Properties

Label 2-825-1.1-c5-0-48
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  − 3.54·2-s − 9·3-s − 19.4·4-s + 31.9·6-s − 15.4·7-s + 182.·8-s + 81·9-s + 121·11-s + 174.·12-s + 9.16·13-s + 54.6·14-s − 24.5·16-s + 1.48e3·17-s − 287.·18-s + 1.05e3·19-s + 138.·21-s − 428.·22-s + 3.00e3·23-s − 1.64e3·24-s − 32.4·26-s − 729·27-s + 299.·28-s + 3.76e3·29-s + 1.37e3·31-s − 5.74e3·32-s − 1.08e3·33-s − 5.25e3·34-s + ⋯
L(s)  = 1  − 0.626·2-s − 0.577·3-s − 0.607·4-s + 0.361·6-s − 0.118·7-s + 1.00·8-s + 0.333·9-s + 0.301·11-s + 0.350·12-s + 0.0150·13-s + 0.0745·14-s − 0.0240·16-s + 1.24·17-s − 0.208·18-s + 0.672·19-s + 0.0686·21-s − 0.188·22-s + 1.18·23-s − 0.581·24-s − 0.00942·26-s − 0.192·27-s + 0.0722·28-s + 0.830·29-s + 0.257·31-s − 0.992·32-s − 0.174·33-s − 0.779·34-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

\(L(3)\) \(\approx\) \(1.224206063\)
\(L(\frac12)\) \(\approx\) \(1.224206063\)
\(L(\frac{7}{2})\) 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 \( 1 + 3.54T + 32T^{2} \)
7 \( 1 + 15.4T + 1.68e4T^{2} \)
13 \( 1 - 9.16T + 3.71e5T^{2} \)
17 \( 1 - 1.48e3T + 1.41e6T^{2} \)
19 \( 1 - 1.05e3T + 2.47e6T^{2} \)
23 \( 1 - 3.00e3T + 6.43e6T^{2} \)
29 \( 1 - 3.76e3T + 2.05e7T^{2} \)
31 \( 1 - 1.37e3T + 2.86e7T^{2} \)
37 \( 1 + 2.01e3T + 6.93e7T^{2} \)
41 \( 1 - 1.65e4T + 1.15e8T^{2} \)
43 \( 1 - 1.36e4T + 1.47e8T^{2} \)
47 \( 1 + 6.63e3T + 2.29e8T^{2} \)
53 \( 1 - 7.71e3T + 4.18e8T^{2} \)
59 \( 1 - 3.91e4T + 7.14e8T^{2} \)
61 \( 1 + 8.22e3T + 8.44e8T^{2} \)
67 \( 1 + 3.82e4T + 1.35e9T^{2} \)
71 \( 1 + 4.84e4T + 1.80e9T^{2} \)
73 \( 1 + 4.93e4T + 2.07e9T^{2} \)
79 \( 1 - 1.64e4T + 3.07e9T^{2} \)
83 \( 1 - 419.T + 3.93e9T^{2} \)
89 \( 1 + 1.70e4T + 5.58e9T^{2} \)
97 \( 1 - 3.46e3T + 8.58e9T^{2} \)
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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

−9.523179958941974971613252535494, −8.761733436053459249393592053119, −7.78399328981927885667116571245, −7.09534166452843455611163065621, −5.91522349651512698045258776155, −5.08532115358202988478096013372, −4.19062881115637815113873514879, −3.02765840735965265349747479827, −1.32160035341704285520744341609, −0.67071606108734986708318034177, 0.67071606108734986708318034177, 1.32160035341704285520744341609, 3.02765840735965265349747479827, 4.19062881115637815113873514879, 5.08532115358202988478096013372, 5.91522349651512698045258776155, 7.09534166452843455611163065621, 7.78399328981927885667116571245, 8.761733436053459249393592053119, 9.523179958941974971613252535494

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