L-function 1-760-760.667-r1-0-0

• Label: 1-760-760.667-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:	$\chi_{760} (667, \cdot )$
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

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

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