Properties

Label 2-825-1.1-c5-0-105
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  + 8.57·2-s + 9·3-s + 41.5·4-s + 77.1·6-s + 178.·7-s + 81.7·8-s + 81·9-s + 121·11-s + 373.·12-s − 361.·13-s + 1.53e3·14-s − 627.·16-s + 934.·17-s + 694.·18-s + 753.·19-s + 1.60e3·21-s + 1.03e3·22-s + 3.23e3·23-s + 735.·24-s − 3.10e3·26-s + 729·27-s + 7.41e3·28-s + 2.60e3·29-s + 662.·31-s − 8.00e3·32-s + 1.08e3·33-s + 8.00e3·34-s + ⋯
L(s)  = 1  + 1.51·2-s + 0.577·3-s + 1.29·4-s + 0.875·6-s + 1.37·7-s + 0.451·8-s + 0.333·9-s + 0.301·11-s + 0.749·12-s − 0.594·13-s + 2.08·14-s − 0.613·16-s + 0.783·17-s + 0.505·18-s + 0.479·19-s + 0.794·21-s + 0.457·22-s + 1.27·23-s + 0.260·24-s − 0.900·26-s + 0.192·27-s + 1.78·28-s + 0.575·29-s + 0.123·31-s − 1.38·32-s + 0.174·33-s + 1.18·34-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

\(L(3)\) \(\approx\) \(8.755234519\)
\(L(\frac12)\) \(\approx\) \(8.755234519\)
\(L(\frac{7}{2})\) 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 \( 1 - 8.57T + 32T^{2} \)
7 \( 1 - 178.T + 1.68e4T^{2} \)
13 \( 1 + 361.T + 3.71e5T^{2} \)
17 \( 1 - 934.T + 1.41e6T^{2} \)
19 \( 1 - 753.T + 2.47e6T^{2} \)
23 \( 1 - 3.23e3T + 6.43e6T^{2} \)
29 \( 1 - 2.60e3T + 2.05e7T^{2} \)
31 \( 1 - 662.T + 2.86e7T^{2} \)
37 \( 1 + 1.29e4T + 6.93e7T^{2} \)
41 \( 1 + 2.53e3T + 1.15e8T^{2} \)
43 \( 1 - 2.20e4T + 1.47e8T^{2} \)
47 \( 1 - 2.08e4T + 2.29e8T^{2} \)
53 \( 1 + 2.74e4T + 4.18e8T^{2} \)
59 \( 1 - 7.75e3T + 7.14e8T^{2} \)
61 \( 1 - 3.84e4T + 8.44e8T^{2} \)
67 \( 1 - 3.55e4T + 1.35e9T^{2} \)
71 \( 1 + 6.26e4T + 1.80e9T^{2} \)
73 \( 1 - 6.89e4T + 2.07e9T^{2} \)
79 \( 1 + 1.73e4T + 3.07e9T^{2} \)
83 \( 1 + 8.90e4T + 3.93e9T^{2} \)
89 \( 1 - 1.29e5T + 5.58e9T^{2} \)
97 \( 1 + 1.36e5T + 8.58e9T^{2} \)
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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

−9.395378890303712976605203432280, −8.534308352084390205789033288481, −7.56175040716609392530523522350, −6.85567473352579741481087114211, −5.54701892096459821252056974984, −4.97176697208473100429513814575, −4.16115881436567396676086060823, −3.18108446384261292693498672593, −2.26210101027825062205978438853, −1.12630357207077386271308885524, 1.12630357207077386271308885524, 2.26210101027825062205978438853, 3.18108446384261292693498672593, 4.16115881436567396676086060823, 4.97176697208473100429513814575, 5.54701892096459821252056974984, 6.85567473352579741481087114211, 7.56175040716609392530523522350, 8.534308352084390205789033288481, 9.395378890303712976605203432280

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