↑

Properties

Label 2-560-112.19-c1-0-7
Degree $2$
Conductor $560$
Sign $-0.880 + 0.474i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
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  + (−1.41 − 0.0650i)2-s + (−0.881 + 3.28i)3-s + (1.99 + 0.183i)4-s + (0.965 − 0.258i)5-s + (1.45 − 4.58i)6-s + (1.65 + 2.06i)7-s + (−2.80 − 0.389i)8-s + (−7.44 − 4.29i)9-s + (−1.38 + 0.302i)10-s + (0.352 − 1.31i)11-s + (−2.36 + 6.38i)12-s + (−0.453 + 0.453i)13-s + (−2.19 − 3.02i)14-s + 3.40i·15-s + (3.93 + 0.732i)16-s + (−5.42 + 3.13i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0460i)2-s + (−0.508 + 1.89i)3-s + (0.995 + 0.0919i)4-s + (0.431 − 0.115i)5-s + (0.595 − 1.87i)6-s + (0.624 + 0.781i)7-s + (−0.990 − 0.137i)8-s + (−2.48 − 1.43i)9-s + (−0.436 + 0.0957i)10-s + (0.106 − 0.396i)11-s + (−0.681 + 1.84i)12-s + (−0.125 + 0.125i)13-s + (−0.587 − 0.809i)14-s + 0.879i·15-s + (0.983 + 0.183i)16-s + (−1.31 + 0.759i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.880 + 0.474i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.880 + 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115410 - 0.457685i\)
\(L(\frac12)\) \(\approx\) \(0.115410 - 0.457685i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0650i)T \)
5 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (-1.65 - 2.06i)T \)
good3 \( 1 + (0.881 - 3.28i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.352 + 1.31i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.453 - 0.453i)T - 13iT^{2} \)
17 \( 1 + (5.42 - 3.13i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.05 - 1.35i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.74 - 3.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.18 + 2.18i)T + 29iT^{2} \)
31 \( 1 + (-3.91 - 6.78i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.117 + 0.438i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.54T + 41T^{2} \)
43 \( 1 + (5.23 + 5.23i)T + 43iT^{2} \)
47 \( 1 + (-1.85 + 3.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.69 + 2.06i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.603 + 0.161i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.04 - 3.91i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-5.39 - 1.44i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 1.17T + 71T^{2} \)
73 \( 1 + (-5.34 - 9.26i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.15 + 2.40i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.2 - 12.2i)T + 83iT^{2} \)
89 \( 1 + (-4.96 + 8.60i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.416iT - 97T^{2} \)
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

−11.01634803766889539000185072027, −10.32518532686387291218971411863, −9.603342418597743606646948954153, −8.605550694222061663149332740465, −8.539486703144472148706213540080, −6.48280996490643502947595504066, −5.78935162792993192498634149501, −4.80086035192377932900187300265, −3.62231251163259108259437189093, −2.19600244701592458134065714974, 0.37043204032920951399936035363, 1.74049462998947891830275106898, 2.49461325269689519193962060516, 4.90642981857373321087132380850, 6.35574757811233844794148860733, 6.64703598706491791414666742382, 7.59274347463051116343430909610, 8.184909341374493990497375281319, 9.166368417404119453537056353240, 10.50594532350427491938513804190

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