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Properties

Label 2-280-56.3-c1-0-8
Degree $2$
Conductor $280$
Sign $0.990 - 0.140i$
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  + (−1.31 + 0.532i)2-s + (−0.502 − 0.290i)3-s + (1.43 − 1.39i)4-s + (0.5 + 0.866i)5-s + (0.813 + 0.112i)6-s + (2.63 + 0.253i)7-s + (−1.13 + 2.59i)8-s + (−1.33 − 2.30i)9-s + (−1.11 − 0.868i)10-s + (0.428 − 0.742i)11-s + (−1.12 + 0.285i)12-s − 2.26·13-s + (−3.58 + 1.07i)14-s − 0.580i·15-s + (0.109 − 3.99i)16-s + (6.65 + 3.84i)17-s + ⋯
L(s)  = 1  + (−0.926 + 0.376i)2-s + (−0.290 − 0.167i)3-s + (0.716 − 0.697i)4-s + (0.223 + 0.387i)5-s + (0.331 + 0.0460i)6-s + (0.995 + 0.0956i)7-s + (−0.401 + 0.915i)8-s + (−0.443 − 0.768i)9-s + (−0.352 − 0.274i)10-s + (0.129 − 0.223i)11-s + (−0.324 + 0.0822i)12-s − 0.627·13-s + (−0.958 + 0.286i)14-s − 0.149i·15-s + (0.0274 − 0.999i)16-s + (1.61 + 0.931i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.990 - 0.140i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.990 - 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.889696 + 0.0629493i\)
\(L(\frac12)\) \(\approx\) \(0.889696 + 0.0629493i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 - 0.532i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.63 - 0.253i)T \)
good3 \( 1 + (0.502 + 0.290i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.428 + 0.742i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.26T + 13T^{2} \)
17 \( 1 + (-6.65 - 3.84i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.17 + 2.98i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.17 + 1.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.76iT - 29T^{2} \)
31 \( 1 + (-4.53 + 7.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.77 - 2.18i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.780iT - 41T^{2} \)
43 \( 1 - 7.36T + 43T^{2} \)
47 \( 1 + (-0.206 - 0.358i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.0 + 6.36i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.74 + 4.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.49 + 4.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.51 - 7.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.69iT - 71T^{2} \)
73 \( 1 + (9.52 + 5.49i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.53 - 2.61i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.58iT - 83T^{2} \)
89 \( 1 + (-5.85 + 3.37i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.09iT - 97T^{2} \)
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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

−11.63237421045834741638396664246, −10.91826025181408782189359414068, −9.904145143761479992573601901070, −9.009941928957062322079009401668, −7.976586436294919213087928200104, −7.13575571441116848987719413344, −6.01374802404240152640636475639, −5.15846186319713961263595645645, −3.01490593673428595951617275180, −1.23993183670409186423182500042, 1.34762174791737843482837578114, 2.91783322750435312607137878261, 4.74004369739133225437241470566, 5.71132094128024841022010516326, 7.50362415284941602158678962178, 7.86920899383361105753461993335, 9.124508315264625841164292541741, 9.982638201657260964682197740123, 10.78278199093795719828964035506, 11.89828475697408645575081897637

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