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Properties

Label 2-280-280.123-c1-0-13
Degree $2$
Conductor $280$
Sign $-0.459 - 0.887i$
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  + (0.949 + 1.04i)2-s + (0.357 − 0.0956i)3-s + (−0.198 + 1.99i)4-s + (−2.11 + 0.723i)5-s + (0.439 + 0.283i)6-s + (1.60 + 2.10i)7-s + (−2.27 + 1.68i)8-s + (−2.47 + 1.43i)9-s + (−2.76 − 1.53i)10-s + (−0.466 + 0.808i)11-s + (0.119 + 0.729i)12-s + (4.27 − 4.27i)13-s + (−0.678 + 3.67i)14-s + (−0.686 + 0.460i)15-s + (−3.92 − 0.789i)16-s + (1.52 + 5.69i)17-s + ⋯
L(s)  = 1  + (0.671 + 0.741i)2-s + (0.206 − 0.0552i)3-s + (−0.0991 + 0.995i)4-s + (−0.946 + 0.323i)5-s + (0.179 + 0.115i)6-s + (0.607 + 0.794i)7-s + (−0.804 + 0.594i)8-s + (−0.826 + 0.477i)9-s + (−0.874 − 0.484i)10-s + (−0.140 + 0.243i)11-s + (0.0345 + 0.210i)12-s + (1.18 − 1.18i)13-s + (−0.181 + 0.983i)14-s + (−0.177 + 0.118i)15-s + (−0.980 − 0.197i)16-s + (0.369 + 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.459 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.459 - 0.887i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.459 - 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.833239 + 1.36994i\)
\(L(\frac12)\) \(\approx\) \(0.833239 + 1.36994i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.949 - 1.04i)T \)
5 \( 1 + (2.11 - 0.723i)T \)
7 \( 1 + (-1.60 - 2.10i)T \)
good3 \( 1 + (-0.357 + 0.0956i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.466 - 0.808i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.27 + 4.27i)T - 13iT^{2} \)
17 \( 1 + (-1.52 - 5.69i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-4.55 + 2.62i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.48 + 0.666i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.15T + 29T^{2} \)
31 \( 1 + (1.23 + 0.711i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.432 + 1.61i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 6.12T + 41T^{2} \)
43 \( 1 + (7.08 + 7.08i)T + 43iT^{2} \)
47 \( 1 + (-2.45 + 9.16i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.06 - 7.69i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.873 - 0.504i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.84 + 1.64i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.526 - 1.96i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 12.5iT - 71T^{2} \)
73 \( 1 + (3.20 - 0.859i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.02 - 3.50i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.21 + 9.21i)T + 83iT^{2} \)
89 \( 1 + (-10.3 + 5.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.41 - 4.41i)T - 97iT^{2} \)
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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

−12.19895425123771351438265165967, −11.47246220085902980086960433270, −10.57159567918561546833576377932, −8.602815393479410844599234397803, −8.305637589572638122675519801504, −7.43390897464947213395097393368, −6.00283101297486167699187107156, −5.24444832596321287847458902298, −3.82634797709429909998839207246, −2.76793431133773984883039712737, 1.08226668926652016755349255522, 3.17377831647986223767923787329, 4.05228760352526722850969789729, 5.09168294323466688068235868592, 6.43334438749196267483451737075, 7.74418949222765620318738450293, 8.795794332656863244152451483057, 9.747620947592093902045666608216, 11.11694892081495750963985027128, 11.52001413461112071946931164419

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