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Properties

Label 1-4235-4235.2977-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.488 + 0.872i$
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  + (−0.0190 + 0.999i)2-s + (−0.406 − 0.913i)3-s + (−0.999 − 0.0380i)4-s + (0.921 − 0.389i)6-s + (0.0570 − 0.998i)8-s + (−0.669 + 0.743i)9-s + (0.371 + 0.928i)12-s + (−0.170 − 0.985i)13-s + (0.997 + 0.0760i)16-s + (0.917 + 0.398i)17-s + (−0.730 − 0.683i)18-s + (0.861 + 0.508i)19-s + (0.971 − 0.235i)23-s + (−0.935 + 0.353i)24-s + (0.988 − 0.151i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.0190 + 0.999i)2-s + (−0.406 − 0.913i)3-s + (−0.999 − 0.0380i)4-s + (0.921 − 0.389i)6-s + (0.0570 − 0.998i)8-s + (−0.669 + 0.743i)9-s + (0.371 + 0.928i)12-s + (−0.170 − 0.985i)13-s + (0.997 + 0.0760i)16-s + (0.917 + 0.398i)17-s + (−0.730 − 0.683i)18-s + (0.861 + 0.508i)19-s + (0.971 − 0.235i)23-s + (−0.935 + 0.353i)24-s + (0.988 − 0.151i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.488 + 0.872i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2977, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.488 + 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.094824770 + 0.6421580848i\)
\(L(\frac12)\) \(\approx\) \(1.094824770 + 0.6421580848i\)
\(L(1)\) \(\approx\) \(0.8540535735 + 0.2168171763i\)
\(L(1)\) \(\approx\) \(0.8540535735 + 0.2168171763i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0190 + 0.999i)T \)
3 \( 1 + (-0.406 - 0.913i)T \)
13 \( 1 + (-0.170 - 0.985i)T \)
17 \( 1 + (0.917 + 0.398i)T \)
19 \( 1 + (0.861 + 0.508i)T \)
23 \( 1 + (0.971 - 0.235i)T \)
29 \( 1 + (0.198 + 0.980i)T \)
31 \( 1 + (-0.0665 + 0.997i)T \)
37 \( 1 + (0.962 + 0.272i)T \)
41 \( 1 + (-0.0855 - 0.996i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + (0.730 - 0.683i)T \)
53 \( 1 + (-0.0760 - 0.997i)T \)
59 \( 1 + (-0.820 - 0.572i)T \)
61 \( 1 + (-0.483 + 0.875i)T \)
67 \( 1 + (0.189 + 0.981i)T \)
71 \( 1 + (-0.736 - 0.676i)T \)
73 \( 1 + (-0.983 - 0.179i)T \)
79 \( 1 + (0.964 - 0.263i)T \)
83 \( 1 + (-0.999 - 0.0285i)T \)
89 \( 1 + (-0.580 + 0.814i)T \)
97 \( 1 + (-0.633 + 0.774i)T \)
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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

−18.45132829833179833846127242679, −17.513027521191555940381797885, −16.9673938439835693305442605057, −16.4353843250124344532043109835, −15.50213136151453370860288152265, −14.84405756700934799668598389822, −14.0525352332921542935198940912, −13.58590060826768200950621855771, −12.54582803226096444200173576810, −11.91425293506264761609120294954, −11.339110329732869308806749035546, −10.8939118860294634239385226899, −9.91209459300773872110797784418, −9.46806430470556811173880791135, −9.0614090128223889352223627671, −8.008860143420461251676854853219, −7.19066663605905782055363683702, −5.98651389542341530952818208996, −5.43968185226273613105886758487, −4.48274131083975149915127892845, −4.221376111847505913090264022541, −3.10744384998441069970383136531, −2.69239303645408673487965105, −1.43535471666699585722392360288, −0.54634188823003266211698197605, 0.839300996720915911985186281776, 1.39323586874633603097314329837, 2.83286310755977698396468310250, 3.47551639389405501425760345357, 4.70011160452793897691649478405, 5.44369827038510880797732891192, 5.77610088658948714153820509254, 6.73171209479483999460478668528, 7.303350458937141314740579722489, 7.91280250954944745927320754529, 8.497548379845156862431172411780, 9.34161188619449823603345126901, 10.287829481397940063740917424526, 10.81337677306791120492294576551, 12.0010305676177668928537664827, 12.48421430392128535206318024046, 13.08538754163500928281912262708, 13.773321824234664611060652254047, 14.51945251552591749654553657303, 14.94991382741893219892000792675, 16.030057119770204776936010157217, 16.45035205252531947260748418212, 17.195727410586627533817921445351, 17.75433863811183023478873868131, 18.30367071100406401436483923267

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