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Properties

Label 2-280-280.107-c1-0-15
Degree $2$
Conductor $280$
Sign $-0.0206 - 0.999i$
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  + (−0.950 + 1.04i)2-s + (1.52 + 0.409i)3-s + (−0.192 − 1.99i)4-s + (0.934 + 2.03i)5-s + (−1.88 + 1.21i)6-s + (1.62 + 2.08i)7-s + (2.26 + 1.69i)8-s + (−0.426 − 0.245i)9-s + (−3.01 − 0.952i)10-s + (−1.76 − 3.05i)11-s + (0.521 − 3.12i)12-s + (1.68 + 1.68i)13-s + (−3.73 − 0.286i)14-s + (0.597 + 3.49i)15-s + (−3.92 + 0.766i)16-s + (0.335 − 1.25i)17-s + ⋯
L(s)  = 1  + (−0.672 + 0.740i)2-s + (0.883 + 0.236i)3-s + (−0.0962 − 0.995i)4-s + (0.418 + 0.908i)5-s + (−0.768 + 0.494i)6-s + (0.613 + 0.789i)7-s + (0.801 + 0.597i)8-s + (−0.142 − 0.0819i)9-s + (−0.953 − 0.301i)10-s + (−0.531 − 0.919i)11-s + (0.150 − 0.901i)12-s + (0.468 + 0.468i)13-s + (−0.997 − 0.0766i)14-s + (0.154 + 0.901i)15-s + (−0.981 + 0.191i)16-s + (0.0814 − 0.304i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0206 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0206 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.0206 - 0.999i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.0206 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.906676 + 0.925636i\)
\(L(\frac12)\) \(\approx\) \(0.906676 + 0.925636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.950 - 1.04i)T \)
5 \( 1 + (-0.934 - 2.03i)T \)
7 \( 1 + (-1.62 - 2.08i)T \)
good3 \( 1 + (-1.52 - 0.409i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.76 + 3.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.68 - 1.68i)T + 13iT^{2} \)
17 \( 1 + (-0.335 + 1.25i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.62 - 3.82i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.50 - 0.940i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.07T + 29T^{2} \)
31 \( 1 + (-2.42 + 1.39i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.340 + 1.27i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.50T + 41T^{2} \)
43 \( 1 + (-7.05 + 7.05i)T - 43iT^{2} \)
47 \( 1 + (-2.14 - 7.98i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.15 + 11.7i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-10.7 + 6.21i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.1 + 6.45i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.866 + 3.23i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.60iT - 71T^{2} \)
73 \( 1 + (-1.56 - 0.420i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.66 + 6.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.94 + 4.94i)T - 83iT^{2} \)
89 \( 1 + (4.71 + 2.71i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.73 + 5.73i)T + 97iT^{2} \)
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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

−11.73484441794924236536808130077, −10.99909201262805937673374414030, −9.854567349426169363892219101312, −9.168356056878383454562702095212, −8.240277795564039144105521683312, −7.54435937165692951760639489549, −6.09625402703358975933908257159, −5.45480261473705808424442188288, −3.44800031753318302155538414097, −2.09575370657587574151350292452, 1.30708456975395742424867971587, 2.58291296458852188918373662345, 4.05211404323255683100897581042, 5.27049159979159496292589045512, 7.33964015010616882133805505085, 7.950811547750919053691390190184, 8.784211699061319263975883930160, 9.665980463520798228880040289243, 10.50871048099309877946739710166, 11.57768235319170425497962635226

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