Properties

Label 2-825-1.1-c5-0-129
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  + 10.7·2-s + 9·3-s + 83.6·4-s + 96.7·6-s + 41.3·7-s + 555.·8-s + 81·9-s + 121·11-s + 752.·12-s − 13.6·13-s + 444.·14-s + 3.29e3·16-s + 1.14e3·17-s + 870.·18-s − 1.42e3·19-s + 372.·21-s + 1.30e3·22-s − 2.53e3·23-s + 4.99e3·24-s − 146.·26-s + 729·27-s + 3.46e3·28-s + 8.59e3·29-s + 434.·31-s + 1.76e4·32-s + 1.08e3·33-s + 1.23e4·34-s + ⋯
L(s)  = 1  + 1.90·2-s + 0.577·3-s + 2.61·4-s + 1.09·6-s + 0.319·7-s + 3.06·8-s + 0.333·9-s + 0.301·11-s + 1.50·12-s − 0.0224·13-s + 0.606·14-s + 3.21·16-s + 0.960·17-s + 0.633·18-s − 0.905·19-s + 0.184·21-s + 0.573·22-s − 0.998·23-s + 1.77·24-s − 0.0425·26-s + 0.192·27-s + 0.834·28-s + 1.89·29-s + 0.0811·31-s + 3.04·32-s + 0.174·33-s + 1.82·34-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

\(L(3)\) \(\approx\) \(11.98220347\)
\(L(\frac12)\) \(\approx\) \(11.98220347\)
\(L(\frac{7}{2})\) 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 \( 1 - 10.7T + 32T^{2} \)
7 \( 1 - 41.3T + 1.68e4T^{2} \)
13 \( 1 + 13.6T + 3.71e5T^{2} \)
17 \( 1 - 1.14e3T + 1.41e6T^{2} \)
19 \( 1 + 1.42e3T + 2.47e6T^{2} \)
23 \( 1 + 2.53e3T + 6.43e6T^{2} \)
29 \( 1 - 8.59e3T + 2.05e7T^{2} \)
31 \( 1 - 434.T + 2.86e7T^{2} \)
37 \( 1 - 1.44e4T + 6.93e7T^{2} \)
41 \( 1 - 1.23e4T + 1.15e8T^{2} \)
43 \( 1 + 2.69e3T + 1.47e8T^{2} \)
47 \( 1 + 1.69e3T + 2.29e8T^{2} \)
53 \( 1 + 9.46e3T + 4.18e8T^{2} \)
59 \( 1 + 4.14e4T + 7.14e8T^{2} \)
61 \( 1 + 1.78e4T + 8.44e8T^{2} \)
67 \( 1 - 5.17e4T + 1.35e9T^{2} \)
71 \( 1 + 1.66e4T + 1.80e9T^{2} \)
73 \( 1 + 4.97e4T + 2.07e9T^{2} \)
79 \( 1 - 7.14e4T + 3.07e9T^{2} \)
83 \( 1 + 3.35e4T + 3.93e9T^{2} \)
89 \( 1 + 7.69e4T + 5.58e9T^{2} \)
97 \( 1 - 1.77e4T + 8.58e9T^{2} \)
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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

−9.689833742579268494298858869025, −8.251334094957907497079390751218, −7.62292567261849491073372216401, −6.50292919402738454974278919063, −5.93910741787431570080146630457, −4.73715783352624150971233220728, −4.18962358338508916765345435022, −3.17217548248427257595919547576, −2.36753245350443828803984972972, −1.28787034865605385716716951962, 1.28787034865605385716716951962, 2.36753245350443828803984972972, 3.17217548248427257595919547576, 4.18962358338508916765345435022, 4.73715783352624150971233220728, 5.93910741787431570080146630457, 6.50292919402738454974278919063, 7.62292567261849491073372216401, 8.251334094957907497079390751218, 9.689833742579268494298858869025

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