L-function 1-760-760.523-r1-0-0

• Label: 1-760-760.523-r1-0-0
• Degree: $1$
• Conductor: $760$
• Sign: $-0.647 - 0.761i$
• Analytic cond.: $81.6733$
• Root an. cond.: $81.6733$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.984 − 0.173i)3^-s + (0.866 − 0.5i)7^-s + (0.939 + 0.342i)9^-s + (−0.5 + 0.866i)11^-s + (−0.984 + 0.173i)13^-s + (0.342 + 0.939i)17^-s + (−0.939 + 0.342i)21^-s + (−0.642 − 0.766i)23^-s + (−0.866 − 0.5i)27^-s + (0.939 + 0.342i)29^-s + (−0.5 − 0.866i)31^-s + (0.642 − 0.766i)33^-s − i·37^-s + 39^-s + (−0.173 + 0.984i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.647 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$760$    =    $2^{3} \cdot 5 \cdot 19$
Sign:	$-0.647 - 0.761i$
Analytic conductor:	$81.6733$
Root analytic conductor:	$81.6733$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{760} (523, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 760,\ (1:\ ),\ -0.647 - 0.761i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.2241036728 - 0.4848945499i$
$L(1)$	$\approx$	$0.7300886581 - 0.05601734690i$

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	$1 + (-0.984 - 0.173i)T$
	7	$1 + (0.866 - 0.5i)T$
	11	$1 + (-0.5 + 0.866i)T$
	13	$1 + (-0.984 + 0.173i)T$
	17	$1 + (0.342 + 0.939i)T$
	23	$1 + (-0.642 - 0.766i)T$
	29	$1 + (0.939 + 0.342i)T$
	31	$1 + (-0.5 - 0.866i)T$
	37	$1 - iT$

Imaginary part of the first few zeros on the critical line

−22.36787169965423255790977564233, −21.70539893185358543704795010575, −21.18511976258059484539207486131, −20.21750331236649157600481077095, −19.0383768260606650820322565677, −18.34412334074639168126856527735, −17.635067483073328823646020011863, −16.932298989924712113462079092352, −15.963777656329836079376657225862, −15.41563875759182636799664889614, −14.325189514960366570379292836, −13.516317008263734716758461407656, −12.2255928338720736625940495362, −11.8787889897060751505950808212, −10.96060805344403562668671318366, −10.19403853157810449863082165846, −9.247266225066499744635940885349, −8.112218152147344683605519692721, −7.311717241346060259707100262030, −6.19562806880067125939912721120, −5.203005020487103109018894045735, −4.90469754265679771727329199206, −3.47537167631601707783792742358, −2.24852243002559790811767181518, −0.9772997041895131307815176315, 0.16186292515882233647105005170, 1.44152865277555904221926496314, 2.33874666334759757269366210077, 4.11666697753907482141927151840, 4.73176180825702905943342093195, 5.58841698525469953694066987105, 6.67796048083866473441271515371, 7.50143200493664000956447237674, 8.183933746074841186425519886033, 9.70526783206552936205887189227, 10.389763025467483104153300651058, 11.07955789537278037473353058271, 12.15131019756000698868241424848, 12.55954787859024527465098057595, 13.66003938243909584425908370483, 14.68463946258807901307138771567, 15.32710992569526359224456473457, 16.605890214960668645792077029774, 16.97379505462419438343379431378, 17.95571380456470881182193186514, 18.31059328994554651012089173633, 19.5622421305614451458108195021, 20.284078916535997101232077474526, 21.38804330204265357678716878691, 21.79297010324585633159547271905

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