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Properties

Label 2-280-280.157-c1-0-39
Degree $2$
Conductor $280$
Sign $-0.937 - 0.347i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−0.5 − 1.86i)3-s − 2i·4-s + (−1.86 − 1.23i)5-s + (2.36 + 1.36i)6-s + (−2.5 + 0.866i)7-s + (2 + 2i)8-s + (−0.633 + 0.366i)9-s + (3.09 − 0.633i)10-s + (0.633 + 0.366i)11-s + (−3.73 + i)12-s + (−3 + 3i)13-s + (1.63 − 3.36i)14-s + (−1.36 + 4.09i)15-s − 4·16-s + (1.09 + 4.09i)17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.288 − 1.07i)3-s − i·4-s + (−0.834 − 0.550i)5-s + (0.965 + 0.557i)6-s + (−0.944 + 0.327i)7-s + (0.707 + 0.707i)8-s + (−0.211 + 0.122i)9-s + (0.979 − 0.200i)10-s + (0.191 + 0.110i)11-s + (−1.07 + 0.288i)12-s + (−0.832 + 0.832i)13-s + (0.436 − 0.899i)14-s + (−0.352 + 1.05i)15-s − 16-s + (0.266 + 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.937 - 0.347i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.937 - 0.347i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
5 \( 1 + (1.86 + 1.23i)T \)
7 \( 1 + (2.5 - 0.866i)T \)
good3 \( 1 + (0.5 + 1.86i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.633 - 0.366i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + (-1.09 - 4.09i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 0.401i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 4.46T + 29T^{2} \)
31 \( 1 + (8.83 + 5.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.46 + 1.73i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.464iT - 41T^{2} \)
43 \( 1 + (2.36 + 2.36i)T + 43iT^{2} \)
47 \( 1 + (-0.633 - 0.169i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3 - 0.803i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (7.73 + 4.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.96 - 8.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.03 + 3.86i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + (8.09 - 2.16i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (11.1 - 6.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.56 - 4.56i)T - 83iT^{2} \)
89 \( 1 + (8.59 + 14.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.26 + 6.26i)T - 97iT^{2} \)
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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

−11.49109338040869838309151368423, −10.18208671651172105480269650128, −9.198517183680155927992841623590, −8.294107361977494219496444995427, −7.32498470385754554009418538853, −6.64746783215604572579625498997, −5.67546771380448751773039459399, −4.13221213250444392625625761669, −1.80849615249347930090697781753, 0, 2.94376085234748517950206501713, 3.76899561126095760582417579529, 4.91998841012592286652111694114, 6.81159856042642961845173788890, 7.60962458049685940924292790643, 8.895628573013395198214926459934, 9.866439647745515534500249071925, 10.42076078882444577262957482322, 11.12582321230534864270327587758

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