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Properties

Label 2-560-35.33-c1-0-11
Degree $2$
Conductor $560$
Sign $-0.153 + 0.988i$
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  + (−3.19 − 0.856i)3-s + (0.672 − 2.13i)5-s + (2.52 − 0.802i)7-s + (6.87 + 3.97i)9-s + (1.05 + 1.82i)11-s + (−1.20 + 1.20i)13-s + (−3.97 + 6.23i)15-s + (0.850 − 3.17i)17-s + (2.36 − 4.09i)19-s + (−8.74 + 0.406i)21-s + (4.00 − 1.07i)23-s + (−4.09 − 2.86i)25-s + (−11.5 − 11.5i)27-s + 3.65i·29-s + (1.63 − 0.946i)31-s + ⋯
L(s)  = 1  + (−1.84 − 0.494i)3-s + (0.300 − 0.953i)5-s + (0.952 − 0.303i)7-s + (2.29 + 1.32i)9-s + (0.317 + 0.550i)11-s + (−0.334 + 0.334i)13-s + (−1.02 + 1.61i)15-s + (0.206 − 0.770i)17-s + (0.542 − 0.939i)19-s + (−1.90 + 0.0886i)21-s + (0.835 − 0.223i)23-s + (−0.819 − 0.573i)25-s + (−2.22 − 2.22i)27-s + 0.679i·29-s + (0.294 − 0.169i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.153 + 0.988i$
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: \((2,\ 560,\ (\ :1/2),\ -0.153 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.578191 - 0.674819i\)
\(L(\frac12)\) \(\approx\) \(0.578191 - 0.674819i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.672 + 2.13i)T \)
7 \( 1 + (-2.52 + 0.802i)T \)
good3 \( 1 + (3.19 + 0.856i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.05 - 1.82i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.20 - 1.20i)T - 13iT^{2} \)
17 \( 1 + (-0.850 + 3.17i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.36 + 4.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.00 + 1.07i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 + (-1.63 + 0.946i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.65 + 9.91i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.826iT - 41T^{2} \)
43 \( 1 + (4.70 + 4.70i)T + 43iT^{2} \)
47 \( 1 + (3.90 - 1.04i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.645 - 2.40i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.15 - 2.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.44 - 0.837i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.2 - 3.28i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + (12.6 + 3.37i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.29 + 4.78i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.47 + 3.47i)T - 83iT^{2} \)
89 \( 1 + (-2.86 + 4.96i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.02 + 1.02i)T + 97iT^{2} \)
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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

−10.75631478871652213259065872779, −9.832183412905184259930536412390, −8.829275088238157988221736876588, −7.41543621811305505105205908164, −6.98587933730137755809738118385, −5.69840599735246428254366588906, −4.95868679915792877177655921043, −4.49563658785402110668186467907, −1.81671746379941018444084192382, −0.73751414748244673619157017847, 1.42616639939951704619659585868, 3.43873595964141747576613737411, 4.70312899959731950057089283486, 5.58031797463589538008572023554, 6.20198070883757159974958424543, 7.11754534290038300858914140438, 8.258055348935886259108083240063, 9.786365655629022356351404284639, 10.25497267976697080040878975988, 11.15442490331151120532526662933

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