L-function 1-693-693.25-r0-0-0

• Label: 1-693-693.25-r0-0-0
• Degree: $1$
• Conductor: $693$
• Sign: $-0.838 - 0.544i$
• Analytic cond.: $3.21827$
• Root an. cond.: $3.21827$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.309 − 0.951i)2^-s + (−0.809 − 0.587i)4^-s + (0.669 − 0.743i)5^-s + (−0.809 + 0.587i)8^-s + (−0.5 − 0.866i)10^-s + (0.669 + 0.743i)13^-s + (0.309 + 0.951i)16^-s + (0.669 − 0.743i)17^-s + (−0.104 − 0.994i)19^-s + (−0.978 + 0.207i)20^-s + (−0.5 − 0.866i)23^-s + (−0.104 − 0.994i)25^-s + (0.913 − 0.406i)26^-s + (−0.104 + 0.994i)29^-s + (0.309 − 0.951i)31^-s + 32^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$693$    =    $3^{2} \cdot 7 \cdot 11$
Sign:	$-0.838 - 0.544i$
Analytic conductor:	$3.21827$
Root analytic conductor:	$3.21827$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{693} (25, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 693,\ (0:\ ),\ -0.838 - 0.544i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.4602371883 - 1.553400029i$
$L(1)$	$\approx$	$0.9210599621 - 0.8599996112i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	7	$1$
	11	$1$
good	2	$1 + (0.309 - 0.951i)T$
	5	$1 + (0.669 - 0.743i)T$
	13	$1 + (0.669 + 0.743i)T$
	17	$1 + (0.669 - 0.743i)T$
	19	$1 + (-0.104 - 0.994i)T$
	23	$1 + (-0.5 - 0.866i)T$
	29	$1 + (-0.104 + 0.994i)T$
	31	$1 + (0.309 - 0.951i)T$
	37	$1 + (0.913 - 0.406i)T$

Imaginary part of the first few zeros on the critical line

−23.11879243378584474815837740441, −22.362914756227076302699014426565, −21.46968705059078138581512033310, −21.00276549203141764101005476014, −19.61904652781902580203305854759, −18.557163394726092803647846921605, −18.04562278604806749155196750084, −17.22836038672286471381853201408, −16.46243153590178100007688900897, −15.44444719009395098870516993833, −14.805508504037972636903786880865, −14.01911151561399068537091571777, −13.299002996802840634937218012690, −12.483039013817819308198551942637, −11.322455955237596821724699374498, −10.14971647243575139776028721372, −9.59761934197427663118466335018, −8.20992532610261965525445250738, −7.77066366745214187865467177483, −6.411494639770004510822786328896, −6.04426337136906226883770503128, −5.09295026848283863044015450975, −3.749050544667117934244606775838, −3.06492815036259034927933483814, −1.50652789291116463477407312514, 0.75212391732114212006095292989, 1.823896730263898838463244534140, 2.75837788700215204727244630836, 4.015707012732671534929729056189, 4.8382958071605580113611694220, 5.683776208200368273483272471453, 6.66780005886360404460354066758, 8.21840483264823412322220317465, 9.09968146186841920200925937832, 9.64329792348385102222923189595, 10.66712675643090071648712414549, 11.52983979627994544105913899878, 12.37501877969586459891839655710, 13.14851391902964557920814301070, 13.87206046022955577176252393476, 14.52810798260261329427036477052, 15.82136196602976871740915904079, 16.67770278381631715308042898304, 17.617816375648770469128319431459, 18.41544862739145292167789951852, 19.10928742089286609671308659700, 20.34091007647004835860872348685, 20.52857941467244708889720244614, 21.54589857880654408986422190530, 22.0339779820790495607477087475

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