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Properties

Label 1-4235-4235.877-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.344 + 0.938i$
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  + (−0.803 + 0.595i)2-s + (−0.994 + 0.104i)3-s + (0.290 − 0.956i)4-s + (0.736 − 0.676i)6-s + (0.336 + 0.941i)8-s + (0.978 − 0.207i)9-s + (−0.189 + 0.981i)12-s + (−0.856 + 0.516i)13-s + (−0.830 − 0.556i)16-s + (0.353 + 0.935i)17-s + (−0.662 + 0.749i)18-s + (0.548 + 0.836i)19-s + (0.618 + 0.786i)23-s + (−0.432 − 0.901i)24-s + (0.380 − 0.924i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.803 + 0.595i)2-s + (−0.994 + 0.104i)3-s + (0.290 − 0.956i)4-s + (0.736 − 0.676i)6-s + (0.336 + 0.941i)8-s + (0.978 − 0.207i)9-s + (−0.189 + 0.981i)12-s + (−0.856 + 0.516i)13-s + (−0.830 − 0.556i)16-s + (0.353 + 0.935i)17-s + (−0.662 + 0.749i)18-s + (0.548 + 0.836i)19-s + (0.618 + 0.786i)23-s + (−0.432 − 0.901i)24-s + (0.380 − 0.924i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.344 + 0.938i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (877, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.344 + 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4252714529 + 0.6093295857i\)
\(L(\frac12)\) \(\approx\) \(0.4252714529 + 0.6093295857i\)
\(L(1)\) \(\approx\) \(0.5270111752 + 0.2242347759i\)
\(L(1)\) \(\approx\) \(0.5270111752 + 0.2242347759i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.803 + 0.595i)T \)
3 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (-0.856 + 0.516i)T \)
17 \( 1 + (0.353 + 0.935i)T \)
19 \( 1 + (0.548 + 0.836i)T \)
23 \( 1 + (0.618 + 0.786i)T \)
29 \( 1 + (-0.362 - 0.931i)T \)
31 \( 1 + (0.123 - 0.992i)T \)
37 \( 1 + (0.572 + 0.820i)T \)
41 \( 1 + (0.870 + 0.491i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (0.662 + 0.749i)T \)
53 \( 1 + (0.556 + 0.830i)T \)
59 \( 1 + (-0.00951 + 0.999i)T \)
61 \( 1 + (0.398 + 0.917i)T \)
67 \( 1 + (0.0950 - 0.995i)T \)
71 \( 1 + (-0.254 + 0.967i)T \)
73 \( 1 + (0.846 - 0.532i)T \)
79 \( 1 + (0.879 - 0.475i)T \)
83 \( 1 + (-0.170 - 0.985i)T \)
89 \( 1 + (0.888 + 0.458i)T \)
97 \( 1 + (-0.825 - 0.564i)T \)
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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

−18.008346025923677893238241121203, −17.71142178149225102697307734983, −16.942059196801704058718343927867, −16.29324597461231176556039668254, −15.87776474414141668107345283057, −14.92715485430889687195983694989, −13.96652526499491407790040235923, −12.99761377786088181865432566097, −12.512430212980021926472234477222, −11.987711783165676447768343022233, −11.075494602948120496116120691977, −10.85827679121077265680567333526, −9.877583522502723772559889000196, −9.46369183664226018938758976319, −8.58831635242830233685070027858, −7.57037574354194214940952647842, −7.157759818277883886010660805623, −6.50784390234145357178689333925, −5.286582185235413026570977489731, −4.89975004637571612969688333237, −3.8512439666696979381197513573, −2.88885553769379017348758571909, −2.22146650050160171821768073488, −1.0454187159286417554598827692, −0.484036375622665677195278940531, 0.8353385100190261555054991674, 1.57917409984105180458718760550, 2.52190301252518903024425553360, 3.91738951611833684846173162779, 4.60503947412093065727440507191, 5.58203417010607414736952833404, 5.900858346506612812787940829652, 6.71414374393454293526374605998, 7.56373523579896834916946227536, 7.86665257592993477575802365886, 9.122131589468570705338975043321, 9.63293671199688108076453830115, 10.25102105774921164463992059344, 10.913485777090286101837267510052, 11.70155263167664051694003571727, 12.145227639557812592791111199617, 13.16166447039352087644017567867, 13.93102499140815390012770109681, 14.98209692223755985156715742921, 15.1599987070311365348230298888, 16.15081241159410172917208667559, 16.701184238626715221841896698798, 17.19048686871653978876134032211, 17.64411083310612739262488387528, 18.61206469105857377033998225809

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