↑

Properties

Label 2-560-35.27-c1-0-2
Degree $2$
Conductor $560$
Sign $-0.835 - 0.549i$
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  + (−1.30 − 1.30i)3-s + (0.158 + 2.23i)5-s + (0.941 + 2.47i)7-s + 0.414i·9-s − 2.82·11-s + (−4.23 − 4.23i)13-s + (2.70 − 3.12i)15-s + (−3.69 + 3.69i)17-s − 1.39·19-s + (2 − 4.46i)21-s + (−0.414 + 0.414i)23-s + (−4.94 + 0.707i)25-s + (−3.37 + 3.37i)27-s − 0.828i·29-s − 1.53i·31-s + ⋯
L(s)  = 1  + (−0.754 − 0.754i)3-s + (0.0708 + 0.997i)5-s + (0.355 + 0.934i)7-s + 0.138i·9-s − 0.852·11-s + (−1.17 − 1.17i)13-s + (0.698 − 0.805i)15-s + (−0.896 + 0.896i)17-s − 0.321·19-s + (0.436 − 0.973i)21-s + (−0.0863 + 0.0863i)23-s + (−0.989 + 0.141i)25-s + (−0.650 + 0.650i)27-s − 0.153i·29-s − 0.274i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.835 - 0.549i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0777641 + 0.259479i\)
\(L(\frac12)\) \(\approx\) \(0.0777641 + 0.259479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.158 - 2.23i)T \)
7 \( 1 + (-0.941 - 2.47i)T \)
good3 \( 1 + (1.30 + 1.30i)T + 3iT^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + (4.23 + 4.23i)T + 13iT^{2} \)
17 \( 1 + (3.69 - 3.69i)T - 17iT^{2} \)
19 \( 1 + 1.39T + 19T^{2} \)
23 \( 1 + (0.414 - 0.414i)T - 23iT^{2} \)
29 \( 1 + 0.828iT - 29T^{2} \)
31 \( 1 + 1.53iT - 31T^{2} \)
37 \( 1 + (-2.58 - 2.58i)T + 37iT^{2} \)
41 \( 1 - 3.69iT - 41T^{2} \)
43 \( 1 + (4 - 4i)T - 43iT^{2} \)
47 \( 1 + (1.08 - 1.08i)T - 47iT^{2} \)
53 \( 1 + (8.24 - 8.24i)T - 53iT^{2} \)
59 \( 1 - 9.23T + 59T^{2} \)
61 \( 1 + 6.43iT - 61T^{2} \)
67 \( 1 + (10.4 + 10.4i)T + 67iT^{2} \)
71 \( 1 - 0.585T + 71T^{2} \)
73 \( 1 + (-4.14 - 4.14i)T + 73iT^{2} \)
79 \( 1 + 5.07iT - 79T^{2} \)
83 \( 1 + (5.31 + 5.31i)T + 83iT^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + (4.59 - 4.59i)T - 97iT^{2} \)
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

−11.15842011214190427925529706260, −10.46242574346063351335957841340, −9.517051352924941992639723190930, −8.158137536213842429885767850516, −7.53808983435230891038374390452, −6.43030641441246130256854489178, −5.85623785866248178324891815614, −4.84131654744645137080243230401, −3.04631809184618544720371026827, −2.06569892124327186625119814008, 0.15666026569211752699146705052, 2.12920080517036602756566411878, 4.16128651108668046704194748071, 4.76245625195990965254346445510, 5.35650446286637207579623064801, 6.80715853620585918509838608480, 7.69303670173144751403876542159, 8.784872923162754410891574243461, 9.724406893266793679202119267753, 10.35340939525528069316095671955

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