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Properties

Label 1-4235-4235.1103-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.212 - 0.977i$
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  + (0.556 + 0.830i)2-s + (−0.406 + 0.913i)3-s + (−0.380 + 0.924i)4-s + (−0.985 + 0.170i)6-s + (−0.980 + 0.198i)8-s + (−0.669 − 0.743i)9-s + (−0.690 − 0.723i)12-s + (−0.825 − 0.564i)13-s + (−0.710 − 0.703i)16-s + (0.992 − 0.123i)17-s + (0.244 − 0.969i)18-s + (0.683 − 0.730i)19-s + (−0.458 − 0.888i)23-s + (0.217 − 0.976i)24-s + (0.00951 − 0.999i)26-s + (0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (0.556 + 0.830i)2-s + (−0.406 + 0.913i)3-s + (−0.380 + 0.924i)4-s + (−0.985 + 0.170i)6-s + (−0.980 + 0.198i)8-s + (−0.669 − 0.743i)9-s + (−0.690 − 0.723i)12-s + (−0.825 − 0.564i)13-s + (−0.710 − 0.703i)16-s + (0.992 − 0.123i)17-s + (0.244 − 0.969i)18-s + (0.683 − 0.730i)19-s + (−0.458 − 0.888i)23-s + (0.217 − 0.976i)24-s + (0.00951 − 0.999i)26-s + (0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.212 - 0.977i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.212 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07510789971 + 0.09323323481i\)
\(L(\frac12)\) \(\approx\) \(-0.07510789971 + 0.09323323481i\)
\(L(1)\) \(\approx\) \(0.7955787214 + 0.6002098768i\)
\(L(1)\) \(\approx\) \(0.7955787214 + 0.6002098768i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.556 + 0.830i)T \)
3 \( 1 + (-0.406 + 0.913i)T \)
13 \( 1 + (-0.825 - 0.564i)T \)
17 \( 1 + (0.992 - 0.123i)T \)
19 \( 1 + (0.683 - 0.730i)T \)
23 \( 1 + (-0.458 - 0.888i)T \)
29 \( 1 + (-0.0855 - 0.996i)T \)
31 \( 1 + (0.879 - 0.475i)T \)
37 \( 1 + (0.647 + 0.761i)T \)
41 \( 1 + (-0.466 - 0.884i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + (0.244 + 0.969i)T \)
53 \( 1 + (-0.703 - 0.710i)T \)
59 \( 1 + (-0.999 + 0.0380i)T \)
61 \( 1 + (-0.0665 + 0.997i)T \)
67 \( 1 + (-0.371 - 0.928i)T \)
71 \( 1 + (0.516 - 0.856i)T \)
73 \( 1 + (-0.780 - 0.625i)T \)
79 \( 1 + (0.398 - 0.917i)T \)
83 \( 1 + (-0.633 + 0.774i)T \)
89 \( 1 + (0.327 + 0.945i)T \)
97 \( 1 + (-0.676 + 0.736i)T \)
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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

−17.878128309327414094835127128788, −17.217073235096622876070976843627, −16.45155726464882974033943450199, −15.661414176562391306037009953692, −14.56090688758853029625044715722, −14.1965874564373994300329209118, −13.626571565946186644514087694286, −12.69871404275952598870729527831, −12.29297072760888528393021065240, −11.73947962129676558934444974404, −11.13140962655812954364221666617, −10.22402901588719806345239586674, −9.70113484524801299540186605444, −8.802284867505537893501838645178, −7.83946686494828727332793318261, −7.213051021880333534474961152603, −6.33670612945737480578829477701, −5.6014071677374337485147237315, −5.120608868287551360324847828770, −4.16955072214471552723954564809, −3.24055296555861723374075546149, −2.55379761064391156391344089298, −1.58805886810982863562737275923, −1.16335705875120077227296374640, −0.01925653323440836076352503672, 0.78625980136537385013701584220, 2.61067141699037105507775468860, 3.063476331074340969565657808667, 4.05439800389714469337199581768, 4.67188930150630559469960643157, 5.25877450059398063455482721656, 5.990600254182784123241734238000, 6.57421482064804732662443569168, 7.671520288982557900401021891238, 8.03557310781877446921849852469, 9.13292141845802718611487630503, 9.64374919750784564673173550674, 10.35560406477645376579641183453, 11.27808254132140705405079079210, 12.09335885843694898769463775668, 12.37451950238940169036694259069, 13.49833012509490625712136121932, 14.05849708502104943026625253253, 14.91336204403506101165349487633, 15.18889774560005117065271258242, 16.037452615075123851820072623167, 16.46785208231451241648247854810, 17.2861342416690031049575734915, 17.57530377683360568794679386992, 18.449176632571528715466844571911

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