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Properties

Label 2-560-5.3-c2-0-8
Degree $2$
Conductor $560$
Sign $0.707 + 0.706i$
Analytic cond. $15.2588$
Root an. cond. $3.90626$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.98 − 2.98i)3-s + (−4.25 + 2.63i)5-s + (−1.87 + 1.87i)7-s + 8.76i·9-s − 11.1·11-s + (7.26 + 7.26i)13-s + (20.5 + 4.81i)15-s + (−2.00 + 2.00i)17-s + 5.59i·19-s + 11.1·21-s + (6.88 + 6.88i)23-s + (11.1 − 22.3i)25-s + (−0.709 + 0.709i)27-s − 46.4i·29-s + 23.7·31-s + ⋯
L(s)  = 1  + (−0.993 − 0.993i)3-s + (−0.850 + 0.526i)5-s + (−0.267 + 0.267i)7-s + 0.973i·9-s − 1.01·11-s + (0.558 + 0.558i)13-s + (1.36 + 0.321i)15-s + (−0.117 + 0.117i)17-s + 0.294i·19-s + 0.530·21-s + (0.299 + 0.299i)23-s + (0.445 − 0.895i)25-s + (−0.0262 + 0.0262i)27-s − 1.60i·29-s + 0.767·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.707 + 0.706i$
Analytic conductor: \(15.2588\)
Root analytic conductor: \(3.90626\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1),\ 0.707 + 0.706i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7290881221\)
\(L(\frac12)\) \(\approx\) \(0.7290881221\)
\(L(2)\) 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 \( 1 + (4.25 - 2.63i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (2.98 + 2.98i)T + 9iT^{2} \)
11 \( 1 + 11.1T + 121T^{2} \)
13 \( 1 + (-7.26 - 7.26i)T + 169iT^{2} \)
17 \( 1 + (2.00 - 2.00i)T - 289iT^{2} \)
19 \( 1 - 5.59iT - 361T^{2} \)
23 \( 1 + (-6.88 - 6.88i)T + 529iT^{2} \)
29 \( 1 + 46.4iT - 841T^{2} \)
31 \( 1 - 23.7T + 961T^{2} \)
37 \( 1 + (14.8 - 14.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 31.9T + 1.68e3T^{2} \)
43 \( 1 + (-3.79 - 3.79i)T + 1.84e3iT^{2} \)
47 \( 1 + (-20.1 + 20.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (73.4 + 73.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 54.9iT - 3.48e3T^{2} \)
61 \( 1 - 96.1T + 3.72e3T^{2} \)
67 \( 1 + (-2.50 + 2.50i)T - 4.48e3iT^{2} \)
71 \( 1 - 104.T + 5.04e3T^{2} \)
73 \( 1 + (-20.1 - 20.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 100. iT - 6.24e3T^{2} \)
83 \( 1 + (-8.15 - 8.15i)T + 6.88e3iT^{2} \)
89 \( 1 - 39.5iT - 7.92e3T^{2} \)
97 \( 1 + (52.3 - 52.3i)T - 9.40e3iT^{2} \)
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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

−10.77793950093445495759317462879, −9.735253484458479431524043854864, −8.321748612287498135775164943171, −7.67451692787963149403619752375, −6.70418781473543118706767897175, −6.12241318291870019892493822556, −4.99922282262296367374555634291, −3.65452337969205498148639087472, −2.25688352199314810656217535672, −0.55926103268316826827887762690, 0.68424650062989193868319319586, 3.10349033526606397397899369354, 4.22691738019844646849480657257, 5.00357114071088098991659320127, 5.76994311195045871895342122879, 7.05028961287026798509862684686, 8.062388672870425650386192440343, 8.982402010287729992474835533808, 10.01347766239781564783586821757, 10.89548293370479302210856076832

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