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Properties

Label 1-572-572.87-r0-0-0
Degree $1$
Conductor $572$
Sign $-0.252 + 0.967i$
Analytic cond. $2.65635$
Root an. cond. $2.65635$
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 + 5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)15-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s − 21-s + (0.5 + 0.866i)23-s + 25-s − 27-s + (0.5 + 0.866i)29-s − 31-s + (−0.5 + 0.866i)35-s + (−0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + 5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)15-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s − 21-s + (0.5 + 0.866i)23-s + 25-s − 27-s + (0.5 + 0.866i)29-s − 31-s + (−0.5 + 0.866i)35-s + (−0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(2.65635\)
Root analytic conductor: \(2.65635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 572,\ (0:\ ),\ -0.252 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.061117673 + 1.373734077i\)
\(L(\frac12)\) \(\approx\) \(1.061117673 + 1.373734077i\)
\(L(1)\) \(\approx\) \(1.191424861 + 0.6225872005i\)
\(L(1)\) \(\approx\) \(1.191424861 + 0.6225872005i\)

Euler product

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

−23.12954964353583713896170627566, −22.22628481183279182694353668116, −21.1638940886707214333050550561, −20.498420824914846025023829879612, −19.57159013404321771498087230271, −18.94760197102924956463639693673, −17.9479484690640524660278443953, −17.219316388042794450971221164426, −16.602871194947939075621845537674, −15.144638580562276044792689511603, −14.366341440048524976305483823100, −13.49010337190877060811051600239, −13.067285235796634367761803901859, −12.186071571070204898960124468340, −10.81931866032155616969173134666, −10.047429874826435571841108425528, −9.06497049290469136402164148435, −8.2345308522674979923126442800, −7.00922621967928283396857646320, −6.535389154538270238235331300283, −5.487094538735731430971461457, −4.05707725562030737006685493804, −2.93331549666745171757703976010, −1.97570453228795473907838770557, −0.840787287744501398185532668618, 1.75722561212273099763800033813, 2.76855282332928285797798056200, 3.57446235388205609989830537487, 5.06226115617945545258843404672, 5.5738478335062641427238671622, 6.692652890150606936491792673023, 8.04389601748598959051026627712, 9.08421331595394916553408472590, 9.54527230917195298942092303925, 10.34820230645066588570371669790, 11.366893473450852751201245501088, 12.56610710905838671112813000974, 13.37355725942257080338156208673, 14.40285159216171616657558924970, 14.86440307901008737506784825644, 16.100172369362878881273951112314, 16.467807015100957930923435331469, 17.64677593701168463441844836624, 18.50858881660944792813954619898, 19.37336784267138519499033091381, 20.3125812301662295906172251722, 21.277006488920902754161253717, 21.55363996278073157326436129731, 22.4662401115403411887287449832, 23.23304776875233960725893490272

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