L-function 1-429-429.5-r0-0-0

• Label: 1-429-429.5-r0-0-0
• Degree: $1$
• Conductor: $429$
• Sign: $0.245 - 0.969i$
• Analytic cond.: $1.99226$
• Root an. cond.: $1.99226$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.587 − 0.809i)2^-s + (−0.309 + 0.951i)4^-s + (0.587 − 0.809i)5^-s + (0.951 + 0.309i)7^-s + (0.951 − 0.309i)8^-s − 10^-s + (−0.309 − 0.951i)14^-s + (−0.809 − 0.587i)16^-s + (−0.809 − 0.587i)17^-s + (0.951 − 0.309i)19^-s + (0.587 + 0.809i)20^-s + 23^-s + (−0.309 − 0.951i)25^-s + (−0.587 + 0.809i)28^-s + (−0.309 + 0.951i)29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign:	$0.245 - 0.969i$
Analytic conductor:	\(1.99226\)
Root analytic conductor:	\(1.99226\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{429} (5, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 429,\ (0:\ ),\ 0.245 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.9456740975 - 0.7358797502i\)
\(L(1)\)	\(\approx\)	\(0.8836566607 - 0.4208016924i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)

	$p$	$F_p(T)$
bad	3	\( 1 \)
	11	\( 1 \)
	13	\( 1 \)
good	2	\( 1 + (-0.587 - 0.809i)T \)
	5	\( 1 + (0.587 - 0.809i)T \)
	7	\( 1 + (0.951 + 0.309i)T \)
	17	\( 1 + (-0.809 - 0.587i)T \)
	19	\( 1 + (0.951 - 0.309i)T \)
	23	\( 1 + T \)
	29	\( 1 + (-0.309 + 0.951i)T \)
	31	\( 1 + (0.587 + 0.809i)T \)
	37	\( 1 + (0.951 + 0.309i)T \)

Imaginary part of the first few zeros on the critical line

−24.58652674809495003882340421275, −23.59721540388432084784380024325, −22.77711075967486432902442798437, −21.9033061183281235938168561179, −20.84658926550291614540263894687, −19.88867914206873169914557331680, −18.771563004815214665252422335446, −18.19954632212933270747385131645, −17.33640918139187964200717898046, −16.82500627964996947215370932187, −15.353562347391720851911784197656, −14.94354814853546692891111006345, −13.918129870644397359822768217411, −13.36783825464298097538328934593, −11.552185027863818794988648852924, −10.79139184121428961826766136856, −9.9695236803842742105529375840, −9.000388394009730326552334247269, −7.91824530190618134660702543086, −7.16172775526316137513846847261, −6.18276160684577440427340912891, −5.282802483447598327907110670222, −4.130366435209077984147950061443, −2.41609458488428676087776386276, −1.27650242969885662553564045594, 1.03337900738791218962056795157, 1.96263404422145288576114620870, 3.097526498329953479126327901030, 4.65376444728967822990228597822, 5.174967524093030195483405661594, 6.85893806036214361464732224522, 8.02854879994791409390631367665, 8.87146010264777833197243673518, 9.47163471936117499826888855397, 10.63589310548952809252326593210, 11.51455268104380019452541160444, 12.27570250090536767784942900888, 13.30340445571690745733576594317, 13.961653434565503169664800040609, 15.34338959444632717729096779191, 16.45586596100585948748186481978, 17.22379688041912411683061944337, 18.024166285141031279177656353441, 18.569424693819963344921606736, 20.09292816901296402613058437710, 20.25769136760221162400442110523, 21.416085724327356901319848198479, 21.755397992483376665819040294597, 22.95034152234589934418496862112, 24.201299427330306380579921211327

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