Properties

Label 1-4235-4235.1137-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.997 - 0.0770i$
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  + (−0.875 + 0.483i)2-s + (−0.994 − 0.104i)3-s + (0.532 − 0.846i)4-s + (0.921 − 0.389i)6-s + (−0.0570 + 0.998i)8-s + (0.978 + 0.207i)9-s + (−0.618 + 0.786i)12-s + (−0.170 − 0.985i)13-s + (−0.432 − 0.901i)16-s + (−0.803 + 0.595i)17-s + (−0.956 + 0.290i)18-s + (0.00951 − 0.999i)19-s + (−0.690 − 0.723i)23-s + (0.161 − 0.986i)24-s + (0.625 + 0.780i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.875 + 0.483i)2-s + (−0.994 − 0.104i)3-s + (0.532 − 0.846i)4-s + (0.921 − 0.389i)6-s + (−0.0570 + 0.998i)8-s + (0.978 + 0.207i)9-s + (−0.618 + 0.786i)12-s + (−0.170 − 0.985i)13-s + (−0.432 − 0.901i)16-s + (−0.803 + 0.595i)17-s + (−0.956 + 0.290i)18-s + (0.00951 − 0.999i)19-s + (−0.690 − 0.723i)23-s + (0.161 − 0.986i)24-s + (0.625 + 0.780i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.997 - 0.0770i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.997 - 0.0770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007569714972 - 0.1962525860i\)
\(L(\frac12)\) \(\approx\) \(0.007569714972 - 0.1962525860i\)
\(L(1)\) \(\approx\) \(0.4595108051 + 0.02704980691i\)
\(L(1)\) \(\approx\) \(0.4595108051 + 0.02704980691i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.875 - 0.483i)T \)
3 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (0.170 + 0.985i)T \)
17 \( 1 + (0.803 - 0.595i)T \)
19 \( 1 + (-0.00951 + 0.999i)T \)
23 \( 1 + (0.690 + 0.723i)T \)
29 \( 1 + (0.198 + 0.980i)T \)
31 \( 1 + (-0.830 - 0.556i)T \)
37 \( 1 + (0.244 + 0.969i)T \)
41 \( 1 + (0.0855 + 0.996i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + (-0.956 - 0.290i)T \)
53 \( 1 + (-0.901 - 0.432i)T \)
59 \( 1 + (0.905 - 0.424i)T \)
61 \( 1 + (-0.999 + 0.0190i)T \)
67 \( 1 + (0.945 + 0.327i)T \)
71 \( 1 + (0.736 + 0.676i)T \)
73 \( 1 + (-0.647 + 0.761i)T \)
79 \( 1 + (-0.710 - 0.703i)T \)
83 \( 1 + (0.999 + 0.0285i)T \)
89 \( 1 + (0.995 + 0.0950i)T \)
97 \( 1 + (-0.633 + 0.774i)T \)
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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

−18.597332896358659870788044236069, −18.05353564300444223799095248249, −17.41462980303697141298667939241, −16.72598979192935160680097670951, −16.284170139220314741007830156461, −15.672198557868666384629714331363, −14.843030472029492006334187451755, −13.72943200642657165149180954874, −13.08415877828095513229022090809, −12.20291063836139294953221329230, −11.695300930685202579194861700287, −11.30133635961620499607817849578, −10.41199604371318778586201511819, −9.84362550097226268240932870411, −9.287282858627696859383717613884, −8.388961737901347208575695404297, −7.59311236632315466151732180412, −6.8146413835962275058577445870, −6.35838603593837553482320341203, −5.372940582672227652494550461373, −4.41584526772757459339275321797, −3.85193316800731143344819702899, −2.78248833236363091745376024875, −1.78217958449013216933311887867, −1.19787779158331603191517735393, 0.11527177938891493668339716305, 0.8135157494279403059038352802, 1.92446109046378865165352183084, 2.62729617666118483826137134405, 4.04805012180908563976104756281, 4.83168211684179029828259003856, 5.63099406260341137474374777717, 6.14835192939825476313138066011, 6.91922542664631791800454552099, 7.48087799352299234132503312610, 8.328929757952894590836073374995, 8.99484395448606557115425680033, 9.90423145572662840989738057442, 10.592593703198297892539955184353, 10.845960584472217367568448209852, 11.83626052443972496185365472913, 12.36537916449373529875114813633, 13.301739822498551009402067780349, 13.96972794149472989426121018088, 15.134611018212019547829853863475, 15.48141343894008399403625573472, 16.03069775499050284402699359757, 17.01096293291599170199819956793, 17.27854538502938739812974580762, 17.97514274403489884887468699282

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