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Properties

Label 2-560-140.83-c2-0-26
Degree $2$
Conductor $560$
Sign $0.855 - 0.518i$
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  + (1.09 + 1.09i)3-s + (0.913 + 4.91i)5-s + (6.93 + 0.925i)7-s − 6.59i·9-s − 4.54i·11-s + (14.2 − 14.2i)13-s + (−4.38 + 6.38i)15-s + (13.7 + 13.7i)17-s − 3.69i·19-s + (6.58 + 8.61i)21-s + (25.9 − 25.9i)23-s + (−23.3 + 8.97i)25-s + (17.0 − 17.0i)27-s + 42.9i·29-s − 10.8·31-s + ⋯
L(s)  = 1  + (0.365 + 0.365i)3-s + (0.182 + 0.983i)5-s + (0.991 + 0.132i)7-s − 0.733i·9-s − 0.413i·11-s + (1.09 − 1.09i)13-s + (−0.292 + 0.425i)15-s + (0.807 + 0.807i)17-s − 0.194i·19-s + (0.313 + 0.410i)21-s + (1.12 − 1.12i)23-s + (−0.933 + 0.359i)25-s + (0.633 − 0.633i)27-s + 1.48i·29-s − 0.350·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.855 - 0.518i$
Analytic conductor: \(15.2588\)
Root analytic conductor: \(3.90626\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1),\ 0.855 - 0.518i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.508269962\)
\(L(\frac12)\) \(\approx\) \(2.508269962\)
\(L(2)\) 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 \( 1 + (-0.913 - 4.91i)T \)
7 \( 1 + (-6.93 - 0.925i)T \)
good3 \( 1 + (-1.09 - 1.09i)T + 9iT^{2} \)
11 \( 1 + 4.54iT - 121T^{2} \)
13 \( 1 + (-14.2 + 14.2i)T - 169iT^{2} \)
17 \( 1 + (-13.7 - 13.7i)T + 289iT^{2} \)
19 \( 1 + 3.69iT - 361T^{2} \)
23 \( 1 + (-25.9 + 25.9i)T - 529iT^{2} \)
29 \( 1 - 42.9iT - 841T^{2} \)
31 \( 1 + 10.8T + 961T^{2} \)
37 \( 1 + (36.7 - 36.7i)T - 1.36e3iT^{2} \)
41 \( 1 - 30.7iT - 1.68e3T^{2} \)
43 \( 1 + (47.4 - 47.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-39.0 + 39.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (17.0 + 17.0i)T + 2.80e3iT^{2} \)
59 \( 1 + 93.6iT - 3.48e3T^{2} \)
61 \( 1 - 51.6iT - 3.72e3T^{2} \)
67 \( 1 + (-54.9 - 54.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 50.1iT - 5.04e3T^{2} \)
73 \( 1 + (1.87 - 1.87i)T - 5.32e3iT^{2} \)
79 \( 1 + 103.T + 6.24e3T^{2} \)
83 \( 1 + (-51.9 - 51.9i)T + 6.88e3iT^{2} \)
89 \( 1 - 67.4T + 7.92e3T^{2} \)
97 \( 1 + (-25.1 - 25.1i)T + 9.40e3iT^{2} \)
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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

−10.70121228512861422837371653312, −9.918292458355690863943093621137, −8.634034620768137604751387051817, −8.261958558329170291505213218192, −6.97109447954878435452726829184, −6.08877395548555350331114504641, −5.07233947184405394440621888434, −3.60786512940924309381216173779, −2.99639711325951354472091446758, −1.27551771708140888528822279426, 1.24528871663748861628954873045, 2.08453900079836165787191905616, 3.87298418245782904646675336590, 4.91360398384257511158999865882, 5.64346982819084001359145379366, 7.17046259976540057334585944159, 7.82272523260261654897695740971, 8.734934701543872684884176073231, 9.329094721699616819998871520741, 10.53734949200678517070660820425

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