↑

Properties

Label 2-560-7.4-c3-0-16
Degree $2$
Conductor $560$
Sign $-0.266 - 0.963i$
Analytic cond. $33.0410$
Root an. cond. $5.74813$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)3-s + (−2.5 − 4.33i)5-s + (14 + 12.1i)7-s + (11.5 + 19.9i)9-s + (−22.5 + 38.9i)11-s + 59·13-s + 10·15-s + (27 − 46.7i)17-s + (−60.5 − 104. i)19-s + (−35 + 12.1i)21-s + (34.5 + 59.7i)23-s + (−12.5 + 21.6i)25-s − 100·27-s − 162·29-s + (−44 + 76.2i)31-s + ⋯
L(s)  = 1  + (−0.192 + 0.333i)3-s + (−0.223 − 0.387i)5-s + (0.755 + 0.654i)7-s + (0.425 + 0.737i)9-s + (−0.616 + 1.06i)11-s + 1.25·13-s + 0.172·15-s + (0.385 − 0.667i)17-s + (−0.730 − 1.26i)19-s + (−0.363 + 0.125i)21-s + (0.312 + 0.541i)23-s + (−0.100 + 0.173i)25-s − 0.712·27-s − 1.03·29-s + (−0.254 + 0.441i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(33.0410\)
Root analytic conductor: \(5.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :3/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.666166635\)
\(L(\frac12)\) \(\approx\) \(1.666166635\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.5 + 4.33i)T \)
7 \( 1 + (-14 - 12.1i)T \)
good3 \( 1 + (1 - 1.73i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (22.5 - 38.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 59T + 2.19e3T^{2} \)
17 \( 1 + (-27 + 46.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (60.5 + 104. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-34.5 - 59.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 162T + 2.43e4T^{2} \)
31 \( 1 + (44 - 76.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-129.5 - 224. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 195T + 6.89e4T^{2} \)
43 \( 1 - 286T + 7.95e4T^{2} \)
47 \( 1 + (-22.5 - 38.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (298.5 - 517. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (180 - 311. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (196 + 339. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (140 - 242. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 48T + 3.57e5T^{2} \)
73 \( 1 + (334 - 578. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-391 - 677. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 768T + 5.71e5T^{2} \)
89 \( 1 + (-597 - 1.03e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 902T + 9.12e5T^{2} \)
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

−10.91796874532800871770764223549, −9.654501098455661572097993301956, −8.910069059219240319585256136259, −7.927833583667584658875731710814, −7.22540348026008709696611574359, −5.77707550062993724905007182985, −4.92662207633951749705784223142, −4.27556526153179904551614671538, −2.60924418659676248201970716497, −1.39978659894588040544829320853, 0.54509718477450021843362418057, 1.71456253059332330903925781118, 3.48954687785082697490935015396, 4.13175858058002997581577793331, 5.76647327660066276511160302097, 6.30667999154460271280950449961, 7.56875381283536026398297967529, 8.118153379539610668149114124838, 9.114871375564382126972698385374, 10.48363156053982916687705063038

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