Properties

Label 2-825-1.1-c1-0-27
Degree $2$
Conductor $825$
Sign $-1$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 7-s + 9-s − 11-s − 2·12-s − 13-s + 4·16-s − 6·17-s − 7·19-s − 21-s + 6·23-s + 27-s + 2·28-s − 6·29-s − 7·31-s − 33-s − 2·36-s + 2·37-s − 39-s − 6·41-s − 43-s + 2·44-s + 4·48-s − 6·49-s − 6·51-s + 2·52-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 0.277·13-s + 16-s − 1.45·17-s − 1.60·19-s − 0.218·21-s + 1.25·23-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 1.25·31-s − 0.174·33-s − 1/3·36-s + 0.328·37-s − 0.160·39-s − 0.937·41-s − 0.152·43-s + 0.301·44-s + 0.577·48-s − 6/7·49-s − 0.840·51-s + 0.277·52-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{825} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
11 \( 1 + T \)
good2 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.567309717603522064679693310635, −8.955732103392000591141257065944, −8.367781725670412632475845953636, −7.30708842123598942599324121736, −6.37594318082116078700438020966, −5.13071370263278861735871300481, −4.31466281424388077222299797352, −3.35971135428784856363070844882, −2.04859260711079521330868103173, 0, 2.04859260711079521330868103173, 3.35971135428784856363070844882, 4.31466281424388077222299797352, 5.13071370263278861735871300481, 6.37594318082116078700438020966, 7.30708842123598942599324121736, 8.367781725670412632475845953636, 8.955732103392000591141257065944, 9.567309717603522064679693310635

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