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Properties

Label 1-4725-4725.1283-r0-0-0
Degree $1$
Conductor $4725$
Sign $-0.285 + 0.958i$
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  + (0.0697 − 0.997i)2-s + (−0.990 − 0.139i)4-s + (−0.207 + 0.978i)8-s + (0.997 + 0.0697i)11-s + (−0.829 + 0.559i)13-s + (0.961 + 0.275i)16-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.139 − 0.990i)22-s + (−0.275 − 0.961i)23-s + (0.5 + 0.866i)26-s + (0.0348 + 0.999i)29-s + (−0.882 + 0.469i)31-s + (0.342 − 0.939i)32-s + (0.241 + 0.970i)34-s + ⋯
L(s)  = 1  + (0.0697 − 0.997i)2-s + (−0.990 − 0.139i)4-s + (−0.207 + 0.978i)8-s + (0.997 + 0.0697i)11-s + (−0.829 + 0.559i)13-s + (0.961 + 0.275i)16-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.139 − 0.990i)22-s + (−0.275 − 0.961i)23-s + (0.5 + 0.866i)26-s + (0.0348 + 0.999i)29-s + (−0.882 + 0.469i)31-s + (0.342 − 0.939i)32-s + (0.241 + 0.970i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4725\)    =    \(3^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.285 + 0.958i$
Analytic conductor: \(21.9428\)
Root analytic conductor: \(21.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4725} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4725,\ (0:\ ),\ -0.285 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002986793473 + 0.004004498634i\)
\(L(\frac12)\) \(\approx\) \(0.002986793473 + 0.004004498634i\)
\(L(1)\) \(\approx\) \(0.7205388711 - 0.3737466536i\)
\(L(1)\) \(\approx\) \(0.7205388711 - 0.3737466536i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.0697 - 0.997i)T \)
11 \( 1 + (0.997 + 0.0697i)T \)
13 \( 1 + (-0.829 + 0.559i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.275 - 0.961i)T \)
29 \( 1 + (0.0348 + 0.999i)T \)
31 \( 1 + (-0.882 + 0.469i)T \)
37 \( 1 + (-0.406 - 0.913i)T \)
41 \( 1 + (-0.559 - 0.829i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (0.469 - 0.882i)T \)
53 \( 1 + (0.743 - 0.669i)T \)
59 \( 1 + (0.559 + 0.829i)T \)
61 \( 1 + (0.438 + 0.898i)T \)
67 \( 1 + (0.529 - 0.848i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (-0.406 + 0.913i)T \)
79 \( 1 + (-0.848 + 0.529i)T \)
83 \( 1 + (-0.788 + 0.615i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.469 + 0.882i)T \)
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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

−18.5367385251777829484902845469, −17.69719295395253607341994295330, −17.15613167072026154616049536593, −16.8138106534654095483038006746, −15.76911266753892268113263454347, −15.42949194396411398865919337526, −14.56310043978677609913534084384, −14.196816983820860248537137616793, −13.34772507807110921935347250128, −12.769515157897597356277909266666, −11.97226483362966719472613640489, −11.32911504178241466411611072721, −10.203334951759328602749789675530, −9.65115445346679075994017297406, −9.01962926580638287530299760633, −8.23749503802507009709120152463, −7.58046485198211755087417663908, −6.9379956599639240120642650926, −6.15445115517398672326174647745, −5.61902429962765447997947873201, −4.71461021493999027220596621943, −4.07059588173334837875349886110, −3.36940510385600489779307993531, −2.254007392665072774331084783566, −1.19921157024987906865524118725, 0.001447057384100672821429663818, 1.10928443160504102234507191655, 2.12542795581225180713777224612, 2.47544608301838489075343615360, 3.69953125444113192807044128999, 4.15563324400012089593850709233, 4.906284879889163020705353512829, 5.65570523224827071155487325078, 6.790816636409161646238463410901, 7.14538530224893206638999026908, 8.58259698597567548050966196664, 8.83036211217667894623359852133, 9.50706625640368524111781521912, 10.35964026305973716875179799230, 10.952904017796937467664236186280, 11.579655206174293706613572751302, 12.342620423223662174092483866133, 12.73286079134942542860156104772, 13.6097301843626880484233193824, 14.32564092397747621232053558050, 14.69741579298614595448075757257, 15.58414923378300113165845032410, 16.60293516563998534213454294793, 17.14110464065632361922562052119, 17.816799767752511897323875633799

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