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Properties

Label 2-280-280.19-c1-0-27
Degree $2$
Conductor $280$
Sign $-0.940 + 0.339i$
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 − 0.670i)2-s + (−0.818 − 1.41i)3-s + (1.10 + 1.66i)4-s + (0.951 + 2.02i)5-s + (0.0691 + 2.31i)6-s + (−2.64 − 0.148i)7-s + (−0.253 − 2.81i)8-s + (0.158 − 0.274i)9-s + (0.171 − 3.15i)10-s + (−2.19 − 3.79i)11-s + (1.46 − 2.92i)12-s − 2.75i·13-s + (3.19 + 1.95i)14-s + (2.09 − 3.00i)15-s + (−1.57 + 3.67i)16-s + (−0.648 − 1.12i)17-s + ⋯
L(s)  = 1  + (−0.880 − 0.473i)2-s + (−0.472 − 0.818i)3-s + (0.550 + 0.834i)4-s + (0.425 + 0.904i)5-s + (0.0282 + 0.945i)6-s + (−0.998 − 0.0561i)7-s + (−0.0895 − 0.995i)8-s + (0.0529 − 0.0916i)9-s + (0.0542 − 0.998i)10-s + (−0.661 − 1.14i)11-s + (0.423 − 0.845i)12-s − 0.764i·13-s + (0.852 + 0.522i)14-s + (0.539 − 0.776i)15-s + (−0.393 + 0.919i)16-s + (−0.157 − 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.940 + 0.339i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.940 + 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0735971 - 0.420973i\)
\(L(\frac12)\) \(\approx\) \(0.0735971 - 0.420973i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 + 0.670i)T \)
5 \( 1 + (-0.951 - 2.02i)T \)
7 \( 1 + (2.64 + 0.148i)T \)
good3 \( 1 + (0.818 + 1.41i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (2.19 + 3.79i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.75iT - 13T^{2} \)
17 \( 1 + (0.648 + 1.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.86 + 2.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.136 - 0.236i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.84iT - 29T^{2} \)
31 \( 1 + (-2.27 - 3.94i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.94 - 6.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.95iT - 41T^{2} \)
43 \( 1 + 4.46iT - 43T^{2} \)
47 \( 1 + (-7.22 - 4.17i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.40 + 9.35i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.31 + 1.91i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.73 - 8.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.32 + 4.80i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.42iT - 71T^{2} \)
73 \( 1 + (-1.84 - 3.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (12.3 + 7.11i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + (3.15 + 1.82i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.0199T + 97T^{2} \)
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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

−11.29827274807424793392247474285, −10.48432002065595922643120543795, −9.788289316974355778917821763584, −8.576038622302204435181487098877, −7.46645023391669116264996160425, −6.57931636055679498981869203906, −5.94758685414719064566057472672, −3.46318676130521978503620476337, −2.45743961793645729732535103776, −0.42668319533253292334532424288, 2.01777192359228269437882825106, 4.35263952857732402818428543306, 5.31651742518167343421884838283, 6.32308878416449656654204022909, 7.45222691670120324420991593043, 8.742378757531362996593290194573, 9.523210923402043390390397311410, 10.14926377854431671939005839810, 10.87775868283818559550798862182, 12.27570997733800001232961635610

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