Properties

Label 2-825-1.1-c3-0-17
Degree $2$
Conductor $825$
Sign $1$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.56·2-s − 3·3-s − 5.56·4-s + 4.68·6-s + 10.2·7-s + 21.1·8-s + 9·9-s − 11·11-s + 16.6·12-s + 40.8·13-s − 16·14-s + 11.4·16-s + 98.7·17-s − 14.0·18-s − 39.6·19-s − 30.7·21-s + 17.1·22-s − 61.6·23-s − 63.5·24-s − 63.8·26-s − 27·27-s − 56.9·28-s − 149.·29-s + 54.7·31-s − 187.·32-s + 33·33-s − 154.·34-s + ⋯
L(s)  = 1  − 0.552·2-s − 0.577·3-s − 0.695·4-s + 0.318·6-s + 0.553·7-s + 0.935·8-s + 0.333·9-s − 0.301·11-s + 0.401·12-s + 0.872·13-s − 0.305·14-s + 0.178·16-s + 1.40·17-s − 0.184·18-s − 0.478·19-s − 0.319·21-s + 0.166·22-s − 0.559·23-s − 0.540·24-s − 0.481·26-s − 0.192·27-s − 0.384·28-s − 0.954·29-s + 0.317·31-s − 1.03·32-s + 0.174·33-s − 0.777·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(48.6765\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.027444351\)
\(L(\frac12)\) \(\approx\) \(1.027444351\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
5 \( 1 \)
11 \( 1 + 11T \)
good2 \( 1 + 1.56T + 8T^{2} \)
7 \( 1 - 10.2T + 343T^{2} \)
13 \( 1 - 40.8T + 2.19e3T^{2} \)
17 \( 1 - 98.7T + 4.91e3T^{2} \)
19 \( 1 + 39.6T + 6.85e3T^{2} \)
23 \( 1 + 61.6T + 1.21e4T^{2} \)
29 \( 1 + 149.T + 2.43e4T^{2} \)
31 \( 1 - 54.7T + 2.97e4T^{2} \)
37 \( 1 + 44.8T + 5.06e4T^{2} \)
41 \( 1 - 336.T + 6.89e4T^{2} \)
43 \( 1 - 2.36T + 7.95e4T^{2} \)
47 \( 1 - 333.T + 1.03e5T^{2} \)
53 \( 1 + 640.T + 1.48e5T^{2} \)
59 \( 1 + 370.T + 2.05e5T^{2} \)
61 \( 1 + 714.T + 2.26e5T^{2} \)
67 \( 1 - 404.T + 3.00e5T^{2} \)
71 \( 1 - 939.T + 3.57e5T^{2} \)
73 \( 1 - 362.T + 3.89e5T^{2} \)
79 \( 1 - 951.T + 4.93e5T^{2} \)
83 \( 1 + 735.T + 5.71e5T^{2} \)
89 \( 1 - 385.T + 7.04e5T^{2} \)
97 \( 1 - 966.T + 9.12e5T^{2} \)
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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

−9.806986566435316226187738075720, −9.068030042095891331729383022713, −8.016825592674984079214629781883, −7.65570917597865097032426039793, −6.23415309639988120884263187498, −5.41503299734325727280676594738, −4.51972008784312345191272966679, −3.53585299880254668214246467297, −1.71164066558505707530058870290, −0.66978612872596675055054712958, 0.66978612872596675055054712958, 1.71164066558505707530058870290, 3.53585299880254668214246467297, 4.51972008784312345191272966679, 5.41503299734325727280676594738, 6.23415309639988120884263187498, 7.65570917597865097032426039793, 8.016825592674984079214629781883, 9.068030042095891331729383022713, 9.806986566435316226187738075720

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