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Properties

Label 1-55-55.48-r1-0-0
Degree $1$
Conductor $55$
Sign $0.401 - 0.916i$
Analytic cond. $5.91057$
Root an. cond. $5.91057$
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  + (0.951 − 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (0.309 − 0.951i)6-s + (0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s − i·12-s + (−0.951 + 0.309i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (−0.587 − 0.809i)18-s + (0.809 + 0.587i)19-s + 21-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (0.309 − 0.951i)6-s + (0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s − i·12-s + (−0.951 + 0.309i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (−0.587 − 0.809i)18-s + (0.809 + 0.587i)19-s + 21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.401 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.401 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.401 - 0.916i$
Analytic conductor: \(5.91057\)
Root analytic conductor: \(5.91057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 55,\ (1:\ ),\ 0.401 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.707974118 - 1.770541307i\)
\(L(\frac12)\) \(\approx\) \(2.707974118 - 1.770541307i\)
\(L(1)\) \(\approx\) \(2.010509378 - 0.8740521592i\)
\(L(1)\) \(\approx\) \(2.010509378 - 0.8740521592i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + iT \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (0.951 + 0.309i)T \)
89 \( 1 - T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−32.98856084270051751682974877560, −32.06824360155896204167259720299, −30.9694373845960468966597706857, −30.151033722391601601814070160676, −28.68536795627635557856742923621, −26.98395677187280388269892873256, −26.33128059022188721253319186672, −24.93362129115886227139948266660, −24.007808607575663651442583849727, −22.54543636843692352261924722193, −21.69579458720232171002468576725, −20.460024069548654974207827477952, −19.826346326391019866049351022005, −17.46038155300414156323913067588, −16.33803739145435748865479265172, −15.11917739858645865246968138043, −14.25604542599703243496958690568, −13.169825139446901921591573677410, −11.42750582819539110062634746170, −10.23553166496727196273848911659, −8.373555550569358350458605813068, −7.09660451208739003267831874638, −5.08425539168836284400349572046, −4.12435399560818841842938896144, −2.545932049011275066614979518405, 1.723758198772869418505426868613, 2.97992395557419224278476627286, 4.90456119209350543290966291353, 6.44015500418437437421037453011, 7.82916659086975939655955205908, 9.51798462482897505199703962805, 11.52983950795412698090652758470, 12.31677251168010655885873954675, 13.64830915552557132050017113315, 14.58529741217483543797274575657, 15.666798026076328082553958688348, 17.70430554696065935721439631452, 18.99680973227963566327431523880, 19.98727178602226004683273893887, 21.14195925722907328762159607574, 22.26284032013705932237979761198, 23.67553876730457443048068877961, 24.58933108698170895767017536496, 25.26563782786901351305311302693, 26.968278032074018810557586885171, 28.66486936651011508887660184115, 29.43954117173196649041801666039, 30.7557184708851293766202306484, 31.27319866070328329500585726727, 32.24239574939017536368366603291

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