Properties

Label 2-560-35.33-c1-0-1
Degree $2$
Conductor $560$
Sign $-0.999 - 0.0445i$
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  + (1.35 + 0.364i)3-s + (−2.13 + 0.666i)5-s + (−2.59 − 0.501i)7-s + (−0.882 − 0.509i)9-s + (−1.86 − 3.23i)11-s + (−4.55 + 4.55i)13-s + (−3.14 + 0.129i)15-s + (−1.58 + 5.91i)17-s + (0.616 − 1.06i)19-s + (−3.34 − 1.62i)21-s + (2.69 − 0.721i)23-s + (4.11 − 2.84i)25-s + (−3.99 − 3.99i)27-s + 2.82i·29-s + (−5.94 + 3.43i)31-s + ⋯
L(s)  = 1  + (0.784 + 0.210i)3-s + (−0.954 + 0.298i)5-s + (−0.981 − 0.189i)7-s + (−0.294 − 0.169i)9-s + (−0.563 − 0.975i)11-s + (−1.26 + 1.26i)13-s + (−0.811 + 0.0333i)15-s + (−0.384 + 1.43i)17-s + (0.141 − 0.244i)19-s + (−0.730 − 0.355i)21-s + (0.561 − 0.150i)23-s + (0.822 − 0.569i)25-s + (−0.769 − 0.769i)27-s + 0.525i·29-s + (−1.06 + 0.616i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.999 - 0.0445i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.999 - 0.0445i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00396341 + 0.178031i\)
\(L(\frac12)\) \(\approx\) \(0.00396341 + 0.178031i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 - 0.666i)T \)
7 \( 1 + (2.59 + 0.501i)T \)
good3 \( 1 + (-1.35 - 0.364i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.86 + 3.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.55 - 4.55i)T - 13iT^{2} \)
17 \( 1 + (1.58 - 5.91i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.616 + 1.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.69 + 0.721i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + (5.94 - 3.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.01 + 7.52i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.56iT - 41T^{2} \)
43 \( 1 + (-1.95 - 1.95i)T + 43iT^{2} \)
47 \( 1 + (-11.1 + 2.98i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.53 - 9.44i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.916 - 1.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.43 + 1.98i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.93 + 2.12i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.570T + 71T^{2} \)
73 \( 1 + (2.03 + 0.546i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (9.47 + 5.47i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.80 - 8.80i)T - 83iT^{2} \)
89 \( 1 + (3.00 - 5.20i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.70 - 3.70i)T + 97iT^{2} \)
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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

−11.02349865209927515969236917872, −10.42037544516024787168494188444, −9.063221024434595823424293576700, −8.850660892488133355121058692442, −7.59910325352184400578628892103, −6.90834918938492781040227865554, −5.77061132768934738813411208624, −4.23707343814560890827048471795, −3.46374376204689692943052712466, −2.53479855307120683158563948173, 0.084696252179127875110906680912, 2.55970292721864505364406064431, 3.17723205596862571484687764643, 4.63822834907929763384634050830, 5.53090345045398539544763264363, 7.28980872571304308090951406538, 7.44451220197092631958406277723, 8.512122182950636109609035747822, 9.439618283528369929573504335657, 10.09313534121637819246703667565

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