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Properties

Label 2-560-35.3-c1-0-20
Degree $2$
Conductor $560$
Sign $-0.837 + 0.546i$
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  + (−0.447 − 1.66i)3-s + (0.849 − 2.06i)5-s + (−2.48 + 0.900i)7-s + (0.0138 − 0.00800i)9-s + (1.10 − 1.91i)11-s + (1.91 − 1.91i)13-s + (−3.83 − 0.491i)15-s + (−4.19 + 1.12i)17-s + (−1.80 − 3.12i)19-s + (2.61 + 3.74i)21-s + (0.100 − 0.375i)23-s + (−3.55 − 3.51i)25-s + (−3.68 − 3.68i)27-s + 6.62i·29-s + (−0.897 − 0.518i)31-s + ⋯
L(s)  = 1  + (−0.258 − 0.963i)3-s + (0.379 − 0.925i)5-s + (−0.940 + 0.340i)7-s + (0.00462 − 0.00266i)9-s + (0.333 − 0.578i)11-s + (0.530 − 0.530i)13-s + (−0.989 − 0.127i)15-s + (−1.01 + 0.272i)17-s + (−0.413 − 0.716i)19-s + (0.570 + 0.817i)21-s + (0.0209 − 0.0783i)23-s + (−0.711 − 0.702i)25-s + (−0.708 − 0.708i)27-s + 1.22i·29-s + (−0.161 − 0.0930i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.837 + 0.546i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.837 + 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.306475 - 1.03031i\)
\(L(\frac12)\) \(\approx\) \(0.306475 - 1.03031i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.849 + 2.06i)T \)
7 \( 1 + (2.48 - 0.900i)T \)
good3 \( 1 + (0.447 + 1.66i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.10 + 1.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.91 + 1.91i)T - 13iT^{2} \)
17 \( 1 + (4.19 - 1.12i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.80 + 3.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.100 + 0.375i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.62iT - 29T^{2} \)
31 \( 1 + (0.897 + 0.518i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.88 + 1.84i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.03iT - 41T^{2} \)
43 \( 1 + (-8.37 - 8.37i)T + 43iT^{2} \)
47 \( 1 + (1.48 - 5.52i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.07 - 1.62i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.71 + 6.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.84 + 4.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.70 + 10.0i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.39T + 71T^{2} \)
73 \( 1 + (3.69 + 13.7i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.38 + 3.11i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.77 + 7.77i)T - 83iT^{2} \)
89 \( 1 + (5.20 + 9.01i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.1 - 12.1i)T + 97iT^{2} \)
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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

−10.46390950187938679226980905790, −9.198571323464741219247491824599, −8.843977122010524349545435951522, −7.68938527937934352802020292094, −6.47384584470744053592627757811, −6.14431714030095213674608688388, −4.91303971387284914000187884748, −3.52946598296288453780496612442, −2.00868905753474267664707957913, −0.62508544887794534781469225650, 2.17817837927299267336124019232, 3.65069289413246266228219991863, 4.28231590281303695444122699244, 5.67291549818388319464795391440, 6.61785739279368143121477893434, 7.24100325506451977511915508984, 8.767325061643169310294765159635, 9.725619712258450866733646725859, 10.12958109839082797125979962253, 10.90903304467941540703555310939

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