Properties

Label 2-825-1.1-c3-0-41
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  − 3.59·2-s + 3·3-s + 4.89·4-s − 10.7·6-s + 16.1·7-s + 11.1·8-s + 9·9-s + 11·11-s + 14.6·12-s + 54.1·13-s − 57.9·14-s − 79.2·16-s + 107.·17-s − 32.3·18-s + 48.7·19-s + 48.4·21-s − 39.4·22-s − 11.9·23-s + 33.4·24-s − 194.·26-s + 27·27-s + 78.9·28-s + 239.·29-s − 82.0·31-s + 195.·32-s + 33·33-s − 384.·34-s + ⋯
L(s)  = 1  − 1.26·2-s + 0.577·3-s + 0.611·4-s − 0.732·6-s + 0.871·7-s + 0.493·8-s + 0.333·9-s + 0.301·11-s + 0.353·12-s + 1.15·13-s − 1.10·14-s − 1.23·16-s + 1.52·17-s − 0.423·18-s + 0.588·19-s + 0.503·21-s − 0.382·22-s − 0.108·23-s + 0.284·24-s − 1.46·26-s + 0.192·27-s + 0.533·28-s + 1.53·29-s − 0.475·31-s + 1.07·32-s + 0.174·33-s − 1.93·34-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

\(L(2)\) \(\approx\) \(1.737398709\)
\(L(\frac12)\) \(\approx\) \(1.737398709\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.59T + 8T^{2} \)
7 \( 1 - 16.1T + 343T^{2} \)
13 \( 1 - 54.1T + 2.19e3T^{2} \)
17 \( 1 - 107.T + 4.91e3T^{2} \)
19 \( 1 - 48.7T + 6.85e3T^{2} \)
23 \( 1 + 11.9T + 1.21e4T^{2} \)
29 \( 1 - 239.T + 2.43e4T^{2} \)
31 \( 1 + 82.0T + 2.97e4T^{2} \)
37 \( 1 - 21.7T + 5.06e4T^{2} \)
41 \( 1 + 124.T + 6.89e4T^{2} \)
43 \( 1 + 224.T + 7.95e4T^{2} \)
47 \( 1 - 186.T + 1.03e5T^{2} \)
53 \( 1 + 233.T + 1.48e5T^{2} \)
59 \( 1 - 232.T + 2.05e5T^{2} \)
61 \( 1 - 163.T + 2.26e5T^{2} \)
67 \( 1 - 876.T + 3.00e5T^{2} \)
71 \( 1 + 733.T + 3.57e5T^{2} \)
73 \( 1 + 1.16e3T + 3.89e5T^{2} \)
79 \( 1 + 588.T + 4.93e5T^{2} \)
83 \( 1 - 1.16e3T + 5.71e5T^{2} \)
89 \( 1 + 1.04e3T + 7.04e5T^{2} \)
97 \( 1 + 1.54e3T + 9.12e5T^{2} \)
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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

−9.784332700739636120047871256374, −8.824538265957090862984442657588, −8.274669889890678787790145893236, −7.70231136854552248755973304148, −6.74517118727960106577098322585, −5.45186028111385934401011142552, −4.31951304257228794079168632437, −3.17039257360102443364282472464, −1.65470999538928771380172920643, −0.978278863960150269743198770832, 0.978278863960150269743198770832, 1.65470999538928771380172920643, 3.17039257360102443364282472464, 4.31951304257228794079168632437, 5.45186028111385934401011142552, 6.74517118727960106577098322585, 7.70231136854552248755973304148, 8.274669889890678787790145893236, 8.824538265957090862984442657588, 9.784332700739636120047871256374

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