Properties

Label 1-4235-4235.299-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.442 + 0.896i$
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  + (−0.179 − 0.983i)2-s + (0.669 + 0.743i)3-s + (−0.935 + 0.353i)4-s + (0.610 − 0.791i)6-s + (0.516 + 0.856i)8-s + (−0.104 + 0.994i)9-s + (−0.888 − 0.458i)12-s + (0.998 + 0.0570i)13-s + (0.749 − 0.662i)16-s + (−0.999 − 0.0380i)17-s + (0.997 − 0.0760i)18-s + (0.345 + 0.938i)19-s + (0.995 − 0.0950i)23-s + (−0.290 + 0.956i)24-s + (−0.123 − 0.992i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (−0.179 − 0.983i)2-s + (0.669 + 0.743i)3-s + (−0.935 + 0.353i)4-s + (0.610 − 0.791i)6-s + (0.516 + 0.856i)8-s + (−0.104 + 0.994i)9-s + (−0.888 − 0.458i)12-s + (0.998 + 0.0570i)13-s + (0.749 − 0.662i)16-s + (−0.999 − 0.0380i)17-s + (0.997 − 0.0760i)18-s + (0.345 + 0.938i)19-s + (0.995 − 0.0950i)23-s + (−0.290 + 0.956i)24-s + (−0.123 − 0.992i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.442 + 0.896i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.442 + 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5084047337 + 0.8177149485i\)
\(L(\frac12)\) \(\approx\) \(0.5084047337 + 0.8177149485i\)
\(L(1)\) \(\approx\) \(0.9790174098 + 0.005383861998i\)
\(L(1)\) \(\approx\) \(0.9790174098 + 0.005383861998i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.179 - 0.983i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.998 + 0.0570i)T \)
17 \( 1 + (-0.999 - 0.0380i)T \)
19 \( 1 + (0.345 + 0.938i)T \)
23 \( 1 + (0.995 - 0.0950i)T \)
29 \( 1 + (-0.897 + 0.441i)T \)
31 \( 1 + (-0.988 - 0.151i)T \)
37 \( 1 + (-0.964 + 0.263i)T \)
41 \( 1 + (-0.0285 + 0.999i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (0.997 + 0.0760i)T \)
53 \( 1 + (-0.749 - 0.662i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (-0.761 - 0.647i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (0.696 + 0.717i)T \)
73 \( 1 + (0.595 - 0.803i)T \)
79 \( 1 + (-0.820 + 0.572i)T \)
83 \( 1 + (0.870 + 0.491i)T \)
89 \( 1 + (-0.928 + 0.371i)T \)
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

−18.26639318162207670923344088279, −17.42039067613743396686236815706, −17.000728598350842829241664319933, −15.92414959363792291960662950743, −15.35258448789443973940136662151, −14.98971230042501489103797567620, −13.83267501147590941193131917363, −13.708407223213309654172917033795, −12.989821063767503149870891633141, −12.323276584334747384050324399072, −11.19971575641701603687540815801, −10.60837666993763359965598930987, −9.39246816820382710522805822144, −8.93086781927233657184712005158, −8.57321873959473830196476447695, −7.461329211208097233256242996488, −7.19941624656389307148593944428, −6.39523720317284077924238706826, −5.7315780341613260560760379085, −4.854556523367048864959346254597, −3.89138666733388521205060454868, −3.262267960107834581006627278867, −2.11125909350848789941264254485, −1.28049967323315337537341269811, −0.25789128907136296712275443749, 1.358893228126572576557432989436, 1.953815221671006821673402185121, 2.98221233953767478978001987633, 3.507779780699213605496722369761, 4.16776154240340361236913364818, 4.948338944589906881775840353841, 5.658445297921241995345754508166, 6.84250876241002157823599585792, 7.87456524096765879327037358248, 8.410624692190716822893088802503, 9.17441671888045373934410805133, 9.51154522056263429876229428908, 10.47785442289706877130998287086, 10.98571958921534183368360076764, 11.446692981829055690094180153000, 12.55695658023199745828245831358, 13.16518359571026799016338468028, 13.73707222710871925255361769438, 14.439432430883756870908835316471, 15.09716251132718421477524679055, 15.92422878648601740696563314824, 16.608363142650843622313605606179, 17.2226937454511583935499911180, 18.24745285092228204660802120388, 18.64235655937435624459591519821

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