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Properties

Label 2-560-20.3-c1-0-16
Degree $2$
Conductor $560$
Sign $0.137 + 0.990i$
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  + (2.31 − 2.31i)3-s + (2.03 − 0.931i)5-s + (−0.707 − 0.707i)7-s − 7.69i·9-s + 6.09i·11-s + (−1.41 − 1.41i)13-s + (2.54 − 6.85i)15-s + (0.291 − 0.291i)17-s + 3.26·19-s − 3.26·21-s + (−5.29 + 5.29i)23-s + (3.26 − 3.78i)25-s + (−10.8 − 10.8i)27-s + 2.37i·29-s + 3.95i·31-s + ⋯
L(s)  = 1  + (1.33 − 1.33i)3-s + (0.909 − 0.416i)5-s + (−0.267 − 0.267i)7-s − 2.56i·9-s + 1.83i·11-s + (−0.393 − 0.393i)13-s + (0.657 − 1.76i)15-s + (0.0707 − 0.0707i)17-s + 0.748·19-s − 0.713·21-s + (−1.10 + 1.10i)23-s + (0.653 − 0.757i)25-s + (−2.08 − 2.08i)27-s + 0.441i·29-s + 0.710i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.137 + 0.990i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.137 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84052 - 1.60290i\)
\(L(\frac12)\) \(\approx\) \(1.84052 - 1.60290i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.03 + 0.931i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-2.31 + 2.31i)T - 3iT^{2} \)
11 \( 1 - 6.09iT - 11T^{2} \)
13 \( 1 + (1.41 + 1.41i)T + 13iT^{2} \)
17 \( 1 + (-0.291 + 0.291i)T - 17iT^{2} \)
19 \( 1 - 3.26T + 19T^{2} \)
23 \( 1 + (5.29 - 5.29i)T - 23iT^{2} \)
29 \( 1 - 2.37iT - 29T^{2} \)
31 \( 1 - 3.95iT - 31T^{2} \)
37 \( 1 + (4.09 - 4.09i)T - 37iT^{2} \)
41 \( 1 - 7.03T + 41T^{2} \)
43 \( 1 + (1.83 - 1.83i)T - 43iT^{2} \)
47 \( 1 + (3.50 + 3.50i)T + 47iT^{2} \)
53 \( 1 + (5.49 + 5.49i)T + 53iT^{2} \)
59 \( 1 - 5.02T + 59T^{2} \)
61 \( 1 - 4.51T + 61T^{2} \)
67 \( 1 + (4.04 + 4.04i)T + 67iT^{2} \)
71 \( 1 + 0.733iT - 71T^{2} \)
73 \( 1 + (2.94 + 2.94i)T + 73iT^{2} \)
79 \( 1 - 1.28T + 79T^{2} \)
83 \( 1 + (-6.84 + 6.84i)T - 83iT^{2} \)
89 \( 1 - 4.21iT - 89T^{2} \)
97 \( 1 + (-2.88 + 2.88i)T - 97iT^{2} \)
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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

−9.990081446360101308510038192440, −9.696458261690976340417232285017, −8.780104501078644637908634584793, −7.72624931136809952843569451881, −7.19593242531579255157206924128, −6.29705329376758715826389685282, −4.96625419059576786335875421237, −3.45006458446921057902495782640, −2.24993918049115502644248414685, −1.45424720314737690306452803138, 2.32769278103683921070287892292, 3.10937706908460659102648441668, 4.06854684790347198438354980839, 5.35901526225202097184270978463, 6.20676264753877877063607323589, 7.72425103008268945164962863353, 8.622029680931578796260870836367, 9.242286079422960332075284178077, 9.958691340821035092382332468277, 10.63482693876500019264017454752

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