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Properties

Label 1-4235-4235.927-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.605 + 0.795i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.263 + 0.964i)2-s + (−0.406 + 0.913i)3-s + (−0.861 − 0.508i)4-s + (−0.774 − 0.633i)6-s + (0.717 − 0.696i)8-s + (−0.669 − 0.743i)9-s + (0.814 − 0.580i)12-s + (0.676 − 0.736i)13-s + (0.483 + 0.875i)16-s + (0.524 + 0.851i)17-s + (0.893 − 0.449i)18-s + (0.380 + 0.924i)19-s + (−0.189 − 0.981i)23-s + (0.345 + 0.938i)24-s + (0.532 + 0.846i)26-s + (0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.263 + 0.964i)2-s + (−0.406 + 0.913i)3-s + (−0.861 − 0.508i)4-s + (−0.774 − 0.633i)6-s + (0.717 − 0.696i)8-s + (−0.669 − 0.743i)9-s + (0.814 − 0.580i)12-s + (0.676 − 0.736i)13-s + (0.483 + 0.875i)16-s + (0.524 + 0.851i)17-s + (0.893 − 0.449i)18-s + (0.380 + 0.924i)19-s + (−0.189 − 0.981i)23-s + (0.345 + 0.938i)24-s + (0.532 + 0.846i)26-s + (0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (927, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4576278772 + 0.9237830129i\)
\(L(\frac12)\) \(\approx\) \(0.4576278772 + 0.9237830129i\)
\(L(1)\) \(\approx\) \(0.5891129204 + 0.4857460691i\)
\(L(1)\) \(\approx\) \(0.5891129204 + 0.4857460691i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.263 + 0.964i)T \)
3 \( 1 + (-0.406 + 0.913i)T \)
13 \( 1 + (0.676 - 0.736i)T \)
17 \( 1 + (0.524 + 0.851i)T \)
19 \( 1 + (0.380 + 0.924i)T \)
23 \( 1 + (-0.189 - 0.981i)T \)
29 \( 1 + (-0.941 - 0.336i)T \)
31 \( 1 + (0.595 + 0.803i)T \)
37 \( 1 + (-0.662 + 0.749i)T \)
41 \( 1 + (0.362 - 0.931i)T \)
43 \( 1 + (-0.989 + 0.142i)T \)
47 \( 1 + (-0.893 - 0.449i)T \)
53 \( 1 + (-0.875 - 0.483i)T \)
59 \( 1 + (-0.625 + 0.780i)T \)
61 \( 1 + (0.710 + 0.703i)T \)
67 \( 1 + (0.458 - 0.888i)T \)
71 \( 1 + (-0.564 - 0.825i)T \)
73 \( 1 + (0.572 + 0.820i)T \)
79 \( 1 + (0.830 - 0.556i)T \)
83 \( 1 + (0.389 - 0.921i)T \)
89 \( 1 + (0.723 - 0.690i)T \)
97 \( 1 + (-0.170 + 0.985i)T \)
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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

−18.10979232926734560868046507902, −17.81837129610772675789901608652, −16.93871527576220327402572723813, −16.42358245407545162896790893189, −15.54954051448732768570328877997, −14.29897342135985633505720961061, −13.85688367434016240589980127018, −13.219707945681414683138548822646, −12.67006804069453116537699285636, −11.77438166122260462627053546836, −11.37781265890450866269638467961, −10.96256324880378584173540610589, −9.78125632994975737919790680639, −9.32755808369082229059757976010, −8.46414148589361104231568412894, −7.73107791032831003177460303558, −7.13430920338488397045548115569, −6.24516489992212092618885471313, −5.3244252418503789706405596500, −4.73540950107423310217797516587, −3.64472205636676246365561648183, −2.95213598862166942644890862069, −2.03379761260820435482312441261, −1.405726427963690960244840085646, −0.52960175596789607677610162350, 0.691477354183313134819907284778, 1.725609709813523538108232433264, 3.345328871454983372757932632024, 3.709967158392619823889010721077, 4.71803689965024790987971163868, 5.298315568468264642407047144677, 6.07904898916580772284641033590, 6.43736109174612974303769036620, 7.6003852573929418624194695938, 8.316515817547309161170651252787, 8.77878370158236456235042809023, 9.72738506191710759218650086783, 10.32965316773160556509648545357, 10.66858335466135804780542226896, 11.76802280264909262834907234362, 12.52266563550288105871919595650, 13.31846398491090823712163666740, 14.19362393964061472105293615362, 14.745431761996675063663539145040, 15.33355009647708143257195331787, 15.97225568258836515092453422115, 16.54749431692449207692987581986, 17.08044320852193380474967860739, 17.76066434840423247264439769655, 18.39857981331363118646110416774

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