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Properties

Label 1-4235-4235.1228-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.141 + 0.989i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.353 − 0.935i)2-s + (0.994 + 0.104i)3-s + (−0.749 + 0.662i)4-s + (−0.254 − 0.967i)6-s + (0.884 + 0.466i)8-s + (0.978 + 0.207i)9-s + (−0.814 + 0.580i)12-s + (−0.113 + 0.993i)13-s + (0.123 − 0.992i)16-s + (0.0760 − 0.997i)17-s + (−0.151 − 0.988i)18-s + (0.761 − 0.647i)19-s + (0.189 + 0.981i)23-s + (0.830 + 0.556i)24-s + (0.969 − 0.244i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.353 − 0.935i)2-s + (0.994 + 0.104i)3-s + (−0.749 + 0.662i)4-s + (−0.254 − 0.967i)6-s + (0.884 + 0.466i)8-s + (0.978 + 0.207i)9-s + (−0.814 + 0.580i)12-s + (−0.113 + 0.993i)13-s + (0.123 − 0.992i)16-s + (0.0760 − 0.997i)17-s + (−0.151 − 0.988i)18-s + (0.761 − 0.647i)19-s + (0.189 + 0.981i)23-s + (0.830 + 0.556i)24-s + (0.969 − 0.244i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.141 + 0.989i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1228, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.141 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9576709810 + 0.8304217028i\)
\(L(\frac12)\) \(\approx\) \(0.9576709810 + 0.8304217028i\)
\(L(1)\) \(\approx\) \(1.071675615 - 0.2824449672i\)
\(L(1)\) \(\approx\) \(1.071675615 - 0.2824449672i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.353 - 0.935i)T \)
3 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (-0.113 + 0.993i)T \)
17 \( 1 + (0.0760 - 0.997i)T \)
19 \( 1 + (0.761 - 0.647i)T \)
23 \( 1 + (0.189 + 0.981i)T \)
29 \( 1 + (0.610 - 0.791i)T \)
31 \( 1 + (-0.953 - 0.299i)T \)
37 \( 1 + (-0.508 - 0.861i)T \)
41 \( 1 + (-0.998 - 0.0570i)T \)
43 \( 1 + (-0.989 + 0.142i)T \)
47 \( 1 + (-0.151 + 0.988i)T \)
53 \( 1 + (-0.992 + 0.123i)T \)
59 \( 1 + (0.548 + 0.836i)T \)
61 \( 1 + (0.161 + 0.986i)T \)
67 \( 1 + (-0.458 + 0.888i)T \)
71 \( 1 + (-0.0285 + 0.999i)T \)
73 \( 1 + (0.956 - 0.290i)T \)
79 \( 1 + (0.345 - 0.938i)T \)
83 \( 1 + (-0.856 + 0.516i)T \)
89 \( 1 + (0.723 - 0.690i)T \)
97 \( 1 + (0.441 + 0.897i)T \)
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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

−18.24544411321987907825621339839, −17.328975652330085764572498933400, −16.6365306258911055292858626103, −15.94009118781763436632793758232, −15.23591930678602121390233751157, −14.78606853590838817223212426393, −14.173524784147169131047420802868, −13.50266917433492699191338274954, −12.77445277413199483166494937841, −12.25892476907272469542811001531, −10.82158661504955394717596983912, −10.21650080746463271441429689826, −9.71304407547899409491528207616, −8.777186191114000576519574552793, −8.25574175270582357564442481888, −7.842367730032914657941377994957, −6.89250355814832683949268360187, −6.427042013366580071327552003010, −5.322276875720064210165949756471, −4.81342220286228421233675300595, −3.64923916088736962673044494264, −3.23985121631056515188757880132, −1.92164514987984411554221780969, −1.26653269887650607457308908107, −0.1825728079361913002048925920, 1.0139341398267089419678912250, 1.76635818908620648048672434213, 2.54896014356184800213531793382, 3.18967454215834145575273780869, 3.93471343019584811665440954457, 4.65631754300603243762302568688, 5.38729337318308851307074274712, 6.87392180538204177985535796565, 7.384296558707330035851613827377, 8.103011736549922440815441163567, 8.99803642626493789889546730772, 9.36440056402957975971170256010, 9.89982160238602297405183603611, 10.77622430084466048329808817909, 11.63453380583795475885872425889, 11.98738908786378953997500086367, 13.110663123202811736384205294873, 13.498402729533641209177234906310, 14.1168211173350981048055760864, 14.74634712324294097386198962010, 15.82015299356575628638680829319, 16.2187470100200250522609352735, 17.186157881648082136060850205, 17.90875375874973675458793654023, 18.57310287859065717405928235903

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