Properties

Label 2-480-1.1-c1-0-0
Degree $2$
Conductor $480$
Sign $1$
Analytic cond. $3.83281$
Root an. cond. $1.95775$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 4·7-s + 9-s + 4·11-s + 6·13-s + 15-s + 2·17-s + 4·19-s + 4·21-s + 25-s − 27-s + 10·29-s − 4·31-s − 4·33-s + 4·35-s − 10·37-s − 6·39-s + 2·41-s − 4·43-s − 45-s + 8·47-s + 9·49-s − 2·51-s + 2·53-s − 4·55-s − 4·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s + 1.85·29-s − 0.718·31-s − 0.696·33-s + 0.676·35-s − 1.64·37-s − 0.960·39-s + 0.312·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.274·53-s − 0.539·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(3.83281\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{480} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.046977174\)
\(L(\frac12)\) \(\approx\) \(1.046977174\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−11.02123424258128403111626112733, −10.13855312007199582968173453548, −9.265808981321240617130460115923, −8.421933311737600437687566687322, −6.97981434955619225320560223720, −6.46117181371757757685905507069, −5.52040688744560586491461101623, −3.95366310025504614794426142117, −3.30373640608268734535981700762, −1.02432023626640801365429192064, 1.02432023626640801365429192064, 3.30373640608268734535981700762, 3.95366310025504614794426142117, 5.52040688744560586491461101623, 6.46117181371757757685905507069, 6.97981434955619225320560223720, 8.421933311737600437687566687322, 9.265808981321240617130460115923, 10.13855312007199582968173453548, 11.02123424258128403111626112733

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