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Properties

Label 1-4235-4235.12-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.895 + 0.444i$
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  + (−0.0950 − 0.995i)2-s + (−0.866 − 0.5i)3-s + (−0.981 + 0.189i)4-s + (−0.415 + 0.909i)6-s + (0.281 + 0.959i)8-s + (0.5 + 0.866i)9-s + (0.945 + 0.327i)12-s + (0.755 − 0.654i)13-s + (0.928 − 0.371i)16-s + (0.458 + 0.888i)17-s + (0.814 − 0.580i)18-s + (−0.888 − 0.458i)19-s + (−0.371 − 0.928i)23-s + (0.235 − 0.971i)24-s + (−0.723 − 0.690i)26-s − i·27-s + ⋯
L(s)  = 1  + (−0.0950 − 0.995i)2-s + (−0.866 − 0.5i)3-s + (−0.981 + 0.189i)4-s + (−0.415 + 0.909i)6-s + (0.281 + 0.959i)8-s + (0.5 + 0.866i)9-s + (0.945 + 0.327i)12-s + (0.755 − 0.654i)13-s + (0.928 − 0.371i)16-s + (0.458 + 0.888i)17-s + (0.814 − 0.580i)18-s + (−0.888 − 0.458i)19-s + (−0.371 − 0.928i)23-s + (0.235 − 0.971i)24-s + (−0.723 − 0.690i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09708587836 - 0.4143956276i\)
\(L(\frac12)\) \(\approx\) \(-0.09708587836 - 0.4143956276i\)
\(L(1)\) \(\approx\) \(0.5154350695 - 0.3683566052i\)
\(L(1)\) \(\approx\) \(0.5154350695 - 0.3683566052i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0950 - 0.995i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.755 - 0.654i)T \)
17 \( 1 + (0.458 + 0.888i)T \)
19 \( 1 + (-0.888 - 0.458i)T \)
23 \( 1 + (-0.371 - 0.928i)T \)
29 \( 1 + (-0.841 + 0.540i)T \)
31 \( 1 + (0.327 + 0.945i)T \)
37 \( 1 + (-0.189 + 0.981i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.281 + 0.959i)T \)
47 \( 1 + (-0.814 - 0.580i)T \)
53 \( 1 + (0.371 - 0.928i)T \)
59 \( 1 + (-0.995 - 0.0950i)T \)
61 \( 1 + (-0.580 + 0.814i)T \)
67 \( 1 + (-0.814 + 0.580i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.618 - 0.786i)T \)
79 \( 1 + (-0.235 - 0.971i)T \)
83 \( 1 + (0.989 - 0.142i)T \)
89 \( 1 + (0.0475 - 0.998i)T \)
97 \( 1 + (0.281 + 0.959i)T \)
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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

−18.678441117688622702803306386542, −17.913308440362445245232615943729, −17.13620022659206648281768058914, −16.79840080538591800468360642934, −16.05917241339354513715219394877, −15.54518352784700887536901371980, −14.97004058318012938440487052610, −14.041968151357079824571503871839, −13.5691143561568346795154526334, −12.61504180572535228064293454290, −11.98134996557546498327395916196, −11.12612287272382679941156507397, −10.51561570913953430908390159343, −9.49187773420896546324639089499, −9.35060821526321318519450218130, −8.30577193408917767096232296161, −7.49581149051469649046852308131, −6.82740285454790723599751807677, −5.98846308694681051436741391743, −5.6620441263248590330181117399, −4.77431682974443757007204206156, −4.01539365569172154579608267166, −3.55047112058325507178531258157, −1.95495455796405612016599174560, −0.8940253950750969270486967549, 0.180690467399253498572447533, 1.26797084878383718681996721574, 1.78461905776109397424946602620, 2.83815244421392366648001351210, 3.62892343949987482201599630781, 4.56186707564764428710353840664, 5.12755686160501019818396575325, 6.07249836845535359058635083282, 6.57741846331965272396624700349, 7.78468159197380433042285082110, 8.28121043210236113992818289948, 9.01172926065206964833532016506, 10.18458424367395480802668997215, 10.47944222151164352049604087137, 11.11593524479615678774765052691, 11.83936414437154532468235916281, 12.4895735138468457410044413916, 13.06836234976867051886008081670, 13.471052940592155983648043889896, 14.508877854428433668365248090963, 15.1641550225262209957447899858, 16.29660065041038458771545443802, 16.78845690102176467514581107620, 17.52030368604404291009537462851, 18.12707252844486015342451439997

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