Properties

Label 2-560-35.33-c1-0-4
Degree $2$
Conductor $560$
Sign $-0.242 - 0.970i$
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  + (0.279 + 0.0749i)3-s + (0.774 + 2.09i)5-s + (−2.64 + 0.126i)7-s + (−2.52 − 1.45i)9-s + (2.81 + 4.87i)11-s + (1.42 − 1.42i)13-s + (0.0593 + 0.645i)15-s + (−1.37 + 5.12i)17-s + (−1.94 + 3.37i)19-s + (−0.749 − 0.162i)21-s + (−1.08 + 0.290i)23-s + (−3.80 + 3.24i)25-s + (−1.21 − 1.21i)27-s + 3.15i·29-s + (3.33 − 1.92i)31-s + ⋯
L(s)  = 1  + (0.161 + 0.0432i)3-s + (0.346 + 0.938i)5-s + (−0.998 + 0.0477i)7-s + (−0.841 − 0.486i)9-s + (0.848 + 1.46i)11-s + (0.396 − 0.396i)13-s + (0.0153 + 0.166i)15-s + (−0.333 + 1.24i)17-s + (−0.446 + 0.773i)19-s + (−0.163 − 0.0355i)21-s + (−0.226 + 0.0606i)23-s + (−0.760 + 0.649i)25-s + (−0.233 − 0.233i)27-s + 0.585i·29-s + (0.598 − 0.345i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.242 - 0.970i$
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: \((2,\ 560,\ (\ :1/2),\ -0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723845 + 0.926715i\)
\(L(\frac12)\) \(\approx\) \(0.723845 + 0.926715i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.774 - 2.09i)T \)
7 \( 1 + (2.64 - 0.126i)T \)
good3 \( 1 + (-0.279 - 0.0749i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.81 - 4.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.42 + 1.42i)T - 13iT^{2} \)
17 \( 1 + (1.37 - 5.12i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.94 - 3.37i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.08 - 0.290i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.15iT - 29T^{2} \)
31 \( 1 + (-3.33 + 1.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.30 + 4.86i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.21iT - 41T^{2} \)
43 \( 1 + (1.85 + 1.85i)T + 43iT^{2} \)
47 \( 1 + (-5.69 + 1.52i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.357 - 1.33i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.73 - 4.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.99 + 2.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.816 - 0.218i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.77T + 71T^{2} \)
73 \( 1 + (-5.42 - 1.45i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.41 - 3.12i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.67 + 5.67i)T - 83iT^{2} \)
89 \( 1 + (-5.96 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.63 - 6.63i)T + 97iT^{2} \)
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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

−10.83067792311240872057603662491, −10.09380756185546739683771779290, −9.415978183890105524197555150479, −8.478618704836369419428839681812, −7.25831114014897542123533368591, −6.36993710242119792670400876806, −5.89776760966281347446457535657, −4.09674365997693732403006813462, −3.26438666517152995040300980615, −2.01084394670280486148939621369, 0.64548591966137465853417544577, 2.52121145505864454395075735572, 3.67249529245420932854785893027, 4.94754731306372885170635919944, 5.98555602510783672537133268584, 6.65700403802769976265080983605, 8.100604863579955806450665834513, 9.002956410530195454303937028129, 9.215894047608027071218632930039, 10.51930613221374777972382791651

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