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Properties

Label 2-560-140.139-c3-0-34
Degree $2$
Conductor $560$
Sign $0.866 + 0.5i$
Analytic cond. $33.0410$
Root an. cond. $5.74813$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.3i·3-s − 11.1·5-s − 18.5i·7-s − 80.9·9-s + 68.2i·11-s − 53.1·13-s − 116. i·15-s − 31.0·17-s + 192.·21-s + 125.·25-s − 561. i·27-s + 239.·29-s − 709.·33-s + 207. i·35-s − 552. i·39-s + ⋯
L(s)  = 1  + 1.99i·3-s − 0.999·5-s − 0.999i·7-s − 2.99·9-s + 1.87i·11-s − 1.13·13-s − 1.99i·15-s − 0.443·17-s + 1.99·21-s + 1.00·25-s − 3.99i·27-s + 1.53·29-s − 3.74·33-s + 0.999i·35-s − 2.26i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(33.0410\)
Root analytic conductor: \(5.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :3/2),\ 0.866 + 0.5i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2368837387\)
\(L(\frac12)\) \(\approx\) \(0.2368837387\)
\(L(\frac{5}{2})\) 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 \( 1 + 11.1T \)
7 \( 1 + 18.5iT \)
good3 \( 1 - 10.3iT - 27T^{2} \)
11 \( 1 - 68.2iT - 1.33e3T^{2} \)
13 \( 1 + 53.1T + 2.19e3T^{2} \)
17 \( 1 + 31.0T + 4.91e3T^{2} \)
19 \( 1 + 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 239.T + 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 + 7.95e4T^{2} \)
47 \( 1 + 154. iT - 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 + 3.00e5T^{2} \)
71 \( 1 - 863. iT - 3.57e5T^{2} \)
73 \( 1 - 523.T + 3.89e5T^{2} \)
79 \( 1 + 896. iT - 4.93e5T^{2} \)
83 \( 1 + 47.6iT - 5.71e5T^{2} \)
89 \( 1 - 7.04e5T^{2} \)
97 \( 1 + 1.65e3T + 9.12e5T^{2} \)
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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

−10.10967360561521731402709889430, −9.786385946253835759449014581552, −8.697843546369582271987696771387, −7.69642262000004152308513021319, −6.77210407651183420738097942612, −4.99065216059562316295679665406, −4.54540875875222827618650225674, −3.90563588124533835514517516343, −2.69521850133026192699114409439, −0.090537650693087819918464115623, 0.901589393106711948610012994204, 2.46064084777312735440937967512, 3.17114338981694314633450286109, 5.14008344433608644875875342491, 6.12473833763701544688026799446, 6.82237859978025640481160343753, 7.87879900573962845921780518987, 8.355297011668528496310762555534, 9.038735481701194239715059969606, 10.93604905358309973023760906546

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