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Properties

Label 2-560-112.109-c1-0-38
Degree $2$
Conductor $560$
Sign $0.386 + 0.922i$
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  + (0.658 + 1.25i)2-s + (−2.30 + 0.617i)3-s + (−1.13 + 1.64i)4-s + (0.965 + 0.258i)5-s + (−2.28 − 2.47i)6-s + (−2.64 − 0.101i)7-s + (−2.80 − 0.329i)8-s + (2.32 − 1.34i)9-s + (0.312 + 1.37i)10-s + (0.0309 + 0.115i)11-s + (1.58 − 4.49i)12-s + (−1.77 − 1.77i)13-s + (−1.61 − 3.37i)14-s − 2.38·15-s + (−1.43 − 3.73i)16-s + (2.01 − 3.49i)17-s + ⋯
L(s)  = 1  + (0.465 + 0.884i)2-s + (−1.32 + 0.356i)3-s + (−0.565 + 0.824i)4-s + (0.431 + 0.115i)5-s + (−0.934 − 1.01i)6-s + (−0.999 − 0.0384i)7-s + (−0.993 − 0.116i)8-s + (0.774 − 0.447i)9-s + (0.0988 + 0.436i)10-s + (0.00934 + 0.0348i)11-s + (0.458 − 1.29i)12-s + (−0.491 − 0.491i)13-s + (−0.431 − 0.902i)14-s − 0.615·15-s + (−0.359 − 0.933i)16-s + (0.489 − 0.847i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.119064 - 0.0791892i\)
\(L(\frac12)\) \(\approx\) \(0.119064 - 0.0791892i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.658 - 1.25i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (2.64 + 0.101i)T \)
good3 \( 1 + (2.30 - 0.617i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.0309 - 0.115i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.77 + 1.77i)T + 13iT^{2} \)
17 \( 1 + (-2.01 + 3.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.216 - 0.808i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.45 - 1.99i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.00872 + 0.00872i)T + 29iT^{2} \)
31 \( 1 + (-1.30 + 2.25i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.12 + 1.10i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 11.2iT - 41T^{2} \)
43 \( 1 + (4.60 - 4.60i)T - 43iT^{2} \)
47 \( 1 + (5.41 + 9.37i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.07 - 7.74i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.540 - 2.01i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.59 - 9.68i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (7.52 - 2.01i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.35iT - 71T^{2} \)
73 \( 1 + (12.1 + 6.99i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.40 + 9.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.55 + 8.55i)T + 83iT^{2} \)
89 \( 1 + (14.6 - 8.43i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.93T + 97T^{2} \)
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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

−10.40303027484020390687700964555, −9.887269677380017962392442535775, −8.888915647424240754507808702151, −7.51067590170188520890980559410, −6.74626562593197051289899924678, −5.79887881808722095546646557114, −5.40455392222486002854163115249, −4.24251226718399571302160401826, −2.98633510764447235135496938510, −0.082700304650639088939162320229, 1.53598471890556829056275592935, 3.04819486479519713468145252276, 4.40237714936409828194186223353, 5.41441109378064207098082965637, 6.19768132232846273543062221310, 6.78373055992417119188082971065, 8.486741810357410032049616143997, 9.703245140612594246301137340947, 10.14746895262259648571578264813, 11.07711054798067540553363723135

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