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Properties

Label 2-560-140.59-c1-0-18
Degree $2$
Conductor $560$
Sign $0.992 + 0.118i$
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  + (2.42 + 1.40i)3-s + (0.792 − 2.09i)5-s + (−0.308 − 2.62i)7-s + (2.43 + 4.22i)9-s + (−3.87 − 2.23i)11-s + 5.47·13-s + (4.85 − 3.96i)15-s + (0.707 − 1.22i)17-s + (3.43 + 5.95i)19-s + (2.93 − 6.81i)21-s + (0.308 + 0.534i)23-s + (−3.74 − 3.31i)25-s + 5.25i·27-s − 3.87·29-s + (−3.43 + 5.95i)31-s + ⋯
L(s)  = 1  + (1.40 + 0.809i)3-s + (0.354 − 0.935i)5-s + (−0.116 − 0.993i)7-s + (0.812 + 1.40i)9-s + (−1.16 − 0.674i)11-s + 1.51·13-s + (1.25 − 1.02i)15-s + (0.171 − 0.297i)17-s + (0.788 + 1.36i)19-s + (0.640 − 1.48i)21-s + (0.0643 + 0.111i)23-s + (−0.748 − 0.663i)25-s + 1.01i·27-s − 0.719·29-s + (−0.617 + 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.992 + 0.118i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.992 + 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37203 - 0.141069i\)
\(L(\frac12)\) \(\approx\) \(2.37203 - 0.141069i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.792 + 2.09i)T \)
7 \( 1 + (0.308 + 2.62i)T \)
good3 \( 1 + (-2.42 - 1.40i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (3.87 + 2.23i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.47T + 13T^{2} \)
17 \( 1 + (-0.707 + 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.43 - 5.95i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.308 - 0.534i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.87T + 29T^{2} \)
31 \( 1 + (3.43 - 5.95i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.24 + 2.44i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + 6.09T + 43T^{2} \)
47 \( 1 + (2.12 - 1.22i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.12 + 1.22i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.56 - 4.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.93 - 6.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.96iT - 71T^{2} \)
73 \( 1 + (-2.12 + 3.67i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.30 - 2.48i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.06iT - 83T^{2} \)
89 \( 1 + (-10.1 + 5.84i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.25T + 97T^{2} \)
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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

−10.40781992436148467722217595470, −9.833547127981191587811553545521, −8.953018821344259399734458407960, −8.193666266819300246389824110032, −7.67788668544285510766303433424, −5.98841602412210671006702744473, −4.94316747116385202094050707481, −3.82000749677122069498682848775, −3.16956552475165718407917156125, −1.43685634937237670300723005765, 1.93075888740821404690422435932, 2.69901065831780455172691508008, 3.55882588685681742560079480813, 5.40362418650721744601524110949, 6.43801752565533856008708788611, 7.35196214426530049250584344402, 8.064651922321446518901868259325, 8.979454930824997476351820993003, 9.627399409566763220506706120709, 10.76207536989476270114158066097

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