Properties

Label 2-720-1.1-c3-0-12
Degree $2$
Conductor $720$
Sign $1$
Analytic cond. $42.4813$
Root an. cond. $6.51777$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s + 18·7-s − 16·11-s − 6·13-s + 6·17-s + 124·19-s + 42·23-s + 25·25-s − 142·29-s + 188·31-s + 90·35-s + 202·37-s − 54·41-s − 66·43-s + 38·47-s − 19·49-s − 738·53-s − 80·55-s + 564·59-s − 262·61-s − 30·65-s + 554·67-s + 140·71-s + 882·73-s − 288·77-s + 1.16e3·79-s + 642·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.971·7-s − 0.438·11-s − 0.128·13-s + 0.0856·17-s + 1.49·19-s + 0.380·23-s + 1/5·25-s − 0.909·29-s + 1.08·31-s + 0.434·35-s + 0.897·37-s − 0.205·41-s − 0.234·43-s + 0.117·47-s − 0.0553·49-s − 1.91·53-s − 0.196·55-s + 1.24·59-s − 0.549·61-s − 0.0572·65-s + 1.01·67-s + 0.234·71-s + 1.41·73-s − 0.426·77-s + 1.65·79-s + 0.849·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(42.4813\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{720} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.567030314\)
\(L(\frac12)\) \(\approx\) \(2.567030314\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5 \( 1 - p T \)
good7 \( 1 - 18 T + p^{3} T^{2} \)
11 \( 1 + 16 T + p^{3} T^{2} \)
13 \( 1 + 6 T + p^{3} T^{2} \)
17 \( 1 - 6 T + p^{3} T^{2} \)
19 \( 1 - 124 T + p^{3} T^{2} \)
23 \( 1 - 42 T + p^{3} T^{2} \)
29 \( 1 + 142 T + p^{3} T^{2} \)
31 \( 1 - 188 T + p^{3} T^{2} \)
37 \( 1 - 202 T + p^{3} T^{2} \)
41 \( 1 + 54 T + p^{3} T^{2} \)
43 \( 1 + 66 T + p^{3} T^{2} \)
47 \( 1 - 38 T + p^{3} T^{2} \)
53 \( 1 + 738 T + p^{3} T^{2} \)
59 \( 1 - 564 T + p^{3} T^{2} \)
61 \( 1 + 262 T + p^{3} T^{2} \)
67 \( 1 - 554 T + p^{3} T^{2} \)
71 \( 1 - 140 T + p^{3} T^{2} \)
73 \( 1 - 882 T + p^{3} T^{2} \)
79 \( 1 - 1160 T + p^{3} T^{2} \)
83 \( 1 - 642 T + p^{3} T^{2} \)
89 \( 1 - 854 T + p^{3} T^{2} \)
97 \( 1 + 478 T + p^{3} T^{2} \)
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

−9.920849793136753121959957260672, −9.287861565403620424961876351027, −8.137261167631183614528329892314, −7.58929732362836523526354771533, −6.45364988355349428546350124599, −5.35454170548207915294112014857, −4.77263737690471452286913231279, −3.35635530913369842802878880504, −2.15682645795474749058403339358, −0.965400042184855393392491537749, 0.965400042184855393392491537749, 2.15682645795474749058403339358, 3.35635530913369842802878880504, 4.77263737690471452286913231279, 5.35454170548207915294112014857, 6.45364988355349428546350124599, 7.58929732362836523526354771533, 8.137261167631183614528329892314, 9.287861565403620424961876351027, 9.920849793136753121959957260672

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