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Properties

Label 2-280-280.67-c1-0-16
Degree $2$
Conductor $280$
Sign $-0.768 - 0.639i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
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  + (0.909 + 1.08i)2-s + (0.835 + 3.11i)3-s + (−0.347 + 1.96i)4-s + (−0.804 − 2.08i)5-s + (−2.61 + 3.73i)6-s + (2.64 − 0.0485i)7-s + (−2.44 + 1.41i)8-s + (−6.42 + 3.70i)9-s + (1.52 − 2.76i)10-s + (1.23 − 2.14i)11-s + (−6.43 + 0.563i)12-s + (−0.837 − 0.837i)13-s + (2.45 + 2.82i)14-s + (5.83 − 4.25i)15-s + (−3.75 − 1.36i)16-s + (6.14 − 1.64i)17-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (0.482 + 1.80i)3-s + (−0.173 + 0.984i)4-s + (−0.359 − 0.932i)5-s + (−1.06 + 1.52i)6-s + (0.999 − 0.0183i)7-s + (−0.865 + 0.500i)8-s + (−2.14 + 1.23i)9-s + (0.483 − 0.875i)10-s + (0.373 − 0.646i)11-s + (−1.85 + 0.162i)12-s + (−0.232 − 0.232i)13-s + (0.656 + 0.754i)14-s + (1.50 − 1.09i)15-s + (−0.939 − 0.341i)16-s + (1.49 − 0.399i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.768 - 0.639i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.768 - 0.639i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663830 + 1.83607i\)
\(L(\frac12)\) \(\approx\) \(0.663830 + 1.83607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.909 - 1.08i)T \)
5 \( 1 + (0.804 + 2.08i)T \)
7 \( 1 + (-2.64 + 0.0485i)T \)
good3 \( 1 + (-0.835 - 3.11i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.23 + 2.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.837 + 0.837i)T + 13iT^{2} \)
17 \( 1 + (-6.14 + 1.64i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.588 - 0.339i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.670 + 2.50i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 3.80T + 29T^{2} \)
31 \( 1 + (-4.91 - 2.83i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.91 + 1.04i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.957T + 41T^{2} \)
43 \( 1 + (4.19 - 4.19i)T - 43iT^{2} \)
47 \( 1 + (0.868 + 0.232i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.66 + 0.712i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-8.10 - 4.67i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.10 - 0.639i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.43 - 1.72i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.973iT - 71T^{2} \)
73 \( 1 + (0.439 + 1.64i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.01 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.2 + 10.2i)T - 83iT^{2} \)
89 \( 1 + (3.10 - 1.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.33 + 1.33i)T + 97iT^{2} \)
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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

−12.08679304324895106886695226547, −11.42566175447014139099858139270, −10.22605118681750178562529752924, −9.077928108006011348559148354481, −8.456983146793599617206647695968, −7.72048528937286860793227644154, −5.64486866375225532944794013285, −4.99060691297329145622959739533, −4.18265466726822011675848665304, −3.18729540062890179010081278530, 1.45223123260258620755169194757, 2.48043303378401527866764473002, 3.74001547783119952396603374645, 5.50376469915991732028361939657, 6.65316576794551249898228630115, 7.46369287564387768299367852934, 8.345012878196875394462958044899, 9.743199074530932522067780428452, 10.99037725063290266941164830819, 11.93071900404651891473264214827

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