Properties

Label 2-825-1.1-c5-0-115
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  + 10.0·2-s + 9·3-s + 68.9·4-s + 90.4·6-s − 29.0·7-s + 370.·8-s + 81·9-s + 121·11-s + 620.·12-s + 1.02e3·13-s − 291.·14-s + 1.52e3·16-s − 1.50e3·17-s + 813.·18-s + 1.64e3·19-s − 261.·21-s + 1.21e3·22-s + 1.47e3·23-s + 3.33e3·24-s + 1.02e4·26-s + 729·27-s − 2.00e3·28-s − 4.57e3·29-s + 7.53e3·31-s + 3.40e3·32-s + 1.08e3·33-s − 1.51e4·34-s + ⋯
L(s)  = 1  + 1.77·2-s + 0.577·3-s + 2.15·4-s + 1.02·6-s − 0.224·7-s + 2.04·8-s + 0.333·9-s + 0.301·11-s + 1.24·12-s + 1.67·13-s − 0.397·14-s + 1.48·16-s − 1.26·17-s + 0.591·18-s + 1.04·19-s − 0.129·21-s + 0.535·22-s + 0.582·23-s + 1.18·24-s + 2.98·26-s + 0.192·27-s − 0.482·28-s − 1.00·29-s + 1.40·31-s + 0.588·32-s + 0.174·33-s − 2.25·34-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

\(L(3)\) \(\approx\) \(10.21416213\)
\(L(\frac12)\) \(\approx\) \(10.21416213\)
\(L(\frac{7}{2})\) 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 \( 1 - 10.0T + 32T^{2} \)
7 \( 1 + 29.0T + 1.68e4T^{2} \)
13 \( 1 - 1.02e3T + 3.71e5T^{2} \)
17 \( 1 + 1.50e3T + 1.41e6T^{2} \)
19 \( 1 - 1.64e3T + 2.47e6T^{2} \)
23 \( 1 - 1.47e3T + 6.43e6T^{2} \)
29 \( 1 + 4.57e3T + 2.05e7T^{2} \)
31 \( 1 - 7.53e3T + 2.86e7T^{2} \)
37 \( 1 - 4.40e3T + 6.93e7T^{2} \)
41 \( 1 - 5.62e3T + 1.15e8T^{2} \)
43 \( 1 + 2.28e3T + 1.47e8T^{2} \)
47 \( 1 - 5.98e3T + 2.29e8T^{2} \)
53 \( 1 - 2.84e4T + 4.18e8T^{2} \)
59 \( 1 - 2.59e4T + 7.14e8T^{2} \)
61 \( 1 + 5.13e4T + 8.44e8T^{2} \)
67 \( 1 + 3.91e4T + 1.35e9T^{2} \)
71 \( 1 - 3.75e4T + 1.80e9T^{2} \)
73 \( 1 - 5.86e4T + 2.07e9T^{2} \)
79 \( 1 + 8.27e3T + 3.07e9T^{2} \)
83 \( 1 + 2.13e4T + 3.93e9T^{2} \)
89 \( 1 - 7.83e4T + 5.58e9T^{2} \)
97 \( 1 - 6.45e4T + 8.58e9T^{2} \)
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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

−9.424380803856558045471652933583, −8.578871614292420473921380973515, −7.45309677572775185912003922113, −6.55098609692131040078874784787, −5.95348259646288810846439704321, −4.84830468399329370587143441427, −3.96792779674101863914753172876, −3.31195730635561776280945142231, −2.35747004406896197428414827142, −1.17457296588602887316471906788, 1.17457296588602887316471906788, 2.35747004406896197428414827142, 3.31195730635561776280945142231, 3.96792779674101863914753172876, 4.84830468399329370587143441427, 5.95348259646288810846439704321, 6.55098609692131040078874784787, 7.45309677572775185912003922113, 8.578871614292420473921380973515, 9.424380803856558045471652933583

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