Properties

Label 2-825-1.1-c5-0-69
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  − 1.08·2-s + 9·3-s − 30.8·4-s − 9.73·6-s + 139.·7-s + 67.9·8-s + 81·9-s − 121·11-s − 277.·12-s + 646.·13-s − 150.·14-s + 913.·16-s + 1.37e3·17-s − 87.5·18-s + 1.90e3·19-s + 1.25e3·21-s + 130.·22-s − 343.·23-s + 611.·24-s − 698.·26-s + 729·27-s − 4.30e3·28-s + 53.5·29-s + 634.·31-s − 3.16e3·32-s − 1.08e3·33-s − 1.49e3·34-s + ⋯
L(s)  = 1  − 0.191·2-s + 0.577·3-s − 0.963·4-s − 0.110·6-s + 1.07·7-s + 0.375·8-s + 0.333·9-s − 0.301·11-s − 0.556·12-s + 1.06·13-s − 0.205·14-s + 0.891·16-s + 1.15·17-s − 0.0637·18-s + 1.21·19-s + 0.621·21-s + 0.0576·22-s − 0.135·23-s + 0.216·24-s − 0.202·26-s + 0.192·27-s − 1.03·28-s + 0.0118·29-s + 0.118·31-s − 0.545·32-s − 0.174·33-s − 0.221·34-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

\(L(3)\) \(\approx\) \(2.781540747\)
\(L(\frac12)\) \(\approx\) \(2.781540747\)
\(L(\frac{7}{2})\) 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 \( 1 + 1.08T + 32T^{2} \)
7 \( 1 - 139.T + 1.68e4T^{2} \)
13 \( 1 - 646.T + 3.71e5T^{2} \)
17 \( 1 - 1.37e3T + 1.41e6T^{2} \)
19 \( 1 - 1.90e3T + 2.47e6T^{2} \)
23 \( 1 + 343.T + 6.43e6T^{2} \)
29 \( 1 - 53.5T + 2.05e7T^{2} \)
31 \( 1 - 634.T + 2.86e7T^{2} \)
37 \( 1 + 1.16e4T + 6.93e7T^{2} \)
41 \( 1 - 1.88e4T + 1.15e8T^{2} \)
43 \( 1 + 1.33e4T + 1.47e8T^{2} \)
47 \( 1 - 2.25e3T + 2.29e8T^{2} \)
53 \( 1 - 8.90e3T + 4.18e8T^{2} \)
59 \( 1 + 1.12e4T + 7.14e8T^{2} \)
61 \( 1 - 1.18e4T + 8.44e8T^{2} \)
67 \( 1 - 5.73e4T + 1.35e9T^{2} \)
71 \( 1 - 3.10e4T + 1.80e9T^{2} \)
73 \( 1 + 5.60e4T + 2.07e9T^{2} \)
79 \( 1 + 883.T + 3.07e9T^{2} \)
83 \( 1 + 9.39e4T + 3.93e9T^{2} \)
89 \( 1 - 2.17e4T + 5.58e9T^{2} \)
97 \( 1 - 1.58e5T + 8.58e9T^{2} \)
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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

−9.404342941132456143639012171175, −8.482215120775309968013891966663, −8.057116307075131438285682697028, −7.25510354805086355741050604983, −5.70828310327807472909894539110, −5.03701218988820549792891851670, −3.99961804413794600935408905284, −3.16457992086396753472234850663, −1.62034891384160577330339635658, −0.843343357606116739670221804741, 0.843343357606116739670221804741, 1.62034891384160577330339635658, 3.16457992086396753472234850663, 3.99961804413794600935408905284, 5.03701218988820549792891851670, 5.70828310327807472909894539110, 7.25510354805086355741050604983, 8.057116307075131438285682697028, 8.482215120775309968013891966663, 9.404342941132456143639012171175

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