L-function 1-760-760.293-r0-0-0

• Label: 1-760-760.293-r0-0-0
• Degree: $1$
• Conductor: $760$
• Sign: $-0.830 - 0.557i$
• Analytic cond.: $3.52942$
• Root an. cond.: $3.52942$
• 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

$\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$760$    =    $2^{3} \cdot 5 \cdot 19$
Sign:	$-0.830 - 0.557i$
Analytic conductor:	$3.52942$
Root analytic conductor:	$3.52942$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{760} (293, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 760,\ (0:\ ),\ -0.830 - 0.557i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3447775743 - 1.131553059i$
$L(1)$	$\approx$	$0.9949010368 - 0.4942926104i$

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

−22.33026406813193350872679287343, −21.88649963534907843751053324852, −21.20111218062436044961807818264, −20.28089658690331054152747110589, −19.664355224529053762315423485071, −18.67075512521047513189908557610, −18.18134655919387196170869888147, −16.96388798671680956404489160471, −15.82271475813851313468728452465, −15.59108092426760238580922894870, −14.60678693315063616080668286628, −13.87349626588912172347864954361, −12.95504916027566889200478802204, −12.105460809180730700541661582495, −11.06768868378479262480274794290, −10.09331190486967818938712577580, −9.32672405091380521601705058745, −8.6120130129740262426250508822, −7.753074336960502213704807429319, −6.8056109619202765545392292371, −5.38797667475817071553136082679, −4.81544482209350207574594191497, −3.61001932513694091777220803746, −2.53550090337567821506350258435, −2.01864929887238291002118664353, 0.4426096937966065867470371425, 1.92210209858526315697390143000, 2.74115415256602762934962404877, 3.84761255571361043606673405025, 4.70700516187075998071537101278, 6.06239057542802206819299610252, 7.08097944808003213779330645155, 7.77778036308314507684515008876, 8.41358905253800485343073147567, 9.65005082657938156062585920193, 10.28300640717824428395501217275, 11.250257006428188018083453458100, 12.60909156262785178388060126564, 12.994026270676652348680747946661, 13.89971639010086249679288467535, 14.60124093547784427349108995770, 15.46070033574133736287362914803, 16.31811405160076797243116189877, 17.49444551831663694259469893067, 17.97836700605118400041632794966, 18.99707119007710548553932069257, 19.85540133246853261839494598808, 20.22788256779962517986287911710, 21.109006203687137831355527171586, 22.003762379462338686189156182844

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