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Properties

Label 1-4560-4560.827-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.986 - 0.164i$
Analytic cond. $21.1765$
Root an. cond. $21.1765$
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 + (−0.866 − 0.5i)11-s + (0.173 + 0.984i)13-s + (0.342 + 0.939i)17-s + (0.642 + 0.766i)23-s + (0.342 − 0.939i)29-s + (−0.5 − 0.866i)31-s + 37-s + (−0.173 + 0.984i)41-s + (−0.766 − 0.642i)43-s + (0.342 − 0.939i)47-s + (0.5 − 0.866i)49-s + (0.766 − 0.642i)53-s + (−0.342 − 0.939i)59-s + (−0.642 − 0.766i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)7-s + (−0.866 − 0.5i)11-s + (0.173 + 0.984i)13-s + (0.342 + 0.939i)17-s + (0.642 + 0.766i)23-s + (0.342 − 0.939i)29-s + (−0.5 − 0.866i)31-s + 37-s + (−0.173 + 0.984i)41-s + (−0.766 − 0.642i)43-s + (0.342 − 0.939i)47-s + (0.5 − 0.866i)49-s + (0.766 − 0.642i)53-s + (−0.342 − 0.939i)59-s + (−0.642 − 0.766i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.986 - 0.164i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.986 - 0.164i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.913746721 - 0.1582656208i\)
\(L(\frac12)\) \(\approx\) \(1.913746721 - 0.1582656208i\)
\(L(1)\) \(\approx\) \(1.176395439 - 0.03853247618i\)
\(L(1)\) \(\approx\) \(1.176395439 - 0.03853247618i\)

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

−18.2705156960278001617630952808, −17.79994613733198219168782288538, −16.92896101169301329800462473657, −16.15210840007064180527960814109, −15.52517433849618677711685171965, −14.883971551026168838770576922536, −14.35792636606959534659721575375, −13.49832325204859476980800151405, −12.763425791929099282994719851325, −12.22324396776441655546825751818, −11.48534808937280956254990053855, −10.548282978673770402809517674310, −10.39304526316839086804606130814, −9.14477125364661396155131523669, −8.7356717923753727606218146047, −7.705220991892395593296797484826, −7.50242523007361369688917550696, −6.40785746334249663059080585620, −5.46469437867318198124981885164, −5.04655000704280030580900758379, −4.38123406195226593081252983227, −3.06874928616054912511944275241, −2.70569422079913495668498779274, −1.68127349816006437599479230861, −0.765222594066027754799072321809, 0.72072380290186515779254589574, 1.67674385784301875047178852027, 2.35090626126126549017015504361, 3.47701032604249313660819088155, 4.089720166276908298296968107096, 4.91674838718847204110596481489, 5.58697610515381878841352950644, 6.408655109969354420946936260581, 7.21414028486383830848454516649, 8.05333007178207152128264580686, 8.32105375471556987737739883274, 9.38806713847742252282788914661, 10.02821380702018001135947891609, 10.926793901471942075553029042878, 11.268324040541870619747028101526, 12.00026370635209989645938195131, 12.966192774857940191763913978471, 13.568996302775885952329962375735, 14.04120585753292649458319413406, 15.04790304403307946082757310369, 15.26404624845193422006193397543, 16.45711092836713737419868687651, 16.78419456292345625809503377594, 17.47633865113940413712553968360, 18.31912833693010529842760245739

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