Properties

Label 2-560-35.33-c1-0-21
Degree $2$
Conductor $560$
Sign $-0.999 - 0.00814i$
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  + (−1.13 − 0.304i)3-s + (0.264 − 2.22i)5-s + (−0.698 − 2.55i)7-s + (−1.40 − 0.810i)9-s + (0.371 + 0.643i)11-s + (−2.05 + 2.05i)13-s + (−0.975 + 2.43i)15-s + (−1.69 + 6.33i)17-s + (0.946 − 1.63i)19-s + (0.0172 + 3.10i)21-s + (−5.11 + 1.36i)23-s + (−4.85 − 1.17i)25-s + (3.83 + 3.83i)27-s − 9.69i·29-s + (−2.96 + 1.71i)31-s + ⋯
L(s)  = 1  + (−0.655 − 0.175i)3-s + (0.118 − 0.992i)5-s + (−0.264 − 0.964i)7-s + (−0.467 − 0.270i)9-s + (0.112 + 0.194i)11-s + (−0.570 + 0.570i)13-s + (−0.251 + 0.629i)15-s + (−0.411 + 1.53i)17-s + (0.217 − 0.375i)19-s + (0.00376 + 0.678i)21-s + (−1.06 + 0.285i)23-s + (−0.971 − 0.235i)25-s + (0.738 + 0.738i)27-s − 1.79i·29-s + (−0.532 + 0.307i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.999 - 0.00814i$
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: \((2,\ 560,\ (\ :1/2),\ -0.999 - 0.00814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00171698 + 0.421679i\)
\(L(\frac12)\) \(\approx\) \(0.00171698 + 0.421679i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.264 + 2.22i)T \)
7 \( 1 + (0.698 + 2.55i)T \)
good3 \( 1 + (1.13 + 0.304i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.371 - 0.643i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.05 - 2.05i)T - 13iT^{2} \)
17 \( 1 + (1.69 - 6.33i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.946 + 1.63i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.11 - 1.36i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.69iT - 29T^{2} \)
31 \( 1 + (2.96 - 1.71i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.691 + 2.58i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.817iT - 41T^{2} \)
43 \( 1 + (1.59 + 1.59i)T + 43iT^{2} \)
47 \( 1 + (4.54 - 1.21i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.29 + 4.81i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.27 + 2.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.25 - 3.03i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.2 + 3.54i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 + (8.54 + 2.29i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.70 + 3.29i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.23 + 9.23i)T - 83iT^{2} \)
89 \( 1 + (3.01 - 5.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.16 + 3.16i)T + 97iT^{2} \)
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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

−10.28943918741147635806751638113, −9.530521261431718487778860248016, −8.567826836405301456150073067311, −7.63153006441529821381163490605, −6.51509173274678090788895926189, −5.79523224875940792026951502661, −4.62144559366337008811908975827, −3.79434849109077991699732383646, −1.81679795777439846888378682503, −0.24687992876359409701186636411, 2.39739210728196568115585709279, 3.24651492974821616045098767004, 4.95955679496998676029952882327, 5.69251701580035347605472797308, 6.54166219917776852297611953345, 7.51355748999879378592474606979, 8.616580813637024781417577135018, 9.631358997810381018796902519021, 10.38152561003319982521360410768, 11.28522817680868666761717083317

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