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Properties

Label 2-560-28.27-c1-0-11
Degree $2$
Conductor $560$
Sign $0.976 + 0.217i$
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  + 2.52·3-s − i·5-s + (2.52 − 0.792i)7-s + 3.37·9-s + 2.52i·11-s + 0.372i·13-s − 2.52i·15-s + 2.37i·17-s − 3.46·19-s + (6.37 − 2i)21-s − 8.51i·23-s − 25-s + 0.939·27-s + 0.372·29-s − 1.87·31-s + ⋯
L(s)  = 1  + 1.45·3-s − 0.447i·5-s + (0.954 − 0.299i)7-s + 1.12·9-s + 0.761i·11-s + 0.103i·13-s − 0.651i·15-s + 0.575i·17-s − 0.794·19-s + (1.39 − 0.436i)21-s − 1.77i·23-s − 0.200·25-s + 0.180·27-s + 0.0691·29-s − 0.337·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.976 + 0.217i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.976 + 0.217i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.45688 - 0.270700i\)
\(L(\frac12)\) \(\approx\) \(2.45688 - 0.270700i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (-2.52 + 0.792i)T \)
good3 \( 1 - 2.52T + 3T^{2} \)
11 \( 1 - 2.52iT - 11T^{2} \)
13 \( 1 - 0.372iT - 13T^{2} \)
17 \( 1 - 2.37iT - 17T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + 8.51iT - 23T^{2} \)
29 \( 1 - 0.372T + 29T^{2} \)
31 \( 1 + 1.87T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 - 8.74iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + 7.86T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + 6.63T + 59T^{2} \)
61 \( 1 - 10.7iT - 61T^{2} \)
67 \( 1 - 6.63iT - 67T^{2} \)
71 \( 1 - 6.63iT - 71T^{2} \)
73 \( 1 - 8.74iT - 73T^{2} \)
79 \( 1 + 15.7iT - 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + 5.48iT - 89T^{2} \)
97 \( 1 + 15.1iT - 97T^{2} \)
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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

−10.56106281740251010036715583723, −9.705364993181331457026315014455, −8.754862963051204092391086176991, −8.211771991947700418062061922223, −7.52016629004559129764747499700, −6.31071130633949858933212068692, −4.69069562103609291861952078326, −4.14754967834791289526797898163, −2.65922370240076701047141694021, −1.64159658977432703057414800838, 1.80230581389703550402901938255, 2.89050329054714452839024785074, 3.80774437283967124485340572419, 5.10711333866453454988407224295, 6.32545120070536415632932634434, 7.73588346225558207034121972194, 7.957011200345813481368218366116, 9.054971014895149539122272085795, 9.583017009635297520997059268291, 10.91254662581792105799162691589

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