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Properties

Label 2-560-20.7-c1-0-9
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: $\chi_{560} (127, \cdot )$
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

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

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