↑

Properties

Label 2-720-80.69-c1-0-2
Degree $2$
Conductor $720$
Sign $-0.0880 - 0.996i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.456 − 1.33i)2-s + (−1.58 + 1.22i)4-s + (−1.65 + 1.50i)5-s + 2.58·7-s + (2.35 + 1.55i)8-s + (2.76 + 1.53i)10-s + (−4.39 − 4.39i)11-s + (0.417 + 0.417i)13-s + (−1.18 − 3.46i)14-s + (1.00 − 3.87i)16-s + 4.40i·17-s + (−4.53 + 4.53i)19-s + (0.787 − 4.40i)20-s + (−3.87 + 7.89i)22-s − 0.281·23-s + ⋯
L(s)  = 1  + (−0.323 − 0.946i)2-s + (−0.791 + 0.611i)4-s + (−0.741 + 0.671i)5-s + 0.978·7-s + (0.834 + 0.551i)8-s + (0.874 + 0.484i)10-s + (−1.32 − 1.32i)11-s + (0.115 + 0.115i)13-s + (−0.316 − 0.926i)14-s + (0.252 − 0.967i)16-s + 1.06i·17-s + (−1.04 + 1.04i)19-s + (0.176 − 0.984i)20-s + (−0.826 + 1.68i)22-s − 0.0586·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0880 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0880 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.0880 - 0.996i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.0880 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.258545 + 0.282393i\)
\(L(\frac12)\) \(\approx\) \(0.258545 + 0.282393i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.456 + 1.33i)T \)
3 \( 1 \)
5 \( 1 + (1.65 - 1.50i)T \)
good7 \( 1 - 2.58T + 7T^{2} \)
11 \( 1 + (4.39 + 4.39i)T + 11iT^{2} \)
13 \( 1 + (-0.417 - 0.417i)T + 13iT^{2} \)
17 \( 1 - 4.40iT - 17T^{2} \)
19 \( 1 + (4.53 - 4.53i)T - 19iT^{2} \)
23 \( 1 + 0.281T + 23T^{2} \)
29 \( 1 + (3.73 - 3.73i)T - 29iT^{2} \)
31 \( 1 + 3.05T + 31T^{2} \)
37 \( 1 + (5.26 - 5.26i)T - 37iT^{2} \)
41 \( 1 - 5.16iT - 41T^{2} \)
43 \( 1 + (2.66 - 2.66i)T - 43iT^{2} \)
47 \( 1 - 7.45iT - 47T^{2} \)
53 \( 1 + (-2.89 + 2.89i)T - 53iT^{2} \)
59 \( 1 + (-4.60 - 4.60i)T + 59iT^{2} \)
61 \( 1 + (0.211 - 0.211i)T - 61iT^{2} \)
67 \( 1 + (7.17 + 7.17i)T + 67iT^{2} \)
71 \( 1 + 15.9iT - 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 - 4.53T + 79T^{2} \)
83 \( 1 + (-4.56 - 4.56i)T + 83iT^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 - 6.78iT - 97T^{2} \)
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

−10.81280569352110705027340522887, −10.20969002551839777061139994372, −8.709275684414533021824933574231, −8.159870373326006915048962213733, −7.66772292948856016488123733627, −6.14337713305505178350871413538, −4.97783343182098567375096931436, −3.87553909088618206528966500659, −3.04102666774562010885371524475, −1.72158858215073871952334488967, 0.21854348729012204766072117993, 2.06880221736389526265631993937, 4.14130753183507584264383945446, 4.92051403981562482008550170183, 5.43459851473249610180925911482, 7.16026256955679021130566448298, 7.44055889434159467805970022635, 8.397328640056397954498652744459, 9.002232282980011550361496356052, 10.05942414418894244659047061046

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