2} \)
89 \( 1 + 6.65e4T + 5.58e9T^{2} \)
97 \( 1 - 1.45e5T + 8.58e9T^{2} \)
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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

−9.408446257530486790753626684196, −8.746586998940824586204002457755, −8.030126930153870969634833228476, −6.66549729687915700787039679264, −6.44766876181142896965701329721, −5.07399512961880828347809531864, −4.08011797031269542164170978887, −2.91286078916180609484240558203, −1.33508644169725273204226593857, −0.51591478659613766708256093230, 0.51591478659613766708256093230, 1.33508644169725273204226593857, 2.91286078916180609484240558203, 4.08011797031269542164170978887, 5.07399512961880828347809531864, 6.44766876181142896965701329721, 6.66549729687915700787039679264, 8.030126930153870969634833228476, 8.746586998940824586204002457755, 9.408446257530486790753626684196

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