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Properties

Label 1-2280-2280.899-r0-0-0
Degree $1$
Conductor $2280$
Sign $0.158 - 0.987i$
Analytic cond. $10.5882$
Root an. cond. $10.5882$
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)7-s + (0.5 − 0.866i)11-s + (0.766 − 0.642i)13-s + (0.173 − 0.984i)17-s + (0.939 + 0.342i)23-s + (0.173 + 0.984i)29-s + (0.5 + 0.866i)31-s + 37-s + (−0.766 − 0.642i)41-s + (0.939 − 0.342i)43-s + (−0.173 − 0.984i)47-s + (−0.5 + 0.866i)49-s + (0.939 + 0.342i)53-s + (−0.173 + 0.984i)59-s + (0.939 + 0.342i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (0.766 − 0.642i)13-s + (0.173 − 0.984i)17-s + (0.939 + 0.342i)23-s + (0.173 + 0.984i)29-s + (0.5 + 0.866i)31-s + 37-s + (−0.766 − 0.642i)41-s + (0.939 − 0.342i)43-s + (−0.173 − 0.984i)47-s + (−0.5 + 0.866i)49-s + (0.939 + 0.342i)53-s + (−0.173 + 0.984i)59-s + (0.939 + 0.342i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.158 - 0.987i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ 0.158 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.273570711 - 1.085201252i\)
\(L(\frac12)\) \(\approx\) \(1.273570711 - 1.085201252i\)
\(L(1)\) \(\approx\) \(1.087279384 - 0.3008889187i\)
\(L(1)\) \(\approx\) \(1.087279384 - 0.3008889187i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−19.65682379894682275310557357233, −19.06609627133459484724177572952, −18.5399570900175094378002263811, −17.62690392498317129387167923184, −16.97698794554314430646615699349, −16.218278600804895288722015686306, −15.397168638499212583169796655582, −14.910722547740039678461411843484, −14.1386731884432470977603987790, −13.01683981330913056979805722783, −12.75881627331222875088643918730, −11.70230457614992749468508499337, −11.29266785360374319543621350930, −10.09967424090673336309682502349, −9.561730310578858754656965196523, −8.77440358186099735786114845565, −8.11618256022289363795176705357, −7.03742490980354159023354200856, −6.291186720077633297458619324024, −5.775788578830183767120085506963, −4.56940434214666930140909842195, −3.97771718432949286257229469431, −2.899394326576414713703650068519, −2.09310798637576828348981369218, −1.13518133163864008981180120100, 0.66407921275869859963499489771, 1.31759342545340911873759968068, 2.89429774911049383847054309242, 3.359063456185865524815368606879, 4.23571776213092317148655365255, 5.25214386161205505837986564500, 6.01875324832110471692034705546, 6.91752500424693760246284665176, 7.4371570979576986346865128523, 8.57094354286179743068643888947, 9.050800026369777402009408089236, 10.089688152489198765800282265124, 10.68615379337028530087973270873, 11.39004309677171700982435222173, 12.211344588017818502381164246771, 13.202789657384645035982442001744, 13.61119777105462242371770760526, 14.30777782046002206679598693114, 15.22268211833020405987062838602, 16.13344015998919668506745074635, 16.47147061564663894452622028286, 17.31308264665636952953175858110, 18.066967399982171548334493668284, 18.832489013664987542949724871727, 19.535093464064624823146271485673

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