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Properties

Label 2-560-35.9-c1-0-20
Degree $2$
Conductor $560$
Sign $0.278 + 0.960i$
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  + (1.5 − 0.866i)3-s + (2.13 − 0.656i)5-s + (−1.13 − 2.38i)7-s + (−2.63 − 4.56i)11-s − 2.62i·13-s + (2.63 − 2.83i)15-s + (0.362 − 0.209i)17-s + (−1.63 + 2.83i)19-s + (−3.77 − 2.59i)21-s + (6.77 + 3.91i)23-s + (4.13 − 2.80i)25-s + 5.19i·27-s − 4.27·29-s + (1.63 + 2.83i)31-s + (−7.91 − 4.56i)33-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.955 − 0.293i)5-s + (−0.429 − 0.902i)7-s + (−0.795 − 1.37i)11-s − 0.728i·13-s + (0.680 − 0.732i)15-s + (0.0879 − 0.0507i)17-s + (−0.375 + 0.650i)19-s + (−0.823 − 0.566i)21-s + (1.41 + 0.815i)23-s + (0.827 − 0.561i)25-s + 0.999i·27-s − 0.793·29-s + (0.294 + 0.509i)31-s + (−1.37 − 0.795i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.278 + 0.960i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.278 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60840 - 1.20822i\)
\(L(\frac12)\) \(\approx\) \(1.60840 - 1.20822i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.13 + 0.656i)T \)
7 \( 1 + (1.13 + 2.38i)T \)
good3 \( 1 + (-1.5 + 0.866i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.63 + 4.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.62iT - 13T^{2} \)
17 \( 1 + (-0.362 + 0.209i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.63 - 2.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.77 - 3.91i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.27T + 29T^{2} \)
31 \( 1 + (-1.63 - 2.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.63 - 4.98i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.72T + 41T^{2} \)
43 \( 1 + 2.15iT - 43T^{2} \)
47 \( 1 + (-5.63 - 3.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.91 + 2.83i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.63 - 2.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.77 + 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.04 - 1.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.54T + 71T^{2} \)
73 \( 1 + (5.63 - 3.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.63 - 6.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.40iT - 83T^{2} \)
89 \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−10.51986371698565435409270665301, −9.677042038107542275657539974186, −8.684202667893400835264797648195, −8.028688412013118826109966399102, −7.12455356976521705168102233679, −5.96817014498255965472221645776, −5.14900490916677743711880298576, −3.46003136619552125626240048643, −2.65773487301925712294831097862, −1.11309280645212246227281576968, 2.26605392336515945470032454479, 2.76272799281168445919596842905, 4.28613190554764124305030702875, 5.34921479461904539665305353664, 6.42390502245169325738949999789, 7.31718229705467604282642512916, 8.675354926258901076182825162769, 9.261358501100817699561473698756, 9.803146259301274500907265969380, 10.67471729301775292300726166571

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