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Properties

Label 1-4235-4235.2973-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.983 - 0.181i$
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  + (0.603 − 0.797i)2-s + (−0.994 + 0.104i)3-s + (−0.272 − 0.962i)4-s + (−0.516 + 0.856i)6-s + (−0.931 − 0.362i)8-s + (0.978 − 0.207i)9-s + (0.371 + 0.928i)12-s + (0.441 − 0.897i)13-s + (−0.851 + 0.524i)16-s + (0.976 − 0.217i)17-s + (0.424 − 0.905i)18-s + (0.749 − 0.662i)19-s + (−0.971 + 0.235i)23-s + (0.964 + 0.263i)24-s + (−0.449 − 0.893i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.603 − 0.797i)2-s + (−0.994 + 0.104i)3-s + (−0.272 − 0.962i)4-s + (−0.516 + 0.856i)6-s + (−0.931 − 0.362i)8-s + (0.978 − 0.207i)9-s + (0.371 + 0.928i)12-s + (0.441 − 0.897i)13-s + (−0.851 + 0.524i)16-s + (0.976 − 0.217i)17-s + (0.424 − 0.905i)18-s + (0.749 − 0.662i)19-s + (−0.971 + 0.235i)23-s + (0.964 + 0.263i)24-s + (−0.449 − 0.893i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.983 - 0.181i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.983 - 0.181i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1289384682 - 1.408856478i\)
\(L(\frac12)\) \(\approx\) \(0.1289384682 - 1.408856478i\)
\(L(1)\) \(\approx\) \(0.8192309108 - 0.6405593089i\)
\(L(1)\) \(\approx\) \(0.8192309108 - 0.6405593089i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.603 - 0.797i)T \)
3 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (0.441 - 0.897i)T \)
17 \( 1 + (0.976 - 0.217i)T \)
19 \( 1 + (0.749 - 0.662i)T \)
23 \( 1 + (-0.971 + 0.235i)T \)
29 \( 1 + (0.870 - 0.491i)T \)
31 \( 1 + (-0.640 + 0.768i)T \)
37 \( 1 + (-0.0380 - 0.999i)T \)
41 \( 1 + (-0.974 - 0.226i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + (-0.424 - 0.905i)T \)
53 \( 1 + (0.524 - 0.851i)T \)
59 \( 1 + (-0.290 + 0.956i)T \)
61 \( 1 + (-0.123 - 0.992i)T \)
67 \( 1 + (-0.189 - 0.981i)T \)
71 \( 1 + (0.993 + 0.113i)T \)
73 \( 1 + (0.132 + 0.991i)T \)
79 \( 1 + (0.935 + 0.353i)T \)
83 \( 1 + (-0.825 + 0.564i)T \)
89 \( 1 + (0.580 - 0.814i)T \)
97 \( 1 + (0.967 - 0.254i)T \)
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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

−18.48419804416593905179040672931, −17.9331659108054904064171420195, −17.07149341240381933497025661615, −16.59524095017676278655274596808, −16.112373735131992544366794838085, −15.49034427774280609820264362780, −14.57358219950238409772166592098, −13.96464272694934865360584134709, −13.353455884625645800278990801623, −12.46279456293594341824911775990, −11.98288255995185019435380265251, −11.5147260632284812931346593219, −10.490104457616492744671382137146, −9.795532174055137328101442549421, −8.924346029578058296061670200450, −7.97216647302879561429169848216, −7.47108920949703142831772276997, −6.57622278475936776479260288402, −6.10762574297280118009553528097, −5.43782624179613648551462024919, −4.72221250417891168107004789121, −3.97299770420873246108647439361, −3.30720402472850567219433952821, −2.03334621418000976464459260572, −1.063651674745042694545294603642, 0.442086935776564384089323856685, 1.16479119682401951004597185641, 2.09504797404153777529155003353, 3.213966479641704913050629800609, 3.71980320367308247178985326931, 4.70230726583218819940376839905, 5.3401284568979335535658493496, 5.806587000489018057707800102546, 6.61466123020353230228455830662, 7.45664505963104656685305604688, 8.42624729026012350413183222052, 9.50047984325901451521288081284, 10.01273049391055126880931676820, 10.63111652934192138367011814517, 11.28430878617946860373038214227, 11.94834941996090867095934174488, 12.436287775589333650137640318476, 13.13816580183268319071017939499, 13.84087792114441399825915652067, 14.52725142183349513804625984496, 15.53624729106404970721101580955, 15.81397280499358974805913911426, 16.669182188019202617773101788601, 17.6222250106290705927735710876, 18.16821732027889482303476970477

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