L-function 1-760-760.189-r1-0-0

• Label: 1-760-760.189-r1-0-0
• Degree: $1$
• Conductor: $760$
• Sign: $1$
• Analytic cond.: $81.6733$
• Root an. cond.: $81.6733$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: yes
• Primitive: yes
• Self-dual: yes
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− 3^-s − 7^-s + 9^-s − 11^-s − 13^-s − 17^-s + 21^-s − 23^-s − 27^-s + 29^-s − 31^-s + 33^-s − 37^-s + 39^-s − 41^-s + 43^-s − 47^-s + 49^-s + 51^-s − 53^-s + 59^-s − 61^-s − 63^-s − 67^-s + 69^-s − 71^-s − 73^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign:	$1$
Analytic conductor:	\(81.6733\)
Root analytic conductor:	\(81.6733\)
Motivic weight:	\(0\)
Rational:	yes
Arithmetic:	yes
Character:	$\chi_{760} (189, \cdot )$
Primitive:	yes
Self-dual:	yes
Analytic rank:	\(0\)
Selberg data:	\((1,\ 760,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.1940667160\)
\(L(1)\)	\(\approx\)	\(0.4558301715\)

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−22.18432215020578754869144461365, −21.754762626835860721602805284237, −20.64716790513832311604197751391, −19.67888134087127380156812712304, −18.95332399330775690183931803404, −18.00247212888284023108677415469, −17.45894811985101010879317874770, −16.396107529633552132209319054958, −15.91689049062292527363933691402, −15.1509251429281546778423391439, −13.83266654155961066709529134864, −12.94134696130773909946777007026, −12.396724913759713120709422175467, −11.51093813704719103665162943609, −10.41371512977792968971054446464, −10.0341108939233834093982022763, −8.94649854679741853521001378083, −7.64155254922616779666729064856, −6.859461508817495173325301165531, −6.03926490753840902449851126109, −5.13482595553884075251814039693, −4.28149275155778167953035824748, −3.01096133806057204720642993207, −1.89308665968452178482368322707, −0.215803916150669430936475482342, 0.215803916150669430936475482342, 1.89308665968452178482368322707, 3.01096133806057204720642993207, 4.28149275155778167953035824748, 5.13482595553884075251814039693, 6.03926490753840902449851126109, 6.859461508817495173325301165531, 7.64155254922616779666729064856, 8.94649854679741853521001378083, 10.0341108939233834093982022763, 10.41371512977792968971054446464, 11.51093813704719103665162943609, 12.396724913759713120709422175467, 12.94134696130773909946777007026, 13.83266654155961066709529134864, 15.1509251429281546778423391439, 15.91689049062292527363933691402, 16.396107529633552132209319054958, 17.45894811985101010879317874770, 18.00247212888284023108677415469, 18.95332399330775690183931803404, 19.67888134087127380156812712304, 20.64716790513832311604197751391, 21.754762626835860721602805284237, 22.18432215020578754869144461365

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