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Properties

Label 2-280-56.27-c1-0-0
Degree $2$
Conductor $280$
Sign $-0.646 + 0.763i$
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  + (−1.24 + 0.678i)2-s + 1.61i·3-s + (1.08 − 1.68i)4-s − 5-s + (−1.09 − 1.99i)6-s + (−2.13 − 1.56i)7-s + (−0.198 + 2.82i)8-s + 0.405·9-s + (1.24 − 0.678i)10-s − 6.01·11-s + (2.71 + 1.73i)12-s − 4.25·13-s + (3.70 + 0.492i)14-s − 1.61i·15-s + (−1.66 − 3.63i)16-s + 5.42i·17-s + ⋯
L(s)  = 1  + (−0.877 + 0.479i)2-s + 0.929i·3-s + (0.540 − 0.841i)4-s − 0.447·5-s + (−0.445 − 0.816i)6-s + (−0.806 − 0.590i)7-s + (−0.0702 + 0.997i)8-s + 0.135·9-s + (0.392 − 0.214i)10-s − 1.81·11-s + (0.782 + 0.502i)12-s − 1.17·13-s + (0.991 + 0.131i)14-s − 0.415i·15-s + (−0.416 − 0.909i)16-s + 1.31i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.646 + 0.763i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.646 + 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0235728 - 0.0508331i\)
\(L(\frac12)\) \(\approx\) \(0.0235728 - 0.0508331i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 - 0.678i)T \)
5 \( 1 + T \)
7 \( 1 + (2.13 + 1.56i)T \)
good3 \( 1 - 1.61iT - 3T^{2} \)
11 \( 1 + 6.01T + 11T^{2} \)
13 \( 1 + 4.25T + 13T^{2} \)
17 \( 1 - 5.42iT - 17T^{2} \)
19 \( 1 + 4.53iT - 19T^{2} \)
23 \( 1 + 5.19iT - 23T^{2} \)
29 \( 1 + 0.376iT - 29T^{2} \)
31 \( 1 + 2.93T + 31T^{2} \)
37 \( 1 - 0.372iT - 37T^{2} \)
41 \( 1 + 5.75iT - 41T^{2} \)
43 \( 1 + 4.96T + 43T^{2} \)
47 \( 1 + 3.86T + 47T^{2} \)
53 \( 1 - 10.2iT - 53T^{2} \)
59 \( 1 - 10.5iT - 59T^{2} \)
61 \( 1 - 5.58T + 61T^{2} \)
67 \( 1 + 0.782T + 67T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 + 8.77iT - 73T^{2} \)
79 \( 1 - 8.74iT - 79T^{2} \)
83 \( 1 + 8.42iT - 83T^{2} \)
89 \( 1 + 1.94iT - 89T^{2} \)
97 \( 1 + 3.14iT - 97T^{2} \)
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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

−12.44859471768342397534053413494, −10.86719230874259190634324927279, −10.41717946353223301650879822837, −9.802219177019109307280108343417, −8.702815240085150187330768088561, −7.63667653645184702051705739238, −6.86957191578765753300813002724, −5.41313985357736642477983481503, −4.38109650965858881488993121899, −2.72679629919912194186494178304, 0.05128421911579737198292046397, 2.17692769994359732893019549406, 3.20637094062334291951636715094, 5.20923661074796746675184382814, 6.74355966025707933840606020657, 7.57606269910839946620910119723, 8.114712138603578844687904153091, 9.583277771595799773612460788019, 10.07527780723354388488941969009, 11.39677636390180562450473064491

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