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Properties

Label 1-4235-4235.1117-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.857 - 0.513i$
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  + (−0.508 − 0.861i)2-s + (−0.743 − 0.669i)3-s + (−0.483 + 0.875i)4-s + (−0.198 + 0.980i)6-s + (0.999 − 0.0285i)8-s + (0.104 + 0.994i)9-s + (0.945 − 0.327i)12-s + (0.996 + 0.0855i)13-s + (−0.532 − 0.846i)16-s + (0.893 − 0.449i)17-s + (0.803 − 0.595i)18-s + (−0.710 − 0.703i)19-s + (0.371 − 0.928i)23-s + (−0.761 − 0.647i)24-s + (−0.432 − 0.901i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.508 − 0.861i)2-s + (−0.743 − 0.669i)3-s + (−0.483 + 0.875i)4-s + (−0.198 + 0.980i)6-s + (0.999 − 0.0285i)8-s + (0.104 + 0.994i)9-s + (0.945 − 0.327i)12-s + (0.996 + 0.0855i)13-s + (−0.532 − 0.846i)16-s + (0.893 − 0.449i)17-s + (0.803 − 0.595i)18-s + (−0.710 − 0.703i)19-s + (0.371 − 0.928i)23-s + (−0.761 − 0.647i)24-s + (−0.432 − 0.901i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.857 - 0.513i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.857 - 0.513i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2716652074 - 0.9824960970i\)
\(L(\frac12)\) \(\approx\) \(0.2716652074 - 0.9824960970i\)
\(L(1)\) \(\approx\) \(0.5587392634 - 0.4448652422i\)
\(L(1)\) \(\approx\) \(0.5587392634 - 0.4448652422i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.508 - 0.861i)T \)
3 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (0.996 + 0.0855i)T \)
17 \( 1 + (0.893 - 0.449i)T \)
19 \( 1 + (-0.710 - 0.703i)T \)
23 \( 1 + (0.371 - 0.928i)T \)
29 \( 1 + (0.774 - 0.633i)T \)
31 \( 1 + (-0.290 - 0.956i)T \)
37 \( 1 + (0.992 + 0.123i)T \)
41 \( 1 + (0.736 - 0.676i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (-0.803 - 0.595i)T \)
53 \( 1 + (0.846 + 0.532i)T \)
59 \( 1 + (0.217 - 0.976i)T \)
61 \( 1 + (-0.00951 + 0.999i)T \)
67 \( 1 + (0.814 + 0.580i)T \)
71 \( 1 + (-0.362 - 0.931i)T \)
73 \( 1 + (-0.938 + 0.345i)T \)
79 \( 1 + (0.380 + 0.924i)T \)
83 \( 1 + (0.717 + 0.696i)T \)
89 \( 1 + (-0.0475 - 0.998i)T \)
97 \( 1 + (0.336 + 0.941i)T \)
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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

−18.28134606167499869103939876743, −17.92264564204699571519717242226, −17.222256360032253821799063018919, −16.435911676562376765963363978838, −16.20038302339944772863207022577, −15.46539681160190682207089279175, −14.669689011646645817112748928919, −14.34169377580129369623480044, −13.200073736666487301320427194983, −12.61570923814339125494840246662, −11.59532931886231612038629820837, −10.89920707500174872585229202198, −10.3602504548714137111389163264, −9.70218236918559173511784370309, −8.9911884994572781176797698862, −8.28448977850804242463511153057, −7.56299199953699071840409649020, −6.56866046340178744286569297341, −6.08316026425217014439853446958, −5.47793552263561507252972854016, −4.707015815847949400882350401921, −3.93954774588702025188055225058, −3.16282774751896595708173265974, −1.535174912031659005159694377896, −0.961982285304516776742581557924, 0.54552017454447940215333938654, 1.06609006260366986265450770962, 2.167618882529539859148139702561, 2.68551290673693515981142507950, 3.834518526376877554492370598066, 4.518730502196319446072723394705, 5.4017004940545148387866870946, 6.25883120193767112583867686577, 6.99411471343697428565960732624, 7.76846847797932807362208696993, 8.41722220140142054799918671514, 9.11577246011242702357873006737, 10.0668063387363772480019546371, 10.66105629178903636626068667475, 11.288442388403718641199811870700, 11.82769407089997458716305996704, 12.5330191704541758770077097599, 13.17003801793677507735048632454, 13.64861900370851233466703778186, 14.510057647037584543740313922398, 15.63951240547328206255291062545, 16.42459319395415999021439681767, 16.87260107959433139513568621737, 17.58200070437779585887562219863, 18.21269676595828302279537311758

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