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Properties

Label 2-560-20.3-c1-0-12
Degree $2$
Conductor $560$
Sign $0.582 + 0.812i$
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  + (−0.206 + 0.206i)3-s + (1.36 − 1.77i)5-s + (−0.707 − 0.707i)7-s + 2.91i·9-s − 5.35i·11-s + (1.26 + 1.26i)13-s + (0.0834 + 0.646i)15-s + (4.90 − 4.90i)17-s − 2.01·19-s + 0.291·21-s + (0.630 − 0.630i)23-s + (−1.26 − 4.83i)25-s + (−1.22 − 1.22i)27-s − 3.20i·29-s + 7.10i·31-s + ⋯
L(s)  = 1  + (−0.119 + 0.119i)3-s + (0.610 − 0.791i)5-s + (−0.267 − 0.267i)7-s + 0.971i·9-s − 1.61i·11-s + (0.351 + 0.351i)13-s + (0.0215 + 0.167i)15-s + (1.18 − 1.18i)17-s − 0.463·19-s + 0.0636·21-s + (0.131 − 0.131i)23-s + (−0.253 − 0.967i)25-s + (−0.234 − 0.234i)27-s − 0.595i·29-s + 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.582 + 0.812i$
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: \((2,\ 560,\ (\ :1/2),\ 0.582 + 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32394 - 0.679938i\)
\(L(\frac12)\) \(\approx\) \(1.32394 - 0.679938i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 + 1.77i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.206 - 0.206i)T - 3iT^{2} \)
11 \( 1 + 5.35iT - 11T^{2} \)
13 \( 1 + (-1.26 - 1.26i)T + 13iT^{2} \)
17 \( 1 + (-4.90 + 4.90i)T - 17iT^{2} \)
19 \( 1 + 2.01T + 19T^{2} \)
23 \( 1 + (-0.630 + 0.630i)T - 23iT^{2} \)
29 \( 1 + 3.20iT - 29T^{2} \)
31 \( 1 - 7.10iT - 31T^{2} \)
37 \( 1 + (-3.03 + 3.03i)T - 37iT^{2} \)
41 \( 1 + 1.34T + 41T^{2} \)
43 \( 1 + (-7.06 + 7.06i)T - 43iT^{2} \)
47 \( 1 + (6.08 + 6.08i)T + 47iT^{2} \)
53 \( 1 + (-6.86 - 6.86i)T + 53iT^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 + 5.70T + 61T^{2} \)
67 \( 1 + (-7.40 - 7.40i)T + 67iT^{2} \)
71 \( 1 - 9.52iT - 71T^{2} \)
73 \( 1 + (1.69 + 1.69i)T + 73iT^{2} \)
79 \( 1 + 9.49T + 79T^{2} \)
83 \( 1 + (2.74 - 2.74i)T - 83iT^{2} \)
89 \( 1 - 12.0iT - 89T^{2} \)
97 \( 1 + (5.08 - 5.08i)T - 97iT^{2} \)
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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

−10.59147181595508076353770346894, −9.798217195575544573796131341772, −8.800978146520058476407405927334, −8.188355833748740564805212312584, −6.99700140677818570575617255370, −5.76001479771246701185171632560, −5.25358901303426860392207244599, −3.97311643230712902872008499652, −2.61450746412638669606964768539, −0.947944060299669314313478650329, 1.66818139242902227431353081753, 3.01278403982323883928647777051, 4.12814149959023378181068146026, 5.62450275306406651913460544352, 6.33256377264650191014158643691, 7.14835274680956687791242439451, 8.163230702112210764160169932610, 9.519500078288231101807879297479, 9.856519771601949695132363171905, 10.77899561742782654634397553377

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