Properties

Label 2-825-1.1-c5-0-102
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.17·2-s − 9·3-s − 5.19·4-s + 46.5·6-s + 123.·7-s + 192.·8-s + 81·9-s − 121·11-s + 46.7·12-s + 500.·13-s − 639.·14-s − 830.·16-s + 422.·17-s − 419.·18-s − 932.·19-s − 1.11e3·21-s + 626.·22-s + 1.22e3·23-s − 1.73e3·24-s − 2.58e3·26-s − 729·27-s − 641.·28-s − 2.11e3·29-s − 159.·31-s − 1.86e3·32-s + 1.08e3·33-s − 2.18e3·34-s + ⋯
L(s)  = 1  − 0.915·2-s − 0.577·3-s − 0.162·4-s + 0.528·6-s + 0.952·7-s + 1.06·8-s + 0.333·9-s − 0.301·11-s + 0.0937·12-s + 0.820·13-s − 0.871·14-s − 0.811·16-s + 0.354·17-s − 0.305·18-s − 0.592·19-s − 0.549·21-s + 0.275·22-s + 0.482·23-s − 0.614·24-s − 0.751·26-s − 0.192·27-s − 0.154·28-s − 0.466·29-s − 0.0298·31-s − 0.321·32-s + 0.174·33-s − 0.324·34-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: $\chi_{825} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 825,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 5.17T + 32T^{2} \)
7 \( 1 - 123.T + 1.68e4T^{2} \)
13 \( 1 - 500.T + 3.71e5T^{2} \)
17 \( 1 - 422.T + 1.41e6T^{2} \)
19 \( 1 + 932.T + 2.47e6T^{2} \)
23 \( 1 - 1.22e3T + 6.43e6T^{2} \)
29 \( 1 + 2.11e3T + 2.05e7T^{2} \)
31 \( 1 + 159.T + 2.86e7T^{2} \)
37 \( 1 + 5.41e3T + 6.93e7T^{2} \)
41 \( 1 + 1.80e4T + 1.15e8T^{2} \)
43 \( 1 + 6.81e3T + 1.47e8T^{2} \)
47 \( 1 - 1.50e4T + 2.29e8T^{2} \)
53 \( 1 + 1.53e4T + 4.18e8T^{2} \)
59 \( 1 - 2.34e4T + 7.14e8T^{2} \)
61 \( 1 - 1.07e4T + 8.44e8T^{2} \)
67 \( 1 + 1.45e4T + 1.35e9T^{2} \)
71 \( 1 + 2.81e4T + 1.80e9T^{2} \)
73 \( 1 - 2.88e4T + 2.07e9T^{2} \)
79 \( 1 + 8.15e3T + 3.07e9T^{2} \)
83 \( 1 - 1.09e5T + 3.93e9T^{2} \)
89 \( 1 - 6.96e4T + 5.58e9T^{2} \)
97 \( 1 + 9.15e4T + 8.58e9T^{2} \)
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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

−8.923843061495281321639171255843, −8.323033958376440858631661080611, −7.56837908225641648548646355406, −6.60856137316361275890159419583, −5.40474164098502051732765924007, −4.73376820302772111932277171051, −3.66114606219011617983742640275, −1.91340040158909911062276212887, −1.09325899895399695595104985822, 0, 1.09325899895399695595104985822, 1.91340040158909911062276212887, 3.66114606219011617983742640275, 4.73376820302772111932277171051, 5.40474164098502051732765924007, 6.60856137316361275890159419583, 7.56837908225641648548646355406, 8.323033958376440858631661080611, 8.923843061495281321639171255843

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