Properties

Label 2-825-1.1-c5-0-13
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  − 8.18·2-s − 9·3-s + 35.0·4-s + 73.6·6-s + 163.·7-s − 24.5·8-s + 81·9-s − 121·11-s − 315.·12-s − 728.·13-s − 1.33e3·14-s − 918.·16-s − 1.47e3·17-s − 663.·18-s − 341.·19-s − 1.47e3·21-s + 990.·22-s − 2.94e3·23-s + 221.·24-s + 5.96e3·26-s − 729·27-s + 5.72e3·28-s − 2.82e3·29-s + 5.16e3·31-s + 8.30e3·32-s + 1.08e3·33-s + 1.20e4·34-s + ⋯
L(s)  = 1  − 1.44·2-s − 0.577·3-s + 1.09·4-s + 0.835·6-s + 1.26·7-s − 0.135·8-s + 0.333·9-s − 0.301·11-s − 0.631·12-s − 1.19·13-s − 1.82·14-s − 0.897·16-s − 1.23·17-s − 0.482·18-s − 0.217·19-s − 0.728·21-s + 0.436·22-s − 1.16·23-s + 0.0784·24-s + 1.73·26-s − 0.192·27-s + 1.38·28-s − 0.623·29-s + 0.964·31-s + 1.43·32-s + 0.174·33-s + 1.79·34-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

\(L(3)\) \(\approx\) \(0.4290884812\)
\(L(\frac12)\) \(\approx\) \(0.4290884812\)
\(L(\frac{7}{2})\) 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 \( 1 + 8.18T + 32T^{2} \)
7 \( 1 - 163.T + 1.68e4T^{2} \)
13 \( 1 + 728.T + 3.71e5T^{2} \)
17 \( 1 + 1.47e3T + 1.41e6T^{2} \)
19 \( 1 + 341.T + 2.47e6T^{2} \)
23 \( 1 + 2.94e3T + 6.43e6T^{2} \)
29 \( 1 + 2.82e3T + 2.05e7T^{2} \)
31 \( 1 - 5.16e3T + 2.86e7T^{2} \)
37 \( 1 + 4.25e3T + 6.93e7T^{2} \)
41 \( 1 - 5.87e3T + 1.15e8T^{2} \)
43 \( 1 + 1.66e3T + 1.47e8T^{2} \)
47 \( 1 - 5.98e3T + 2.29e8T^{2} \)
53 \( 1 + 2.75e4T + 4.18e8T^{2} \)
59 \( 1 - 8.24e3T + 7.14e8T^{2} \)
61 \( 1 + 2.34e3T + 8.44e8T^{2} \)
67 \( 1 + 4.94e4T + 1.35e9T^{2} \)
71 \( 1 + 2.46e4T + 1.80e9T^{2} \)
73 \( 1 - 2.30e4T + 2.07e9T^{2} \)
79 \( 1 - 8.72e4T + 3.07e9T^{2} \)
83 \( 1 + 9.05e4T + 3.93e9T^{2} \)
89 \( 1 - 8.62e4T + 5.58e9T^{2} \)
97 \( 1 + 1.30e4T + 8.58e9T^{2} \)
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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

−9.498973499944493568795340246805, −8.603189500050774268512270537751, −7.86166352090701694175797688675, −7.26467522991358564288711691417, −6.23697988148978293775697045596, −4.96898869440547700130906896149, −4.36734554318463253070638415890, −2.34740040754293296222443322203, −1.64258465624372901292740152231, −0.38285967482634164063268485726, 0.38285967482634164063268485726, 1.64258465624372901292740152231, 2.34740040754293296222443322203, 4.36734554318463253070638415890, 4.96898869440547700130906896149, 6.23697988148978293775697045596, 7.26467522991358564288711691417, 7.86166352090701694175797688675, 8.603189500050774268512270537751, 9.498973499944493568795340246805

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