Properties

Degree $2$
Conductor $3630$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s − 6·13-s − 15-s + 16-s − 2·17-s − 18-s + 4·19-s + 20-s + 24-s + 25-s + 6·26-s − 27-s + 10·29-s + 30-s − 32-s + 2·34-s + 36-s + 6·37-s − 4·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 1.66·13-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 1.85·29-s + 0.182·30-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + 0.986·37-s − 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3630\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{3630} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3630,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 \)
good7 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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

−8.085934361283387711379640695620, −7.49041755614925508343340066706, −6.66337611341046156940754000004, −6.17955098032914215865508254198, −5.02793023545440757252689567911, −4.70717282494171643247853802761, −3.18706730735035014552502351321, −2.36646738318065634335521514301, −1.27992569263552166640931532540, 0, 1.27992569263552166640931532540, 2.36646738318065634335521514301, 3.18706730735035014552502351321, 4.70717282494171643247853802761, 5.02793023545440757252689567911, 6.17955098032914215865508254198, 6.66337611341046156940754000004, 7.49041755614925508343340066706, 8.085934361283387711379640695620

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