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Properties

Label 1-4725-4725.3296-r0-0-0
Degree $1$
Conductor $4725$
Sign $0.223 + 0.974i$
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  + (−0.559 − 0.829i)2-s + (−0.374 + 0.927i)4-s + (0.978 − 0.207i)8-s + (−0.438 + 0.898i)11-s + (−0.559 + 0.829i)13-s + (−0.719 − 0.694i)16-s + (0.669 + 0.743i)17-s + (0.978 − 0.207i)19-s + (0.990 − 0.139i)22-s + (0.241 + 0.970i)23-s + 26-s + (−0.0348 + 0.999i)29-s + (−0.848 + 0.529i)31-s + (−0.173 + 0.984i)32-s + (0.241 − 0.970i)34-s + ⋯
L(s)  = 1  + (−0.559 − 0.829i)2-s + (−0.374 + 0.927i)4-s + (0.978 − 0.207i)8-s + (−0.438 + 0.898i)11-s + (−0.559 + 0.829i)13-s + (−0.719 − 0.694i)16-s + (0.669 + 0.743i)17-s + (0.978 − 0.207i)19-s + (0.990 − 0.139i)22-s + (0.241 + 0.970i)23-s + 26-s + (−0.0348 + 0.999i)29-s + (−0.848 + 0.529i)31-s + (−0.173 + 0.984i)32-s + (0.241 − 0.970i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4725\)    =    \(3^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.223 + 0.974i$
Analytic conductor: \(21.9428\)
Root analytic conductor: \(21.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4725} (3296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4725,\ (0:\ ),\ 0.223 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7733537728 + 0.6158882983i\)
\(L(\frac12)\) \(\approx\) \(0.7733537728 + 0.6158882983i\)
\(L(1)\) \(\approx\) \(0.7677508503 + 0.03044578813i\)
\(L(1)\) \(\approx\) \(0.7677508503 + 0.03044578813i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.559 - 0.829i)T \)
11 \( 1 + (-0.438 + 0.898i)T \)
13 \( 1 + (-0.559 + 0.829i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (0.241 + 0.970i)T \)
29 \( 1 + (-0.0348 + 0.999i)T \)
31 \( 1 + (-0.848 + 0.529i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (0.559 - 0.829i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.848 + 0.529i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.438 + 0.898i)T \)
61 \( 1 + (0.997 - 0.0697i)T \)
67 \( 1 + (0.0348 + 0.999i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (0.0348 - 0.999i)T \)
83 \( 1 + (-0.615 + 0.788i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (0.882 - 0.469i)T \)
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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

−18.089105192435588082597373067821, −17.1176418535573438263445120224, −16.81729577628386872442294753851, −15.95552728835007291219809289395, −15.56943651531351086143780442326, −14.7197137593343623293242167282, −14.13762531735426084988208386576, −13.51635373622159783450286421015, −12.786424593748308705642750502744, −11.848069756627005773661704467, −11.09368999818543022439838282291, −10.34873907418287897789939797080, −9.77394684623463593062720731513, −9.12228257298229729016939091477, −8.19483174752027495705778427794, −7.819054075175578820236002589733, −7.113845549919891621117033922937, −6.2602862966354448306316473108, −5.44996229225527013720709982259, −5.17166780317574405607966532299, −4.0895378432405462457479005671, −3.07649698148089213440675284330, −2.32273888392494784208032323977, −1.02819719626432382438342072230, −0.40572473404657428536231147295, 1.083300838682782641354036555022, 1.80478179222707721814093283402, 2.5293950973078678228096267810, 3.435688871260297611876622764619, 4.06797930451901199762418949862, 4.97979666938654625874220532833, 5.58709753624103624993753890872, 7.006125767306567958938441810698, 7.319118935985564163166440349880, 8.04198312578684612628302174197, 9.04004334757999470543076926222, 9.45775624932517193070407329539, 10.12012628318917137147593987250, 10.841944307643730930023395054462, 11.47572671006045310786274428593, 12.315307009713146942455852218477, 12.6106176369456107554750286080, 13.43513640917808258851026258005, 14.27164313866389985472725122224, 14.80284898810636605394055814009, 15.95792180968111239818152704701, 16.29366923348175282261843302608, 17.24968466859699231344026908742, 17.70438490686154468467849608643, 18.27585687420217733533712182155

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