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Properties

Label 2-280-8.5-c1-0-12
Degree $2$
Conductor $280$
Sign $0.523 + 0.852i$
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  + (−1.39 + 0.258i)2-s − 0.861i·3-s + (1.86 − 0.718i)4-s − i·5-s + (0.222 + 1.19i)6-s + 7-s + (−2.40 + 1.48i)8-s + 2.25·9-s + (0.258 + 1.39i)10-s + 2.22i·11-s + (−0.618 − 1.60i)12-s − 5.81i·13-s + (−1.39 + 0.258i)14-s − 0.861·15-s + (2.96 − 2.68i)16-s − 3.57·17-s + ⋯
L(s)  = 1  + (−0.983 + 0.182i)2-s − 0.497i·3-s + (0.933 − 0.359i)4-s − 0.447i·5-s + (0.0907 + 0.488i)6-s + 0.377·7-s + (−0.852 + 0.523i)8-s + 0.752·9-s + (0.0816 + 0.439i)10-s + 0.671i·11-s + (−0.178 − 0.463i)12-s − 1.61i·13-s + (−0.371 + 0.0690i)14-s − 0.222·15-s + (0.742 − 0.670i)16-s − 0.866·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.523 + 0.852i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.523 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.776609 - 0.434336i\)
\(L(\frac12)\) \(\approx\) \(0.776609 - 0.434336i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.258i)T \)
5 \( 1 + iT \)
7 \( 1 - T \)
good3 \( 1 + 0.861iT - 3T^{2} \)
11 \( 1 - 2.22iT - 11T^{2} \)
13 \( 1 + 5.81iT - 13T^{2} \)
17 \( 1 + 3.57T + 17T^{2} \)
19 \( 1 + 1.87iT - 19T^{2} \)
23 \( 1 - 6.40T + 23T^{2} \)
29 \( 1 + 4.57iT - 29T^{2} \)
31 \( 1 - 2.83T + 31T^{2} \)
37 \( 1 + 10.4iT - 37T^{2} \)
41 \( 1 - 6.46T + 41T^{2} \)
43 \( 1 - 9.49iT - 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 - 5.91iT - 53T^{2} \)
59 \( 1 - 7.32iT - 59T^{2} \)
61 \( 1 - 6.56iT - 61T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 + 3.22T + 71T^{2} \)
73 \( 1 + 1.22T + 73T^{2} \)
79 \( 1 + 4.79T + 79T^{2} \)
83 \( 1 - 0.872iT - 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 - 8.62T + 97T^{2} \)
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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

−11.54492458953745271208469508508, −10.63499162550087629988833287393, −9.747173311962816169316014439532, −8.773293545215406753235847518216, −7.77031763357276931299975290412, −7.15691088177453427451072422058, −5.92719035740584533784102393789, −4.63818999640788757056671684473, −2.53599450831846821537135210145, −1.02803012809067791134411119992, 1.73573752159366410820734580130, 3.36722436755373923514256879143, 4.69858227106110903328623676746, 6.48010298517004411873123976028, 7.11689082610012543970936921198, 8.428390484386517988422020439849, 9.217466177989252922475651289575, 10.08262451903495166872454502418, 11.05650981810484032369057825824, 11.49453709756554258091226638503

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