L-function 1-760-760.653-r1-0-0

• Label: 1-760-760.653-r1-0-0
• Degree: $1$
• Conductor: $760$
• Sign: $-0.277 + 0.960i$
• 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

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$760$    =    $2^{3} \cdot 5 \cdot 19$
Sign:	$-0.277 + 0.960i$
Analytic conductor:	$81.6733$
Root analytic conductor:	$81.6733$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{760} (653, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 760,\ (1:\ ),\ -0.277 + 0.960i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.200722709 + 1.595930472i$
$L(1)$	$\approx$	$1.240580343 + 0.3027254113i$

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	$1 + (0.866 + 0.5i)T$
	7	$1 - iT$
	11	$1 - T$
	13	$1 + (-0.866 + 0.5i)T$
	17	$1 + (0.866 + 0.5i)T$
	23	$1 + (0.866 - 0.5i)T$
	29	$1 + (-0.5 - 0.866i)T$
	31	$1 + T$
	37	$1 - iT$

Imaginary part of the first few zeros on the critical line

−21.81386383654984557793149836525, −21.03460463277668755926391157087, −20.45680605210619699487689830957, −19.355666153132614044267651596467, −18.86875829107009889304563232709, −18.12011456205094628612284498492, −17.33650889352618504873041617261, −16.01069949454811101484123551630, −15.3522924080870332221588024830, −14.64138549531333231062654975879, −13.83200363718616916684686714485, −12.712973066562831297755317557400, −12.46525229033199896506399475623, −11.319395102434398252093727595064, −10.074557383545649428959274854442, −9.37049663202854322124111180111, −8.4587028044616497907810821473, −7.69824365204344645710239406188, −6.95305346528506333746603774775, −5.63246606390315533255782078594, −4.962271193430562212332265141186, −3.36118393042038155293492321592, −2.73216448786843693509378207903, −1.833554493411624165000373028662, −0.39743628650775769226510391989, 1.17922822501620431759927976442, 2.48346437954756947158192840459, 3.28636927619505565052359224318, 4.40440407116437081582472564528, 4.980094633859345953743402380475, 6.43522212883918696264740195507, 7.63110901581602765941733828684, 7.93903463007944535195341814886, 9.167622527455673524689133155921, 10.07920045433603140153940013234, 10.4741044313210733951262107803, 11.62423687750792522244756718219, 12.89766969081935751125207703733, 13.44802432842891267455693972275, 14.39875254909777814986369478852, 14.94760911664404339600085590791, 15.93517356758533943976254596031, 16.7219232261966259618105150995, 17.38850031413762598575730562106, 18.765110178588335528694994793762, 19.22627786439771417527640679125, 20.13516815991213297833273440226, 20.91165212771226810154862594702, 21.313276226843791511417691876793, 22.414362713418213594503476498876

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