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Properties

Label 1-5328-5328.5-r0-0-0
Degree $1$
Conductor $5328$
Sign $-0.709 + 0.704i$
Analytic cond. $24.7431$
Root an. cond. $24.7431$
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  + (0.766 + 0.642i)5-s + (−0.173 + 0.984i)7-s + i·11-s + (−0.939 + 0.342i)13-s + (0.984 − 0.173i)17-s + (0.766 + 0.642i)19-s − i·23-s + (0.173 + 0.984i)25-s + 29-s + (−0.866 + 0.5i)31-s + (−0.766 + 0.642i)35-s + (0.939 − 0.342i)41-s + (0.5 − 0.866i)43-s + (0.5 + 0.866i)47-s + (−0.939 − 0.342i)49-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)5-s + (−0.173 + 0.984i)7-s + i·11-s + (−0.939 + 0.342i)13-s + (0.984 − 0.173i)17-s + (0.766 + 0.642i)19-s − i·23-s + (0.173 + 0.984i)25-s + 29-s + (−0.866 + 0.5i)31-s + (−0.766 + 0.642i)35-s + (0.939 − 0.342i)41-s + (0.5 − 0.866i)43-s + (0.5 + 0.866i)47-s + (−0.939 − 0.342i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5328\)    =    \(2^{4} \cdot 3^{2} \cdot 37\)
Sign: $-0.709 + 0.704i$
Analytic conductor: \(24.7431\)
Root analytic conductor: \(24.7431\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5328} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5328,\ (0:\ ),\ -0.709 + 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7848922103 + 1.903721157i\)
\(L(\frac12)\) \(\approx\) \(0.7848922103 + 1.903721157i\)
\(L(1)\) \(\approx\) \(1.101850880 + 0.5592232556i\)
\(L(1)\) \(\approx\) \(1.101850880 + 0.5592232556i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( 1 \)
good5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + iT \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 - iT \)
29 \( 1 + T \)
31 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.342 + 0.939i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.642 + 0.766i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + T \)
79 \( 1 + (0.984 + 0.173i)T \)
83 \( 1 + (0.342 - 0.939i)T \)
89 \( 1 + (-0.342 + 0.939i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−17.630229251940669610343270435645, −16.85971604024180974266121848671, −16.54108085338664486398620576321, −15.98551328081708970650439836319, −14.89119227377621077167179973019, −14.13468383835924997598043298333, −13.854150072392163001636512573877, −12.98198364797896891093432746359, −12.56010957392299457050773688150, −11.7212487400694453013485889028, −10.82141450847400235475583313330, −10.2905309284792507844186163713, −9.60192148892237602762085089806, −9.06524946842038910395587002381, −8.0689581830114449611468546218, −7.64575777649506844586433079806, −6.64811297483958859232635107378, −6.05916692385106365613102672991, −5.18670095104492400205332670494, −4.73069381165126678403377839730, −3.733560544369604478308230057078, −2.98089054769718209787979225555, −2.18518439841217988920333561811, −1.00195509683496657833650659225, −0.6069441371874781908077301597, 1.30169119101846599306562400851, 2.035903547605654261239991558264, 2.7135671205089551640463838816, 3.35476314643642923033087186253, 4.43344266291511561027848813521, 5.431010681757588471219270884230, 5.62066295402214432851941565469, 6.6015266408650229346889457484, 7.34455455759814865304006567617, 7.781521022081650058818752858299, 9.054642374836364615338314432753, 9.47938990165722886457049025983, 9.98548758198037730405787069399, 10.69267101788391157351108893245, 11.64949574544528911737921834738, 12.35300671045307555877382189736, 12.55459930263418323967033652792, 13.7843845474547896850794184878, 14.21265020421371651643173384657, 14.8288575020935415546419523314, 15.44780390245080891044758984491, 16.15164477490413695472337440993, 16.984057432006440085735697067456, 17.65977380813379997009161018942, 18.099815658908203157857495070840

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