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Properties

Label 1-4235-4235.1087-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.638 + 0.769i$
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  + (−0.893 + 0.449i)2-s + (0.743 − 0.669i)3-s + (0.595 − 0.803i)4-s + (−0.362 + 0.931i)6-s + (−0.170 + 0.985i)8-s + (0.104 − 0.994i)9-s + (−0.0950 − 0.995i)12-s + (0.491 + 0.870i)13-s + (−0.290 − 0.956i)16-s + (−0.983 − 0.179i)17-s + (0.353 + 0.935i)18-s + (−0.879 + 0.475i)19-s + (0.945 + 0.327i)23-s + (0.532 + 0.846i)24-s + (−0.830 − 0.556i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.893 + 0.449i)2-s + (0.743 − 0.669i)3-s + (0.595 − 0.803i)4-s + (−0.362 + 0.931i)6-s + (−0.170 + 0.985i)8-s + (0.104 − 0.994i)9-s + (−0.0950 − 0.995i)12-s + (0.491 + 0.870i)13-s + (−0.290 − 0.956i)16-s + (−0.983 − 0.179i)17-s + (0.353 + 0.935i)18-s + (−0.879 + 0.475i)19-s + (0.945 + 0.327i)23-s + (0.532 + 0.846i)24-s + (−0.830 − 0.556i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.638 + 0.769i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.638 + 0.769i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3246674423 + 0.6909337860i\)
\(L(\frac12)\) \(\approx\) \(0.3246674423 + 0.6909337860i\)
\(L(1)\) \(\approx\) \(0.8478398717 + 0.02332847567i\)
\(L(1)\) \(\approx\) \(0.8478398717 + 0.02332847567i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.893 + 0.449i)T \)
3 \( 1 + (0.743 - 0.669i)T \)
13 \( 1 + (0.491 + 0.870i)T \)
17 \( 1 + (-0.983 - 0.179i)T \)
19 \( 1 + (-0.879 + 0.475i)T \)
23 \( 1 + (0.945 + 0.327i)T \)
29 \( 1 + (0.564 + 0.825i)T \)
31 \( 1 + (0.749 + 0.662i)T \)
37 \( 1 + (-0.299 + 0.953i)T \)
41 \( 1 + (-0.254 - 0.967i)T \)
43 \( 1 + (-0.989 + 0.142i)T \)
47 \( 1 + (0.353 - 0.935i)T \)
53 \( 1 + (0.956 + 0.290i)T \)
59 \( 1 + (0.710 + 0.703i)T \)
61 \( 1 + (0.548 + 0.836i)T \)
67 \( 1 + (-0.998 + 0.0475i)T \)
71 \( 1 + (0.610 - 0.791i)T \)
73 \( 1 + (0.875 - 0.483i)T \)
79 \( 1 + (0.969 + 0.244i)T \)
83 \( 1 + (-0.996 - 0.0855i)T \)
89 \( 1 + (-0.235 - 0.971i)T \)
97 \( 1 + (-0.884 - 0.466i)T \)
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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

−17.92374958014267046809477531699, −17.351339377935344729464439726423, −16.69435614557099954962011235679, −15.88463359379028915214511735005, −15.33222534253585049936455685851, −14.926402901769693337718960262904, −13.740588085064936644811315798517, −13.13227604016125598181792571388, −12.61402582880260517070937387568, −11.405436946332673593013281392467, −10.985013480850684823152484041769, −10.315018852600363507681536300683, −9.707815917505154405837559655211, −8.93074312962773310959492664071, −8.397807873692027265246023006489, −7.93265738302425261229400773721, −6.92824684730134025196249986385, −6.254981233186868278887900323393, −5.05170043511111246402946617525, −4.21949259140272101761200439859, −3.57218612261120264037782965272, −2.62120080567978419789864084400, −2.280306576789325639136838114258, −1.09288054602570946689646969167, −0.145828566375167233567436822852, 0.963956258874637163798588947013, 1.66458755294634697202618058000, 2.35463927145390310697739629703, 3.22864144459375285896288318414, 4.25124070709725098157627559780, 5.207856631392181769204117704155, 6.25787086005257162272645518818, 6.82651318831632063046769696922, 7.167492362044250844226248985027, 8.312326701256920931653137566848, 8.66767598563150005619887306227, 9.13079077841569817385030644292, 10.096318349782181536312067447548, 10.72796014317241009708873535868, 11.64592513312743316341432797780, 12.168490561836324606536271877503, 13.26470182330915678055585552324, 13.7431639997815864906585764287, 14.47020164436670622179954516286, 15.20049917870583621400797091120, 15.612213638340487647045872979099, 16.61034305309757176525266652689, 17.09191321389849058601073684421, 18.03737592221495059682001519339, 18.307866568941380526779813550358

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