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Properties

Label 1-4725-4725.1472-r0-0-0
Degree $1$
Conductor $4725$
Sign $0.888 + 0.459i$
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  + (−0.970 + 0.241i)2-s + (0.882 − 0.469i)4-s + (−0.743 + 0.669i)8-s + (0.241 + 0.970i)11-s + (−0.275 − 0.961i)13-s + (0.559 − 0.829i)16-s + (−0.951 − 0.309i)17-s + (−0.309 + 0.951i)19-s + (−0.469 − 0.882i)22-s + (−0.829 + 0.559i)23-s + (0.5 + 0.866i)26-s + (−0.615 − 0.788i)29-s + (0.990 + 0.139i)31-s + (−0.342 + 0.939i)32-s + (0.997 + 0.0697i)34-s + ⋯
L(s)  = 1  + (−0.970 + 0.241i)2-s + (0.882 − 0.469i)4-s + (−0.743 + 0.669i)8-s + (0.241 + 0.970i)11-s + (−0.275 − 0.961i)13-s + (0.559 − 0.829i)16-s + (−0.951 − 0.309i)17-s + (−0.309 + 0.951i)19-s + (−0.469 − 0.882i)22-s + (−0.829 + 0.559i)23-s + (0.5 + 0.866i)26-s + (−0.615 − 0.788i)29-s + (0.990 + 0.139i)31-s + (−0.342 + 0.939i)32-s + (0.997 + 0.0697i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4725\)    =    \(3^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.888 + 0.459i$
Analytic conductor: \(21.9428\)
Root analytic conductor: \(21.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4725} (1472, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4725,\ (0:\ ),\ 0.888 + 0.459i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7934446365 + 0.1931603020i\)
\(L(\frac12)\) \(\approx\) \(0.7934446365 + 0.1931603020i\)
\(L(1)\) \(\approx\) \(0.6432919012 + 0.07380218085i\)
\(L(1)\) \(\approx\) \(0.6432919012 + 0.07380218085i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.970 + 0.241i)T \)
11 \( 1 + (0.241 + 0.970i)T \)
13 \( 1 + (-0.275 - 0.961i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (-0.829 + 0.559i)T \)
29 \( 1 + (-0.615 - 0.788i)T \)
31 \( 1 + (0.990 + 0.139i)T \)
37 \( 1 + (0.994 - 0.104i)T \)
41 \( 1 + (-0.961 + 0.275i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (-0.139 - 0.990i)T \)
53 \( 1 + (0.207 - 0.978i)T \)
59 \( 1 + (0.961 - 0.275i)T \)
61 \( 1 + (-0.719 + 0.694i)T \)
67 \( 1 + (0.927 - 0.374i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (0.994 + 0.104i)T \)
79 \( 1 + (0.374 - 0.927i)T \)
83 \( 1 + (-0.999 + 0.0348i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.139 + 0.990i)T \)
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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

−18.31385113684513671088198676690, −17.36192093005989042417366823912, −16.88142606741948677096771817265, −16.29974733386971362849548932758, −15.60644461088216773472869121018, −14.94889885970098959650389334595, −14.05715481034881000366575354318, −13.36386947779424060568853341145, −12.58219760473336097932583621038, −11.77482860512132458005816853025, −11.23820038101040590475819192178, −10.73744484803477409184065569926, −9.83894297148493803643136314509, −9.201224118737471908135274838891, −8.58046034138690102188159304490, −8.08314599677261758525802903532, −7.0192505220322197718788500386, −6.56244938178453652915471236529, −5.91045893967127507911765220589, −4.68386145160038197576910661663, −3.97694263884510514986478329483, −3.03419482883587730023676670468, −2.30877231132245300646568015277, −1.54780163008539082807413649724, −0.50597558756228649699287689364, 0.56960711093672805376224824924, 1.76966584716772809010781439547, 2.21894426132919028874740782020, 3.23398241267026362814095247113, 4.20617879229047563527069047844, 5.12678378825673824186692338855, 5.881597936073458671295427621884, 6.62981853587936769688740596143, 7.26661723886349268499173494087, 8.06359067695959257260110824360, 8.45315799812676490140117587831, 9.52885451428029021805149965662, 9.95981817023400485577235932881, 10.44706881235315116453079185443, 11.524907345091875163681266473124, 11.85113815670442606460728865444, 12.76609486775917858386224406474, 13.49477460924597302466826724806, 14.471474944160502993946070108155, 15.13378533728209862761492001204, 15.474049676326927421108677247259, 16.3105803187904830774082303872, 17.04833973313164894854780669402, 17.54070372705681978892979008086, 18.15869247887237168753719913969

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