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Properties

Label 2-560-112.109-c1-0-19
Degree $2$
Conductor $560$
Sign $-0.284 - 0.958i$
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  + (1.26 + 0.633i)2-s + (−0.378 + 0.101i)3-s + (1.19 + 1.60i)4-s + (0.965 + 0.258i)5-s + (−0.543 − 0.111i)6-s + (−2.31 + 1.28i)7-s + (0.500 + 2.78i)8-s + (−2.46 + 1.42i)9-s + (1.05 + 0.938i)10-s + (−0.463 − 1.73i)11-s + (−0.616 − 0.485i)12-s + (4.33 + 4.33i)13-s + (−3.73 + 0.160i)14-s − 0.392·15-s + (−1.12 + 3.83i)16-s + (−1.53 + 2.65i)17-s + ⋯
L(s)  = 1  + (0.894 + 0.447i)2-s + (−0.218 + 0.0586i)3-s + (0.598 + 0.800i)4-s + (0.431 + 0.115i)5-s + (−0.221 − 0.0455i)6-s + (−0.874 + 0.485i)7-s + (0.176 + 0.984i)8-s + (−0.821 + 0.474i)9-s + (0.334 + 0.296i)10-s + (−0.139 − 0.521i)11-s + (−0.177 − 0.140i)12-s + (1.20 + 1.20i)13-s + (−0.999 + 0.0428i)14-s − 0.101·15-s + (−0.282 + 0.959i)16-s + (−0.372 + 0.644i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.284 - 0.958i$
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: \((2,\ 560,\ (\ :1/2),\ -0.284 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25324 + 1.67887i\)
\(L(\frac12)\) \(\approx\) \(1.25324 + 1.67887i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 0.633i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (2.31 - 1.28i)T \)
good3 \( 1 + (0.378 - 0.101i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.463 + 1.73i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-4.33 - 4.33i)T + 13iT^{2} \)
17 \( 1 + (1.53 - 2.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0621 + 0.231i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.42 + 2.55i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.02 + 2.02i)T + 29iT^{2} \)
31 \( 1 + (-1.39 + 2.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.68 - 1.25i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.68iT - 41T^{2} \)
43 \( 1 + (-7.41 + 7.41i)T - 43iT^{2} \)
47 \( 1 + (2.63 + 4.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.64 + 13.5i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.76 - 14.0i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.06 - 11.4i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (4.23 - 1.13i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.70iT - 71T^{2} \)
73 \( 1 + (-4.54 - 2.62i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.91 + 11.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.33 + 9.33i)T + 83iT^{2} \)
89 \( 1 + (-10.5 + 6.11i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.47T + 97T^{2} \)
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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

−11.30098870049934145670199184901, −10.37300282501532531632925508011, −8.926405307000885401872321821416, −8.515291193556211112006286966683, −7.08150170433130659886929166418, −6.08620042845715289476730750102, −5.85621607652320903450579604471, −4.49691692672439500047569929771, −3.33923078931117979895527594036, −2.27746219297619152635877867178, 0.949061733822345634859409075676, 2.80370049348189840269696485870, 3.55394901894975935917494509547, 4.91432543515660714290310241282, 5.88594232376278301397174106785, 6.47343873175685139498406567029, 7.57951217245137761286082280813, 9.071327434141713648832704403265, 9.754149680450963701744708616885, 10.84430860068059659255791527048

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