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Properties

Label 2-280-280.59-c1-0-8
Degree $2$
Conductor $280$
Sign $-0.586 - 0.810i$
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  + (−0.634 + 1.26i)2-s + (0.653 − 1.13i)3-s + (−1.19 − 1.60i)4-s + (−1.67 + 1.48i)5-s + (1.01 + 1.54i)6-s + (−1.03 + 2.43i)7-s + (2.78 − 0.494i)8-s + (0.646 + 1.11i)9-s + (−0.816 − 3.05i)10-s + (0.168 − 0.292i)11-s + (−2.59 + 0.305i)12-s + 1.29i·13-s + (−2.42 − 2.85i)14-s + (0.588 + 2.86i)15-s + (−1.14 + 3.83i)16-s + (−3.41 + 5.92i)17-s + ⋯
L(s)  = 1  + (−0.448 + 0.893i)2-s + (0.377 − 0.653i)3-s + (−0.597 − 0.801i)4-s + (−0.747 + 0.664i)5-s + (0.414 + 0.630i)6-s + (−0.391 + 0.920i)7-s + (0.984 − 0.174i)8-s + (0.215 + 0.373i)9-s + (−0.258 − 0.966i)10-s + (0.0508 − 0.0880i)11-s + (−0.749 + 0.0882i)12-s + 0.360i·13-s + (−0.646 − 0.762i)14-s + (0.151 + 0.739i)15-s + (−0.285 + 0.958i)16-s + (−0.829 + 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.586 - 0.810i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.586 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.357398 + 0.699794i\)
\(L(\frac12)\) \(\approx\) \(0.357398 + 0.699794i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.634 - 1.26i)T \)
5 \( 1 + (1.67 - 1.48i)T \)
7 \( 1 + (1.03 - 2.43i)T \)
good3 \( 1 + (-0.653 + 1.13i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.168 + 0.292i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.29iT - 13T^{2} \)
17 \( 1 + (3.41 - 5.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.551 - 0.318i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.51 - 2.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.90iT - 29T^{2} \)
31 \( 1 + (1.08 - 1.87i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.30 + 7.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.28iT - 41T^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 + (-7.78 + 4.49i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.63 + 6.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.50 - 4.33i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.97 + 6.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.9 - 6.31i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.53iT - 71T^{2} \)
73 \( 1 + (5.93 - 10.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.96 + 5.17i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.76T + 83T^{2} \)
89 \( 1 + (6.43 - 3.71i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.03T + 97T^{2} \)
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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

−12.38065156064061627389200021451, −11.07137037968824697521595011955, −10.25312848628530254975357145324, −8.842642939691440597926470275706, −8.376427922734397623801792239353, −7.20863262498588973764488296838, −6.65882816593688379071248770507, −5.42474686476888603163951606403, −3.86614547569631076281027975624, −2.06792561710204122888945975365, 0.67959540965697530984391193066, 2.97617827641454541661746330273, 4.08729583528079883313749171123, 4.71237194296814741273661362381, 6.94129223534653321954581400148, 7.926256634411307215929552892607, 8.994549503516406236619003625360, 9.595482029648103353992738380316, 10.52270278701028642844198148258, 11.43288805392297872348069825127

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