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Properties

Label 1-280-280.19-r0-0-0
Degree $1$
Conductor $280$
Sign $-0.991 + 0.126i$
Analytic cond. $1.30031$
Root an. cond. $1.30031$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s − 13-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + 27-s − 29-s + (−0.5 + 0.866i)31-s + (−0.5 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (0.5 − 0.866i)39-s − 41-s − 43-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s − 13-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + 27-s − 29-s + (−0.5 + 0.866i)31-s + (−0.5 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (0.5 − 0.866i)39-s − 41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(1.30031\)
Root analytic conductor: \(1.30031\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 280,\ (0:\ ),\ -0.991 + 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02734143746 + 0.4308439628i\)
\(L(\frac12)\) \(\approx\) \(0.02734143746 + 0.4308439628i\)
\(L(1)\) \(\approx\) \(0.6005490047 + 0.2879223795i\)
\(L(1)\) \(\approx\) \(0.6005490047 + 0.2879223795i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 - T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.95220586574291119573153177761, −24.15560941324810278561853972480, −23.66815475019695993312426013937, −22.32055308242956676993725939230, −21.95553819949793063919407992635, −20.48281784783358167940769573234, −19.58778135896479816379358447314, −18.69904894206120884807448819676, −17.93183851591031688061060460105, −17.00563494612992508762146772333, −16.14490484380503414511278212768, −14.98537085702623614543918150481, −13.61345046177573162241523761198, −13.26674341009373400051897481476, −11.85466096038088334014824881650, −11.38410834451820804379083071119, −10.12256002272518009530436469762, −8.87332844667524518499387614028, −7.671735592860267238691160002313, −6.96748973520194094194199894272, −5.70207934476828951523735082517, −4.90885491510208152790452628037, −3.11397996458246582108901421355, −1.96083855720034756288596843967, −0.28783213264829737455382430715, 2.01600502134313340689949630364, 3.52255179963235700825301063107, 4.61389929370599544823926232276, 5.47306947368288332870084522485, 6.677549554617123373892560139599, 7.899872367143083342789814456144, 9.18907506549123357425635233317, 10.10243863464653323228885969478, 10.78045633080932712753341832743, 12.06974055172984291987690053157, 12.70474632124955428413728700809, 14.30252688412161814919074008976, 15.026046568259624625627086904268, 15.91968162065586808926270940426, 16.882663128625981374460215806732, 17.62382906547421646181317470022, 18.61236051610488279974444480915, 19.99702273885202665107285847128, 20.58156392438757677127266567738, 21.68988567220235213247078505759, 22.33823515593768834254913223972, 23.20689819537814008951392267534, 24.11571587092547312697525241221, 25.227177814593657975890558062173, 26.33247197182337890313308444234

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