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Properties

Label 2-560-35.12-c1-0-13
Degree $2$
Conductor $560$
Sign $-0.247 + 0.968i$
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  + (−0.523 + 1.95i)3-s + (−2.03 − 0.935i)5-s + (−1.83 + 1.90i)7-s + (−0.941 − 0.543i)9-s + (−2.01 − 3.49i)11-s + (0.204 + 0.204i)13-s + (2.89 − 3.47i)15-s + (−1.97 − 0.527i)17-s + (3.10 − 5.37i)19-s + (−2.75 − 4.58i)21-s + (−1.17 − 4.38i)23-s + (3.24 + 3.80i)25-s + (−2.73 + 2.73i)27-s − 7.15i·29-s + (−6.33 + 3.65i)31-s + ⋯
L(s)  = 1  + (−0.302 + 1.12i)3-s + (−0.908 − 0.418i)5-s + (−0.695 + 0.718i)7-s + (−0.313 − 0.181i)9-s + (−0.609 − 1.05i)11-s + (0.0568 + 0.0568i)13-s + (0.746 − 0.897i)15-s + (−0.477 − 0.128i)17-s + (0.711 − 1.23i)19-s + (−0.600 − 1.00i)21-s + (−0.244 − 0.914i)23-s + (0.649 + 0.760i)25-s + (−0.526 + 0.526i)27-s − 1.32i·29-s + (−1.13 + 0.656i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.247 + 0.968i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.247 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.145243 - 0.187041i\)
\(L(\frac12)\) \(\approx\) \(0.145243 - 0.187041i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 + 0.935i)T \)
7 \( 1 + (1.83 - 1.90i)T \)
good3 \( 1 + (0.523 - 1.95i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (2.01 + 3.49i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.204 - 0.204i)T + 13iT^{2} \)
17 \( 1 + (1.97 + 0.527i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.10 + 5.37i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.17 + 4.38i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 7.15iT - 29T^{2} \)
31 \( 1 + (6.33 - 3.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.46 - 1.19i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.58iT - 41T^{2} \)
43 \( 1 + (-4.97 + 4.97i)T - 43iT^{2} \)
47 \( 1 + (-0.0815 - 0.304i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (8.00 + 2.14i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.427 + 0.740i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.99 + 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.817 + 3.05i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.12T + 71T^{2} \)
73 \( 1 + (2.98 - 11.1i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.39 - 2.53i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.85 - 3.85i)T + 83iT^{2} \)
89 \( 1 + (1.53 - 2.66i)T + (-44.5 - 77.0i)T^{2} \)
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

−10.67813039510558370186636231292, −9.515190361448021950921277134751, −8.937183138674533372485611296563, −8.053689201900603690385693316152, −6.83094709492222502833407527252, −5.60764145455175425251285179436, −4.86267556546111068830687777737, −3.83261058060139339999090969153, −2.83986970100312740770763953479, −0.13925918715815855019687531435, 1.62368355607200278714912942136, 3.24365552257987782779891418773, 4.25985137562282152946934165816, 5.72347785256727189212972088783, 6.80109301784115771938176022985, 7.43827925204311956348425882360, 7.81928152277486706290385524428, 9.328523806308788069135921267231, 10.29906626262610573650004822245, 11.05172461743063299678144416045

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