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Properties

Label 1-2280-2280.1283-r0-0-0
Degree $1$
Conductor $2280$
Sign $-0.505 - 0.862i$
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  + (0.866 − 0.5i)7-s + (0.5 − 0.866i)11-s + (−0.984 + 0.173i)13-s + (−0.342 − 0.939i)17-s + (0.642 + 0.766i)23-s + (−0.939 − 0.342i)29-s + (−0.5 − 0.866i)31-s − i·37-s + (0.173 − 0.984i)41-s + (−0.642 + 0.766i)43-s + (−0.342 + 0.939i)47-s + (0.5 − 0.866i)49-s + (−0.642 − 0.766i)53-s + (0.939 − 0.342i)59-s + (−0.766 + 0.642i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)7-s + (0.5 − 0.866i)11-s + (−0.984 + 0.173i)13-s + (−0.342 − 0.939i)17-s + (0.642 + 0.766i)23-s + (−0.939 − 0.342i)29-s + (−0.5 − 0.866i)31-s − i·37-s + (0.173 − 0.984i)41-s + (−0.642 + 0.766i)43-s + (−0.342 + 0.939i)47-s + (0.5 − 0.866i)49-s + (−0.642 − 0.766i)53-s + (0.939 − 0.342i)59-s + (−0.766 + 0.642i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.505 - 0.862i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ -0.505 - 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6085409368 - 1.062359132i\)
\(L(\frac12)\) \(\approx\) \(0.6085409368 - 1.062359132i\)
\(L(1)\) \(\approx\) \(0.9996016536 - 0.2678134161i\)
\(L(1)\) \(\approx\) \(0.9996016536 - 0.2678134161i\)

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 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (-0.342 + 0.939i)T \)
53 \( 1 + (-0.642 - 0.766i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (0.984 + 0.173i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (-0.342 - 0.939i)T \)
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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

−20.12129634431800637874596239811, −19.12075936489803933990700541838, −18.45660888724872009439235326231, −17.634401983445666263114329257680, −17.17751352781078995358913723991, −16.45127349094051552293910714868, −15.2526642397989811081292430672, −14.880016679950520338480369350005, −14.42758168961831311279830138466, −13.28663027275983387474935232187, −12.51401133135201751628618670260, −12.00325435323458303431625510624, −11.15632582201872491970343031811, −10.40911563012694585369017052236, −9.59405597720366532045218925380, −8.78607677697392179483795793716, −8.13327108093832160459468078844, −7.217477807387054415921946144622, −6.5897489357626611326168658433, −5.47440596599808024864871736003, −4.83759308772780374094256397905, −4.13279874721176592024885582468, −2.98240058522745322533985369310, −2.038072962401010987822332434325, −1.39918348770257097753918917672, 0.38859536321133494323110661423, 1.50675946544778245029079356532, 2.40499084813861077725126875562, 3.44941508203396483009194495016, 4.2764284716282449527621948072, 5.086009927407195146803719704940, 5.78957487796922679319937913810, 6.930736934960445264893566171960, 7.47402955733447711512067737933, 8.236545964225876595717156048704, 9.24632612195013117600575585276, 9.66755316576659496319354794658, 10.95457898488191451972235580736, 11.246750472237189076235915716300, 11.9833577865314434069349632585, 12.99893398735302718226940278224, 13.717059512295719614173265351044, 14.37992767362364632538144609234, 14.92732470802135870727041253602, 15.878022638337015239872188849546, 16.70971664019944984753447592562, 17.197186360858452121986094937820, 17.917081514929106952724124685610, 18.723354839593343433380348814493, 19.484120947114017190337465557102

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