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Properties

Label 2-560-35.13-c1-0-16
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

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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: $\chi_{560} (433, \cdot )$
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 \( 1 + (4.59 + 4.59i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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