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Properties

Label 1-4235-4235.2719-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.982 + 0.187i$
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  + (−0.761 + 0.647i)2-s + (−0.978 + 0.207i)3-s + (0.161 − 0.986i)4-s + (0.610 − 0.791i)6-s + (0.516 + 0.856i)8-s + (0.913 − 0.406i)9-s + (0.0475 + 0.998i)12-s + (0.998 + 0.0570i)13-s + (−0.948 − 0.318i)16-s + (0.532 − 0.846i)17-s + (−0.432 + 0.901i)18-s + (0.640 − 0.768i)19-s + (−0.580 − 0.814i)23-s + (−0.683 − 0.730i)24-s + (−0.797 + 0.603i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (−0.761 + 0.647i)2-s + (−0.978 + 0.207i)3-s + (0.161 − 0.986i)4-s + (0.610 − 0.791i)6-s + (0.516 + 0.856i)8-s + (0.913 − 0.406i)9-s + (0.0475 + 0.998i)12-s + (0.998 + 0.0570i)13-s + (−0.948 − 0.318i)16-s + (0.532 − 0.846i)17-s + (−0.432 + 0.901i)18-s + (0.640 − 0.768i)19-s + (−0.580 − 0.814i)23-s + (−0.683 − 0.730i)24-s + (−0.797 + 0.603i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.982 + 0.187i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.982 + 0.187i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8961889527 + 0.08458006196i\)
\(L(\frac12)\) \(\approx\) \(0.8961889527 + 0.08458006196i\)
\(L(1)\) \(\approx\) \(0.6200665718 + 0.1260644111i\)
\(L(1)\) \(\approx\) \(0.6200665718 + 0.1260644111i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.761 + 0.647i)T \)
3 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (0.998 + 0.0570i)T \)
17 \( 1 + (0.532 - 0.846i)T \)
19 \( 1 + (0.640 - 0.768i)T \)
23 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (-0.897 + 0.441i)T \)
31 \( 1 + (0.625 - 0.780i)T \)
37 \( 1 + (0.710 + 0.703i)T \)
41 \( 1 + (-0.0285 + 0.999i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (-0.432 - 0.901i)T \)
53 \( 1 + (0.948 - 0.318i)T \)
59 \( 1 + (0.851 - 0.524i)T \)
61 \( 1 + (-0.179 + 0.983i)T \)
67 \( 1 + (-0.723 - 0.690i)T \)
71 \( 1 + (0.696 + 0.717i)T \)
73 \( 1 + (0.398 + 0.917i)T \)
79 \( 1 + (0.905 + 0.424i)T \)
83 \( 1 + (0.870 + 0.491i)T \)
89 \( 1 + (0.786 + 0.618i)T \)
97 \( 1 + (0.974 - 0.226i)T \)
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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

−18.2649745908125731853705317893, −17.81429755160824682790760360681, −17.12528313679199067272705166760, −16.495857052631418995262783927194, −15.964789987472735900034689880142, −15.28126539865617919228846961738, −14.05646285372911152874347947512, −13.31116987401929172697704621985, −12.73082727693103664338346817660, −11.90764551062511378696759063221, −11.63417521366934513888083364523, −10.648115649924121437478804892607, −10.371468946559354722497395401031, −9.54923074654492625669670387608, −8.75516342423261561931190249372, −7.80128135995759271798812972612, −7.50543623236037143340909144516, −6.37335981213262521294461691399, −5.89234434454434466486467740069, −4.972601267191391257682738490842, −3.788715821943935308496022170685, −3.575141128835033018586829081232, −2.13228116225758154369213667387, −1.498314486328935713335737208076, −0.73611571717946874546997098632, 0.61699847651071141739354183937, 1.220153956373489150612345047755, 2.33667247639859455093659106595, 3.51837203948110714374949865329, 4.564624275843345322355738583102, 5.14991664159397341960199370371, 5.91328686641596922026651539505, 6.50247463184144755870499917807, 7.14189876777386433981867223097, 7.94767160578208125346249646162, 8.68801818031112744454089263782, 9.65506662495950073331505917663, 9.90913911266052389906665287007, 10.86720816645133410944886622477, 11.42951578724389896410542266129, 11.88887850756513320165283451427, 13.106815960864365444544737945901, 13.609991504349829678666312875479, 14.60406006993032831901655964500, 15.25300994236398211933761721492, 15.90773701430443860101080092041, 16.61092991114834738678650507161, 16.723063942146813863991385630761, 17.88289351789013486065099506511, 18.253709804909053073464455578419

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