Properties

Label 2-825-1.1-c5-0-22
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  + 0.0402·2-s − 9·3-s − 31.9·4-s − 0.362·6-s − 37.9·7-s − 2.57·8-s + 81·9-s + 121·11-s + 287.·12-s + 338.·13-s − 1.52·14-s + 1.02e3·16-s − 57.1·17-s + 3.26·18-s − 1.25e3·19-s + 341.·21-s + 4.87·22-s − 3.45e3·23-s + 23.1·24-s + 13.6·26-s − 729·27-s + 1.21e3·28-s + 7.48e3·29-s + 3.77e3·31-s + 123.·32-s − 1.08e3·33-s − 2.30·34-s + ⋯
L(s)  = 1  + 0.00711·2-s − 0.577·3-s − 0.999·4-s − 0.00410·6-s − 0.292·7-s − 0.0142·8-s + 0.333·9-s + 0.301·11-s + 0.577·12-s + 0.556·13-s − 0.00208·14-s + 0.999·16-s − 0.0479·17-s + 0.00237·18-s − 0.795·19-s + 0.168·21-s + 0.00214·22-s − 1.36·23-s + 0.00821·24-s + 0.00395·26-s − 0.192·27-s + 0.292·28-s + 1.65·29-s + 0.705·31-s + 0.0213·32-s − 0.174·33-s − 0.000341·34-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

\(L(3)\) \(\approx\) \(0.8475827965\)
\(L(\frac12)\) \(\approx\) \(0.8475827965\)
\(L(\frac{7}{2})\) 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 \( 1 - 0.0402T + 32T^{2} \)
7 \( 1 + 37.9T + 1.68e4T^{2} \)
13 \( 1 - 338.T + 3.71e5T^{2} \)
17 \( 1 + 57.1T + 1.41e6T^{2} \)
19 \( 1 + 1.25e3T + 2.47e6T^{2} \)
23 \( 1 + 3.45e3T + 6.43e6T^{2} \)
29 \( 1 - 7.48e3T + 2.05e7T^{2} \)
31 \( 1 - 3.77e3T + 2.86e7T^{2} \)
37 \( 1 + 4.95e3T + 6.93e7T^{2} \)
41 \( 1 + 191.T + 1.15e8T^{2} \)
43 \( 1 + 2.78e3T + 1.47e8T^{2} \)
47 \( 1 + 2.17e4T + 2.29e8T^{2} \)
53 \( 1 + 2.74e4T + 4.18e8T^{2} \)
59 \( 1 + 3.07e4T + 7.14e8T^{2} \)
61 \( 1 - 2.91e4T + 8.44e8T^{2} \)
67 \( 1 + 1.17e4T + 1.35e9T^{2} \)
71 \( 1 - 3.43e4T + 1.80e9T^{2} \)
73 \( 1 + 6.50e4T + 2.07e9T^{2} \)
79 \( 1 - 5.76e4T + 3.07e9T^{2} \)
83 \( 1 - 8.74e4T + 3.93e9T^{2} \)
89 \( 1 + 1.01e5T + 5.58e9T^{2} \)
97 \( 1 + 1.06e5T + 8.58e9T^{2} \)
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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

−9.627880621386183213265715153546, −8.549208570498317265260637394122, −8.029632833776491874818485435362, −6.59633597836075812322972273223, −6.06967210335749610159163057593, −4.91829344504401261014732199970, −4.23218693547831422758166387640, −3.22854222492915645762120126479, −1.60881395679846757565073384401, −0.44727689307770138813179281320, 0.44727689307770138813179281320, 1.60881395679846757565073384401, 3.22854222492915645762120126479, 4.23218693547831422758166387640, 4.91829344504401261014732199970, 6.06967210335749610159163057593, 6.59633597836075812322972273223, 8.029632833776491874818485435362, 8.549208570498317265260637394122, 9.627880621386183213265715153546

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