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Properties

Label 2-280-40.27-c1-0-29
Degree $2$
Conductor $280$
Sign $-0.682 + 0.730i$
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  + (−0.700 + 1.22i)2-s + (0.311 − 0.311i)3-s + (−1.01 − 1.72i)4-s + (−1.92 − 1.14i)5-s + (0.164 + 0.600i)6-s + (−0.707 + 0.707i)7-s + (2.82 − 0.0450i)8-s + 2.80i·9-s + (2.74 − 1.56i)10-s − 5.26·11-s + (−0.852 − 0.218i)12-s + (−2.51 − 2.51i)13-s + (−0.373 − 1.36i)14-s + (−0.953 + 0.243i)15-s + (−1.92 + 3.50i)16-s + (−4.97 − 4.97i)17-s + ⋯
L(s)  = 1  + (−0.495 + 0.868i)2-s + (0.179 − 0.179i)3-s + (−0.509 − 0.860i)4-s + (−0.859 − 0.510i)5-s + (0.0670 + 0.245i)6-s + (−0.267 + 0.267i)7-s + (0.999 − 0.0159i)8-s + 0.935i·9-s + (0.869 − 0.494i)10-s − 1.58·11-s + (−0.246 − 0.0631i)12-s + (−0.696 − 0.696i)13-s + (−0.0997 − 0.364i)14-s + (−0.246 + 0.0628i)15-s + (−0.481 + 0.876i)16-s + (−1.20 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.682 + 0.730i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.682 + 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0341624 - 0.0786683i\)
\(L(\frac12)\) \(\approx\) \(0.0341624 - 0.0786683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.700 - 1.22i)T \)
5 \( 1 + (1.92 + 1.14i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-0.311 + 0.311i)T - 3iT^{2} \)
11 \( 1 + 5.26T + 11T^{2} \)
13 \( 1 + (2.51 + 2.51i)T + 13iT^{2} \)
17 \( 1 + (4.97 + 4.97i)T + 17iT^{2} \)
19 \( 1 + 0.888iT - 19T^{2} \)
23 \( 1 + (1.26 + 1.26i)T + 23iT^{2} \)
29 \( 1 - 1.37T + 29T^{2} \)
31 \( 1 - 5.24iT - 31T^{2} \)
37 \( 1 + (-4.60 + 4.60i)T - 37iT^{2} \)
41 \( 1 - 1.29T + 41T^{2} \)
43 \( 1 + (5.98 - 5.98i)T - 43iT^{2} \)
47 \( 1 + (1.47 - 1.47i)T - 47iT^{2} \)
53 \( 1 + (7.34 + 7.34i)T + 53iT^{2} \)
59 \( 1 - 1.92iT - 59T^{2} \)
61 \( 1 + 6.47iT - 61T^{2} \)
67 \( 1 + (-4.76 - 4.76i)T + 67iT^{2} \)
71 \( 1 - 15.1iT - 71T^{2} \)
73 \( 1 + (-8.78 + 8.78i)T - 73iT^{2} \)
79 \( 1 - 1.86T + 79T^{2} \)
83 \( 1 + (-0.380 + 0.380i)T - 83iT^{2} \)
89 \( 1 + 7.38iT - 89T^{2} \)
97 \( 1 + (-7.22 - 7.22i)T + 97iT^{2} \)
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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

−11.30251484284310593536202668186, −10.46402281149385118915672737621, −9.395178928088087079514514311500, −8.278240593623284510574096651290, −7.80730761611172055445277060347, −6.89129673011455645277125219742, −5.23722704992408294100694146701, −4.77230000304998480129217657124, −2.59952038206198186017495795132, −0.06998328264184466659757120574, 2.45273390945462499212044077844, 3.63389314175500286175987220675, 4.55912162501964769340472775103, 6.53463570380449882037844737959, 7.65175206858641965845058034050, 8.427477777449210356049027151602, 9.562033615727330375874654801892, 10.39520963548918487347202400042, 11.14663242689653431351704938367, 12.08021474243205524918555268681

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