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Properties

Label 1-4235-4235.1047-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.627 + 0.778i$
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  + (0.962 − 0.272i)2-s + (0.207 + 0.978i)3-s + (0.851 − 0.524i)4-s + (0.466 + 0.884i)6-s + (0.676 − 0.736i)8-s + (−0.913 + 0.406i)9-s + (0.690 + 0.723i)12-s + (−0.791 + 0.610i)13-s + (0.449 − 0.893i)16-s + (0.424 + 0.905i)17-s + (−0.768 + 0.640i)18-s + (0.123 − 0.992i)19-s + (0.458 + 0.888i)23-s + (0.861 + 0.508i)24-s + (−0.595 + 0.803i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.962 − 0.272i)2-s + (0.207 + 0.978i)3-s + (0.851 − 0.524i)4-s + (0.466 + 0.884i)6-s + (0.676 − 0.736i)8-s + (−0.913 + 0.406i)9-s + (0.690 + 0.723i)12-s + (−0.791 + 0.610i)13-s + (0.449 − 0.893i)16-s + (0.424 + 0.905i)17-s + (−0.768 + 0.640i)18-s + (0.123 − 0.992i)19-s + (0.458 + 0.888i)23-s + (0.861 + 0.508i)24-s + (−0.595 + 0.803i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.627 + 0.778i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1047, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.627 + 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.220851876 + 1.541245484i\)
\(L(\frac12)\) \(\approx\) \(3.220851876 + 1.541245484i\)
\(L(1)\) \(\approx\) \(1.977640218 + 0.3698119221i\)
\(L(1)\) \(\approx\) \(1.977640218 + 0.3698119221i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.962 - 0.272i)T \)
3 \( 1 + (0.207 + 0.978i)T \)
13 \( 1 + (-0.791 + 0.610i)T \)
17 \( 1 + (0.424 + 0.905i)T \)
19 \( 1 + (0.123 - 0.992i)T \)
23 \( 1 + (0.458 + 0.888i)T \)
29 \( 1 + (0.516 - 0.856i)T \)
31 \( 1 + (-0.179 - 0.983i)T \)
37 \( 1 + (0.0760 + 0.997i)T \)
41 \( 1 + (-0.897 - 0.441i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + (0.768 + 0.640i)T \)
53 \( 1 + (0.893 - 0.449i)T \)
59 \( 1 + (0.830 + 0.556i)T \)
61 \( 1 + (0.969 - 0.244i)T \)
67 \( 1 + (0.371 + 0.928i)T \)
71 \( 1 + (0.974 + 0.226i)T \)
73 \( 1 + (0.263 + 0.964i)T \)
79 \( 1 + (0.749 + 0.662i)T \)
83 \( 1 + (-0.931 - 0.362i)T \)
89 \( 1 + (0.327 + 0.945i)T \)
97 \( 1 + (-0.491 - 0.870i)T \)
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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

−18.2196019931400643728191131265, −17.61956589968194130849474001896, −16.76404684467901573180642374279, −16.31958218686350180462799143174, −15.37144801191234055690987164643, −14.59662920988311684737709018420, −14.23762744523210359173921503643, −13.605263939222214843626851192549, −12.718510665596301847332591864475, −12.33838059661870591965787664937, −11.880529102084470207306243973280, −10.89682085608414533456866694220, −10.23423612151173113982805376708, −9.08924846769040201956431391177, −8.29417713448977911910023400295, −7.64084899335791692086042668797, −7.02195876551638742247064857379, −6.50330357369679154675653963789, −5.38372300983432426923353446985, −5.24211788118896620583983104648, −3.99969483197915783783717344219, −3.14904456310240985037667623603, −2.60082032767991055312364851638, −1.78439274586022643811439296157, −0.72782973390905347313347319574, 1.01912238931153262293867619413, 2.2863427789766500186690275001, 2.690234843640875514505445067755, 3.7706417748062933295880714680, 4.13833263861593926481828137671, 5.02777904135479084832880442658, 5.50305376226983076016655732759, 6.39026861172969545438502210425, 7.20050171461214182446712157046, 8.03682052336803209233833899388, 8.98040718303161998995535778630, 9.83533380734752934500612849678, 10.12243847623808034630372366972, 11.217620616213657723521751516712, 11.461243684454570303555405730195, 12.31128615458477403423229675917, 13.16806180744414815446992440914, 13.79461419551733210303098709272, 14.42451867405914943062237919228, 15.1756850971258713467174178279, 15.42232010638865245564088870070, 16.27153787647058369134988623482, 17.02123879682627527597531453281, 17.429727607588934149175450783548, 18.8651196771419058131631504291

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