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Properties

Label 2-280-8.5-c1-0-16
Degree $2$
Conductor $280$
Sign $0.999 + 0.0380i$
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  + (1.21 + 0.722i)2-s − 1.52i·3-s + (0.955 + 1.75i)4-s − i·5-s + (1.10 − 1.85i)6-s + 7-s + (−0.107 + 2.82i)8-s + 0.666·9-s + (0.722 − 1.21i)10-s − 1.22i·11-s + (2.68 − 1.45i)12-s − 1.21i·13-s + (1.21 + 0.722i)14-s − 1.52·15-s + (−2.17 + 3.35i)16-s + 2.59·17-s + ⋯
L(s)  = 1  + (0.859 + 0.510i)2-s − 0.881i·3-s + (0.477 + 0.878i)4-s − 0.447i·5-s + (0.450 − 0.758i)6-s + 0.377·7-s + (−0.0380 + 0.999i)8-s + 0.222·9-s + (0.228 − 0.384i)10-s − 0.368i·11-s + (0.774 − 0.421i)12-s − 0.337i·13-s + (0.324 + 0.193i)14-s − 0.394·15-s + (−0.543 + 0.839i)16-s + 0.629·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0380i$
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: \((2,\ 280,\ (\ :1/2),\ 0.999 + 0.0380i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12274 - 0.0404014i\)
\(L(\frac12)\) \(\approx\) \(2.12274 - 0.0404014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.21 - 0.722i)T \)
5 \( 1 + iT \)
7 \( 1 - T \)
good3 \( 1 + 1.52iT - 3T^{2} \)
11 \( 1 + 1.22iT - 11T^{2} \)
13 \( 1 + 1.21iT - 13T^{2} \)
17 \( 1 - 2.59T + 17T^{2} \)
19 \( 1 - 0.616iT - 19T^{2} \)
23 \( 1 + 8.36T + 23T^{2} \)
29 \( 1 - 9.00iT - 29T^{2} \)
31 \( 1 + 1.50T + 31T^{2} \)
37 \( 1 - 4.36iT - 37T^{2} \)
41 \( 1 + 4.58T + 41T^{2} \)
43 \( 1 - 0.301iT - 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + 4.62iT - 53T^{2} \)
59 \( 1 - 5.34iT - 59T^{2} \)
61 \( 1 - 3.54iT - 61T^{2} \)
67 \( 1 + 9.16iT - 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 + 6.69T + 79T^{2} \)
83 \( 1 + 9.02iT - 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 - 2.91T + 97T^{2} \)
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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

−12.25222524816111598207514623981, −11.33677360563325618798552022040, −10.03131997807006675123451894124, −8.445225301144100967303499267887, −7.86771042679125532723234462904, −6.86543786503268419461567399077, −5.86450425392030608901272921038, −4.82783258522631529943790073337, −3.48413275480058388266510242464, −1.76527437070451126860463029881, 2.05845054307306030927924716250, 3.64435334303989113901043885726, 4.42390161858218829740228707514, 5.52833797672393727625807435816, 6.68270395836039406260981088938, 7.919825977853398888727813099518, 9.629822958477217674697598112823, 10.01227350145641101909526486787, 11.00341100630354283701858984939, 11.77864511907210783635018164001

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