L-function 1-680-680.173-r0-0-0

• Label: 1-680-680.173-r0-0-0
• Degree: $1$
• Conductor: $680$
• Sign: $0.0253 - 0.999i$
• Analytic cond.: $3.15790$
• Root an. cond.: $3.15790$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.382 − 0.923i)3^-s + (−0.923 + 0.382i)7^-s + (−0.707 − 0.707i)9^-s + (0.923 − 0.382i)11^-s + 13^-s + (0.707 − 0.707i)19^-s − i·21^-s + (0.382 + 0.923i)23^-s + (−0.923 + 0.382i)27^-s + (0.382 − 0.923i)29^-s + (−0.923 − 0.382i)31^-s − i·33^-s + (−0.382 + 0.923i)37^-s + (0.382 − 0.923i)39^-s + (−0.382 − 0.923i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0253 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$680$    =    $2^{3} \cdot 5 \cdot 17$
Sign:	$0.0253 - 0.999i$
Analytic conductor:	$3.15790$
Root analytic conductor:	$3.15790$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{680} (173, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 680,\ (0:\ ),\ 0.0253 - 0.999i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.063573066 - 1.036935100i$
$L(1)$	$\approx$	$1.077920334 - 0.4430141769i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	17	$1$
good	3	$1 + (0.382 - 0.923i)T$
	7	$1 + (-0.923 + 0.382i)T$
	11	$1 + (0.923 - 0.382i)T$
	13	$1 + T$
	19	$1 + (0.707 - 0.707i)T$
	23	$1 + (0.382 + 0.923i)T$
	29	$1 + (0.382 - 0.923i)T$
	31	$1 + (-0.923 - 0.382i)T$
	37	$1 + (-0.382 + 0.923i)T$

Imaginary part of the first few zeros on the critical line

−22.742608780744832249518335966447, −22.21440778913920059788253125167, −21.274970159684185357802201451375, −20.352317803089201964530787506544, −19.936303838592230037697627078493, −19.01530594141250968019275048484, −18.03551268393171464366371280007, −16.84579954335381338752644135315, −16.32654612087276461860003591074, −15.663407792889907201490890711096, −14.5658739719539669222701740275, −14.04147907926248811409694030629, −13.011553382300510333440476720001, −12.08003855268062612908954177800, −10.90502478591197141965217623791, −10.31381803671322232900432348153, −9.289686783539868745825671448053, −8.84882420072892008417307993917, −7.591854915587268033479812847887, −6.54483536801872912777999657078, −5.632702794369386338824777495480, −4.38115668622828886049553072642, −3.6686623580599469630208040359, −2.87402228761289969255235344940, −1.323042953007801768813863532602, 0.76835104855024536708920930405, 1.924724173457023108125348986514, 3.14518874483105415189860353247, 3.74802824680753046320237347142, 5.48088727947558180760999780427, 6.314138915378437597454618318721, 6.96353599101638842920356961947, 8.03075287290665801456339657140, 9.0506367921105496544651199409, 9.44865618375920429721643872715, 10.97149132570294016311452311437, 11.80146100659898169847713476317, 12.54059643542604815430560220509, 13.60424790048205353937281486723, 13.813197424373014932270059339367, 15.16118985505565088037789122029, 15.81511267668144892905162080818, 16.9248330389792241163601215070, 17.67686906856542522394897757475, 18.72282383717853777897464232524, 19.115937285805378663633964580854, 19.9555802195576511292502592288, 20.69157230451040246129273779097, 21.88482720704507862919100342932, 22.55125369020828264059036387487

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