↑

Properties

Label 2-560-7.6-c4-0-16
Degree $2$
Conductor $560$
Sign $0.101 - 0.994i$
Analytic cond. $57.8871$
Root an. cond. $7.60836$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7.67i·3-s − 11.1i·5-s + (−48.7 − 4.95i)7-s + 22.0·9-s − 54.9·11-s − 34.6i·13-s + 85.8·15-s − 305. i·17-s + 129. i·19-s + (38.0 − 374. i)21-s + 503.·23-s − 125.·25-s + 791. i·27-s − 161.·29-s + 334. i·31-s + ⋯
L(s)  = 1  + 0.852i·3-s − 0.447i·5-s + (−0.994 − 0.101i)7-s + 0.272·9-s − 0.454·11-s − 0.205i·13-s + 0.381·15-s − 1.05i·17-s + 0.358i·19-s + (0.0862 − 0.848i)21-s + 0.952·23-s − 0.200·25-s + 1.08i·27-s − 0.191·29-s + 0.347i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.101 - 0.994i$
Analytic conductor: \(57.8871\)
Root analytic conductor: \(7.60836\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :2),\ 0.101 - 0.994i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.476568207\)
\(L(\frac12)\) \(\approx\) \(1.476568207\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 11.1iT \)
7 \( 1 + (48.7 + 4.95i)T \)
good3 \( 1 - 7.67iT - 81T^{2} \)
11 \( 1 + 54.9T + 1.46e4T^{2} \)
13 \( 1 + 34.6iT - 2.85e4T^{2} \)
17 \( 1 + 305. iT - 8.35e4T^{2} \)
19 \( 1 - 129. iT - 1.30e5T^{2} \)
23 \( 1 - 503.T + 2.79e5T^{2} \)
29 \( 1 + 161.T + 7.07e5T^{2} \)
31 \( 1 - 334. iT - 9.23e5T^{2} \)
37 \( 1 - 212.T + 1.87e6T^{2} \)
41 \( 1 + 199. iT - 2.82e6T^{2} \)
43 \( 1 - 291.T + 3.41e6T^{2} \)
47 \( 1 - 3.32e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.99e3T + 7.89e6T^{2} \)
59 \( 1 + 2.30e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.63e3iT - 1.38e7T^{2} \)
67 \( 1 + 4.49e3T + 2.01e7T^{2} \)
71 \( 1 - 6.80e3T + 2.54e7T^{2} \)
73 \( 1 - 5.45e3iT - 2.83e7T^{2} \)
79 \( 1 - 146.T + 3.89e7T^{2} \)
83 \( 1 + 3.76e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.69e3iT - 6.27e7T^{2} \)
97 \( 1 - 8.71e3iT - 8.85e7T^{2} \)
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

−10.24577788477605480658212348857, −9.514755944547885547260439064560, −8.947397216765886724870521521211, −7.66620472733726829565008388901, −6.78891191666028735511170917507, −5.56648903126371356618553683489, −4.74101720709603497506073498337, −3.72314494842980732533971335420, −2.72232181924271999728424948042, −0.919379972782288458363255875981, 0.46806747046164163989750886721, 1.88294023374825223660005193025, 2.96676133002007191473869078346, 4.10636172383842752798978416987, 5.57648748825645905456058688152, 6.55115170289523698269582089052, 7.08025626496760762139979368324, 8.029327059279059027306005620896, 9.076091560920827493326430198842, 10.03766407019655538099847180155

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