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Properties

Label 1-4235-4235.277-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.00830 + 0.999i$
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  + (−0.424 + 0.905i)2-s + (0.207 + 0.978i)3-s + (−0.640 − 0.768i)4-s + (−0.974 − 0.226i)6-s + (0.967 − 0.254i)8-s + (−0.913 + 0.406i)9-s + (0.618 − 0.786i)12-s + (0.717 + 0.696i)13-s + (−0.179 + 0.983i)16-s + (0.999 − 0.00951i)17-s + (0.0190 − 0.999i)18-s + (0.953 − 0.299i)19-s + (−0.690 − 0.723i)23-s + (0.449 + 0.893i)24-s + (−0.935 + 0.353i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.424 + 0.905i)2-s + (0.207 + 0.978i)3-s + (−0.640 − 0.768i)4-s + (−0.974 − 0.226i)6-s + (0.967 − 0.254i)8-s + (−0.913 + 0.406i)9-s + (0.618 − 0.786i)12-s + (0.717 + 0.696i)13-s + (−0.179 + 0.983i)16-s + (0.999 − 0.00951i)17-s + (0.0190 − 0.999i)18-s + (0.953 − 0.299i)19-s + (−0.690 − 0.723i)23-s + (0.449 + 0.893i)24-s + (−0.935 + 0.353i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00830 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00830 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.00830 + 0.999i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.00830 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.081251068 + 1.072312120i\)
\(L(\frac12)\) \(\approx\) \(1.081251068 + 1.072312120i\)
\(L(1)\) \(\approx\) \(0.7677934696 + 0.5837849033i\)
\(L(1)\) \(\approx\) \(0.7677934696 + 0.5837849033i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.424 + 0.905i)T \)
3 \( 1 + (0.207 + 0.978i)T \)
13 \( 1 + (0.717 + 0.696i)T \)
17 \( 1 + (0.999 - 0.00951i)T \)
19 \( 1 + (0.953 - 0.299i)T \)
23 \( 1 + (-0.690 - 0.723i)T \)
29 \( 1 + (0.993 + 0.113i)T \)
31 \( 1 + (0.999 - 0.0380i)T \)
37 \( 1 + (-0.997 - 0.0665i)T \)
41 \( 1 + (0.921 - 0.389i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + (-0.0190 - 0.999i)T \)
53 \( 1 + (-0.983 + 0.179i)T \)
59 \( 1 + (-0.123 - 0.992i)T \)
61 \( 1 + (-0.820 - 0.572i)T \)
67 \( 1 + (-0.945 - 0.327i)T \)
71 \( 1 + (0.198 + 0.980i)T \)
73 \( 1 + (-0.524 - 0.851i)T \)
79 \( 1 + (0.988 + 0.151i)T \)
83 \( 1 + (0.791 + 0.610i)T \)
89 \( 1 + (0.995 + 0.0950i)T \)
97 \( 1 + (0.0570 - 0.998i)T \)
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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

−18.20212779956163563370838004867, −17.77704432728591367365465669054, −17.25622369884806449528332325368, −16.28912608069728519601291385540, −15.64268348688833341575465705584, −14.47261149095656449928304965760, −13.82493904795536772653079265674, −13.469746642698193378952833693522, −12.5340205559865016334999495168, −12.08664352279415244162070155690, −11.56562543575500905403447032702, −10.63347798249766924796767953341, −10.03981277731402595514416596155, −9.19650676560515606672574081410, −8.5212575166261865943291915942, −7.690683336781089095449048286663, −7.56595659053685371943309138784, −6.26251068795094215768269866472, −5.64577442141265956813540741296, −4.65723402782038648188904623953, −3.469292389197577470008697984386, −3.16013619286913077085887641452, −2.247373485492993177835841168615, −1.27443565492100422538727633705, −0.87553995440394650943222595784, 0.66660068003995135694626458808, 1.74600453120613490720227847670, 2.94757478006863266335364921508, 3.76797413992256068237427883890, 4.56857801441198913679080119710, 5.12384870219353829441977520048, 6.00672985835279790897075016452, 6.52830367145939262581916553930, 7.6167866722176210661330572163, 8.17314865446692766443496584264, 8.90432901666612533571830605904, 9.46524960738188898133131643970, 10.15598680138034484063982878823, 10.69207543153122034478020279402, 11.55901085355049412989460137934, 12.362270899359291308801299636982, 13.56861409143675740557068545392, 14.11216778919428104795403543499, 14.41784713585516854081343942302, 15.43247323722152382733171160989, 15.90767413242113138910437090357, 16.33701658808564550115050374812, 17.017276855094412442937809200949, 17.73981669726440702085518246424, 18.40920056111379610506855037710

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