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Properties

Label 2-280-56.19-c1-0-16
Degree $2$
Conductor $280$
Sign $-0.197 + 0.980i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−1.5 + 0.866i)3-s − 2i·4-s + (−0.5 + 0.866i)5-s + (0.633 − 2.36i)6-s + (−0.866 + 2.5i)7-s + (2 + 2i)8-s + (−0.366 − 1.36i)10-s + (−2.73 − 4.73i)11-s + (1.73 + 3i)12-s − 1.26·13-s + (−1.63 − 3.36i)14-s − 1.73i·15-s − 4·16-s + (4.09 − 2.36i)17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.866 + 0.499i)3-s − i·4-s + (−0.223 + 0.387i)5-s + (0.258 − 0.965i)6-s + (−0.327 + 0.944i)7-s + (0.707 + 0.707i)8-s + (−0.115 − 0.431i)10-s + (−0.823 − 1.42i)11-s + (0.499 + 0.866i)12-s − 0.351·13-s + (−0.436 − 0.899i)14-s − 0.447i·15-s − 16-s + (0.993 − 0.573i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.197 + 0.980i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.197 + 0.980i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.866 - 2.5i)T \)
good3 \( 1 + (1.5 - 0.866i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.73 + 4.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.26T + 13T^{2} \)
17 \( 1 + (-4.09 + 2.36i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.90 + 1.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.86 + 3.96i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.53iT - 29T^{2} \)
31 \( 1 + (-4.09 - 7.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.73 - i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 + 9.92T + 43T^{2} \)
47 \( 1 + (4.73 - 8.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.83 - 5.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.09 - 0.633i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.86 - 6.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.69 - 8.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.73iT - 71T^{2} \)
73 \( 1 + (4.90 - 2.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.09 + 0.633i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.66iT - 83T^{2} \)
89 \( 1 + (-2.30 - 1.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.46iT - 97T^{2} \)
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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

−11.36371916758799169089474686085, −10.49667058478195244349522643468, −9.860681449651809871547997402257, −8.555486439842126471054043189548, −7.893780494438213361211865683266, −6.37421838365332261072671762417, −5.76279013121502496092617392208, −4.85079335238427475029557264149, −2.77494325968640988911933112943, 0, 1.68517863689454721011176667104, 3.60869616102761753444986059970, 4.85780221458947423922527835508, 6.40660556810252431110409381198, 7.49070242268475714322864488203, 8.077371125147485741836240099910, 9.803251194063441923130431728283, 10.06414072931174097860296277798, 11.25704845396704398800159276642

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