↑

Properties

Label 2-720-9.4-c1-0-22
Degree $2$
Conductor $720$
Sign $-0.974 + 0.226i$
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  + (0.403 − 1.68i)3-s + (−0.5 + 0.866i)5-s + (−0.596 − 1.03i)7-s + (−2.67 − 1.35i)9-s + (−1.66 − 2.87i)11-s + (−0.853 + 1.47i)13-s + (1.25 + 1.19i)15-s − 6.34·17-s − 1.32·19-s + (−1.98 + 0.588i)21-s + (3.43 − 5.94i)23-s + (−0.499 − 0.866i)25-s + (−3.36 + 3.95i)27-s + (−1.01 − 1.75i)29-s + (−1.33 + 2.32i)31-s + ⋯
L(s)  = 1  + (0.232 − 0.972i)3-s + (−0.223 + 0.387i)5-s + (−0.225 − 0.390i)7-s + (−0.891 − 0.452i)9-s + (−0.500 − 0.867i)11-s + (−0.236 + 0.410i)13-s + (0.324 + 0.307i)15-s − 1.53·17-s − 0.303·19-s + (−0.432 + 0.128i)21-s + (0.715 − 1.23i)23-s + (−0.0999 − 0.173i)25-s + (−0.648 + 0.761i)27-s + (−0.188 − 0.326i)29-s + (−0.240 + 0.416i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.974 + 0.226i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.974 + 0.226i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0837976 - 0.731753i\)
\(L(\frac12)\) \(\approx\) \(0.0837976 - 0.731753i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.403 + 1.68i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (0.596 + 1.03i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.66 + 2.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.853 - 1.47i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.34T + 17T^{2} \)
19 \( 1 + 1.32T + 19T^{2} \)
23 \( 1 + (-3.43 + 5.94i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.01 + 1.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.33 - 2.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.32T + 37T^{2} \)
41 \( 1 + (1.16 - 2.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.17 + 5.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.38 + 11.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.02T + 53T^{2} \)
59 \( 1 + (5.83 - 10.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.86 - 8.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.28 - 9.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.06T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 + (-0.707 - 1.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.91 + 10.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11T + 89T^{2} \)
97 \( 1 + (8.12 + 14.0i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.12684399285656713903966906283, −8.751643659207049674103055279878, −8.445178587171499449697667917208, −7.13481794722201834412558914716, −6.77720985585551163532273266741, −5.75776094710384609721414470164, −4.38875773122656827541422039399, −3.13635317851626567319248955011, −2.15887507240626667393385120048, −0.34357217104186711540791382052, 2.20828205464276030176226976106, 3.35582713800095172587092019550, 4.55485084616701942663406850090, 5.11286124913026223312518543680, 6.27553565204935958337204025427, 7.52673988347490010777701073991, 8.348620981584695959257762375736, 9.355301355328427982761787077232, 9.656869188931337081959883361257, 10.89908962442816338355653392531

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