Properties

Label 2-560-112.109-c1-0-16
Degree $2$
Conductor $560$
Sign $0.796 - 0.604i$
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  + (−1.10 − 0.882i)2-s + (0.378 − 0.101i)3-s + (0.442 + 1.95i)4-s + (−0.965 − 0.258i)5-s + (−0.507 − 0.221i)6-s + (2.62 − 0.358i)7-s + (1.23 − 2.54i)8-s + (−2.46 + 1.42i)9-s + (0.839 + 1.13i)10-s + (1.57 + 5.86i)11-s + (0.365 + 0.693i)12-s + (−0.728 − 0.728i)13-s + (−3.21 − 1.91i)14-s − 0.391·15-s + (−3.60 + 1.72i)16-s + (−2.14 + 3.71i)17-s + ⋯
L(s)  = 1  + (−0.781 − 0.623i)2-s + (0.218 − 0.0585i)3-s + (0.221 + 0.975i)4-s + (−0.431 − 0.115i)5-s + (−0.207 − 0.0905i)6-s + (0.990 − 0.135i)7-s + (0.435 − 0.900i)8-s + (−0.821 + 0.474i)9-s + (0.265 + 0.359i)10-s + (0.473 + 1.76i)11-s + (0.105 + 0.200i)12-s + (−0.202 − 0.202i)13-s + (−0.858 − 0.512i)14-s − 0.101·15-s + (−0.902 + 0.431i)16-s + (−0.520 + 0.902i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.796 - 0.604i$
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: \((2,\ 560,\ (\ :1/2),\ 0.796 - 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869736 + 0.292897i\)
\(L(\frac12)\) \(\approx\) \(0.869736 + 0.292897i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 + 0.882i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 + (-2.62 + 0.358i)T \)
good3 \( 1 + (-0.378 + 0.101i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.57 - 5.86i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.728 + 0.728i)T + 13iT^{2} \)
17 \( 1 + (2.14 - 3.71i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.657 + 2.45i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.0492 + 0.0284i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.910 - 0.910i)T + 29iT^{2} \)
31 \( 1 + (2.92 - 5.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-10.8 - 2.91i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.81iT - 41T^{2} \)
43 \( 1 + (0.446 - 0.446i)T - 43iT^{2} \)
47 \( 1 + (-2.77 - 4.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.35 - 5.04i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.48 + 9.26i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.05 + 11.3i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-5.44 + 1.46i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 11.1iT - 71T^{2} \)
73 \( 1 + (-7.11 - 4.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.851 + 1.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.60 + 8.60i)T + 83iT^{2} \)
89 \( 1 + (4.82 - 2.78i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.08T + 97T^{2} \)
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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

−11.01155733311108249443112070472, −9.958634842445316030531014552245, −9.091296356535895985846659903964, −8.203075035669453083977260128019, −7.65678476933733050322336871120, −6.69503950831535256238400973963, −4.92415433600125249446643020053, −4.13602445212679227672206496533, −2.63203172185086358837374791249, −1.56453242478378216336839552472, 0.69533585692195055341054903673, 2.52593270328699386075514468817, 4.01159986385554582097154825589, 5.43490090412177904848953171591, 6.10897216910240299556142209069, 7.29471032535086376178588888945, 8.180385182720104459013417482644, 8.755592597991348425048847449559, 9.438043027771588396180067210017, 10.78457673542820087770760480956

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