Properties

Degree $2$
Conductor $525$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s − 6-s − 7-s + 3·8-s + 9-s − 6·11-s − 12-s − 2·13-s + 14-s − 16-s + 4·17-s − 18-s − 6·19-s − 21-s + 6·22-s + 3·24-s + 2·26-s + 27-s + 28-s − 2·29-s − 10·31-s − 5·32-s − 6·33-s − 4·34-s − 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 1.37·19-s − 0.218·21-s + 1.27·22-s + 0.612·24-s + 0.392·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s − 1.79·31-s − 0.883·32-s − 1.04·33-s − 0.685·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{525} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−19.64623121600341, −18.99272549006296, −18.47979183607997, −17.84462502140675, −16.80884825073454, −16.33077993442403, −15.27187198252556, −14.61825076046658, −13.62536309866576, −12.98714631288735, −12.40050696846965, −10.75753147480618, −10.32936245971432, −9.483915259822821, −8.662795051661331, −7.841207836065022, −7.258754890589428, −5.650153460363262, −4.692941810490154, −3.408579770839836, −2.079380669215899, 0, 2.079380669215899, 3.408579770839836, 4.692941810490154, 5.650153460363262, 7.258754890589428, 7.841207836065022, 8.662795051661331, 9.483915259822821, 10.32936245971432, 10.75753147480618, 12.40050696846965, 12.98714631288735, 13.62536309866576, 14.61825076046658, 15.27187198252556, 16.33077993442403, 16.80884825073454, 17.84462502140675, 18.47979183607997, 18.99272549006296, 19.64623121600341

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