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Properties

Label 1-4725-4725.41-r0-0-0
Degree $1$
Conductor $4725$
Sign $0.830 - 0.557i$
Analytic cond. $21.9428$
Root an. cond. $21.9428$
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  + (0.615 + 0.788i)2-s + (−0.241 + 0.970i)4-s + (−0.913 + 0.406i)8-s + (−0.990 + 0.139i)11-s + (0.615 − 0.788i)13-s + (−0.882 − 0.469i)16-s + (−0.104 + 0.994i)17-s + (−0.913 + 0.406i)19-s + (−0.719 − 0.694i)22-s + (−0.848 − 0.529i)23-s + 26-s + (−0.559 + 0.829i)29-s + (0.997 − 0.0697i)31-s + (−0.173 − 0.984i)32-s + (−0.848 + 0.529i)34-s + ⋯
L(s)  = 1  + (0.615 + 0.788i)2-s + (−0.241 + 0.970i)4-s + (−0.913 + 0.406i)8-s + (−0.990 + 0.139i)11-s + (0.615 − 0.788i)13-s + (−0.882 − 0.469i)16-s + (−0.104 + 0.994i)17-s + (−0.913 + 0.406i)19-s + (−0.719 − 0.694i)22-s + (−0.848 − 0.529i)23-s + 26-s + (−0.559 + 0.829i)29-s + (0.997 − 0.0697i)31-s + (−0.173 − 0.984i)32-s + (−0.848 + 0.529i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4725\)    =    \(3^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.830 - 0.557i$
Analytic conductor: \(21.9428\)
Root analytic conductor: \(21.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4725} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4725,\ (0:\ ),\ 0.830 - 0.557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7632394860 - 0.2323749946i\)
\(L(\frac12)\) \(\approx\) \(0.7632394860 - 0.2323749946i\)
\(L(1)\) \(\approx\) \(0.9605562001 + 0.4829040055i\)
\(L(1)\) \(\approx\) \(0.9605562001 + 0.4829040055i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.615 + 0.788i)T \)
11 \( 1 + (-0.990 + 0.139i)T \)
13 \( 1 + (0.615 - 0.788i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (-0.848 - 0.529i)T \)
29 \( 1 + (-0.559 + 0.829i)T \)
31 \( 1 + (0.997 - 0.0697i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (-0.615 + 0.788i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.997 - 0.0697i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.990 + 0.139i)T \)
61 \( 1 + (0.374 - 0.927i)T \)
67 \( 1 + (0.559 + 0.829i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (0.978 + 0.207i)T \)
79 \( 1 + (0.559 - 0.829i)T \)
83 \( 1 + (0.961 + 0.275i)T \)
89 \( 1 + (-0.978 - 0.207i)T \)
97 \( 1 + (-0.438 - 0.898i)T \)
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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

−18.2617729216153651496008077708, −17.88444635198036074743589097810, −16.86901820033484677808658507513, −15.90381830454673201161538069771, −15.59888918474620822481500378391, −14.7744621284163847541584874452, −13.87669468463541893116190187950, −13.56476818479467995495808010273, −12.93790219269136886948330552871, −12.04562152665246086627617349947, −11.531873465074421130355712430270, −10.89681917456784925195137150713, −10.19301250850593942902282447330, −9.55138536236510413957769196291, −8.78753615691066618180114904480, −8.05315563811912538851653683999, −7.00913689883616149889452281040, −6.323107857733579507037228674219, −5.55052591508959436706655660064, −4.86491699690928757957594775440, −4.14111034800574847186091630433, −3.43986260620314526196322380148, −2.48126706555326682853519013925, −2.03016591520355573408189936638, −0.91809903851159779511493142091, 0.18487991564918646935556030286, 1.7487076361636605816492559015, 2.5981291590074648612830005088, 3.50773611409188499720939430840, 4.04679049330860645720274035803, 5.01788843533802427044722010297, 5.5435630330061028960772141671, 6.33522084366039986539562814423, 6.85904388991449008945334966293, 8.01981953884723348461216918570, 8.17335464346033958702146869859, 8.875500265017851048340809987338, 10.15245644419960752171233051136, 10.49284221700505331529577765056, 11.47000003882428621482385960367, 12.32222847251851791469816027240, 12.907952684577962414648696256324, 13.30684731808977603382895676806, 14.14233890820986261932364633491, 14.8995480821096518449784458884, 15.344528482660171872049918539942, 15.99979259820913425944441160034, 16.628439158959741982460598310860, 17.36509361818256506392379332372, 17.95903258134674824468902151233

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