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Properties

Label 1-4725-4725.1913-r0-0-0
Degree $1$
Conductor $4725$
Sign $0.999 + 0.00411i$
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  + (0.139 + 0.990i)2-s + (−0.961 + 0.275i)4-s + (−0.406 − 0.913i)8-s + (−0.990 + 0.139i)11-s + (0.927 − 0.374i)13-s + (0.848 − 0.529i)16-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.275 − 0.961i)22-s + (0.529 − 0.848i)23-s + (0.5 + 0.866i)26-s + (−0.997 − 0.0697i)29-s + (0.559 + 0.829i)31-s + (0.642 + 0.766i)32-s + (0.882 + 0.469i)34-s + ⋯
L(s)  = 1  + (0.139 + 0.990i)2-s + (−0.961 + 0.275i)4-s + (−0.406 − 0.913i)8-s + (−0.990 + 0.139i)11-s + (0.927 − 0.374i)13-s + (0.848 − 0.529i)16-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.275 − 0.961i)22-s + (0.529 − 0.848i)23-s + (0.5 + 0.866i)26-s + (−0.997 − 0.0697i)29-s + (0.559 + 0.829i)31-s + (0.642 + 0.766i)32-s + (0.882 + 0.469i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4725\)    =    \(3^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.999 + 0.00411i$
Analytic conductor: \(21.9428\)
Root analytic conductor: \(21.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4725} (1913, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4725,\ (0:\ ),\ 0.999 + 0.00411i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.384998299 + 0.002846773344i\)
\(L(\frac12)\) \(\approx\) \(1.384998299 + 0.002846773344i\)
\(L(1)\) \(\approx\) \(0.9434291476 + 0.3597104347i\)
\(L(1)\) \(\approx\) \(0.9434291476 + 0.3597104347i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.139 + 0.990i)T \)
11 \( 1 + (-0.990 + 0.139i)T \)
13 \( 1 + (0.927 - 0.374i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (0.529 - 0.848i)T \)
29 \( 1 + (-0.997 - 0.0697i)T \)
31 \( 1 + (0.559 + 0.829i)T \)
37 \( 1 + (0.743 - 0.669i)T \)
41 \( 1 + (0.374 + 0.927i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (0.829 + 0.559i)T \)
53 \( 1 + (-0.994 + 0.104i)T \)
59 \( 1 + (-0.374 - 0.927i)T \)
61 \( 1 + (-0.615 - 0.788i)T \)
67 \( 1 + (-0.898 - 0.438i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (0.743 + 0.669i)T \)
79 \( 1 + (-0.438 - 0.898i)T \)
83 \( 1 + (-0.970 + 0.241i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.829 - 0.559i)T \)
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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

−18.52323321561707767140462063559, −17.63418733219447573760308383018, −16.997166125046822210064440583400, −16.14838897674894843444215318478, −15.344264962355959848370645914107, −14.79463701996096894325337718098, −13.79952244993881943903085849943, −13.36353426572529393663774763094, −12.89605081149882037620164615542, −11.96961398172003616862570564000, −11.35329148485790268440051014928, −10.82740371252959962674225202196, −10.109538074039676917738461495516, −9.413558325747298963119546708904, −8.76606311170251357507818334244, −7.969689757403218457273886383814, −7.342341568234136208345319093359, −6.02338788263686710945789834277, −5.629201633356175222760119830934, −4.75599235221658449964722120837, −3.97937914123986686901855089007, −3.23853654053608085224708857236, −2.61641326809758088913811583175, −1.60626179838561079938082594738, −0.94493848563366401848499671101, 0.43653659081720428033271582478, 1.42915397945199833503732384128, 2.84853770667309870576552940785, 3.33566257603267526045808462147, 4.348272477738383471755073975130, 5.06910961205135628898675410459, 5.682093131676119669425976484306, 6.30701894806171074903271455296, 7.27456318509861903868741409608, 7.74805227361926178600183681924, 8.38163018419950028907263218779, 9.191099698123834440162117725344, 9.84309634653822962339478093410, 10.60597309428579816728948068832, 11.379039588870662929921377909915, 12.48134383101400248832744760121, 12.77107930602102407415635169299, 13.73431985424291180012866672016, 14.04925226053952190500468613472, 14.950151360960346185180592463696, 15.54266159503788050781635828017, 16.18670308062515157625140979201, 16.56637493791031794532582039783, 17.47169753769102997338034707935, 18.257095281443276545658914475646

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