L-function 1-1309-1309.395-r0-0-0

• Label: 1-1309-1309.395-r0-0-0
• Degree: $1$
• Conductor: $1309$
• Sign: $0.710 - 0.704i$
• Analytic cond.: $6.07897$
• Root an. cond.: $6.07897$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.5 + 0.866i)2^-s + (0.866 − 0.5i)3^-s + (−0.5 − 0.866i)4^-s + (−0.866 − 0.5i)5^-s + i·6^-s + 8^-s + (0.5 − 0.866i)9^-s + (0.866 − 0.5i)10^-s + (−0.866 − 0.5i)12^-s + 13^-s − 15^-s + (−0.5 + 0.866i)16^-s + (0.5 + 0.866i)18^-s + (0.5 − 0.866i)19^-s + i·20^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1309$    =    $7 \cdot 11 \cdot 17$
Sign:	$0.710 - 0.704i$
Analytic conductor:	$6.07897$
Root analytic conductor:	$6.07897$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1309} (395, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1309,\ (0:\ ),\ 0.710 - 0.704i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.245997484 - 0.5130016881i$
$L(1)$	$\approx$	$0.9949595285 - 0.03918033862i$

Euler product

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

	$p$	$F_p(T)$
bad	7	$1$
	11	$1$
	17	$1$
good	2	$1 + (-0.5 + 0.866i)T$
	3	$1 + (0.866 - 0.5i)T$
	5	$1 + (-0.866 - 0.5i)T$
	13	$1 + T$
	19	$1 + (0.5 - 0.866i)T$
	23	$1 + (-0.866 - 0.5i)T$
	29	$1 + iT$
	31	$1 + (0.866 - 0.5i)T$
	37	$1 + (0.866 + 0.5i)T$

Imaginary part of the first few zeros on the critical line

−20.87983017670895400756043344744, −20.280770159529092319408478395840, −19.52752839737884019395594744987, −18.98918512083174150489343791598, −18.346773004402377509832325638945, −17.47839920483180233038305485121, −16.10503066338755438588210980231, −16.02553366457259963339756561615, −14.91244849215244952352488666504, −13.9998234851530708067174745784, −13.47578650575160234918731953110, −12.35497639083560095312493612684, −11.651443447353071193693246091352, −10.796037916194037454177916895377, −10.22051211025839405130258677667, −9.37287066947878562633153814548, −8.511638296449414678862991886349, −7.903912872647843152834352303397, −7.28733817688936766895692472375, −5.849837203554390057874341595891, −4.34079105723743800052142221926, −3.92383435333345850129932770364, −3.13285061051091237828051303438, −2.320439034634394733797898262018, −1.148897214307540121707805157539, 0.680040230215921529737736224798, 1.52793849744011152077151398352, 2.91820007256507304987487129001, 4.011547321723262371543463648515, 4.72306692267163076534228448089, 5.98411734824089349420358590857, 6.76708589226365395758812725889, 7.65507152360728498269819029999, 8.16567022155980972588310665099, 8.88364211672084488904525893831, 9.480099164365555600716442779663, 10.61763792625873172725517526317, 11.582959108175345441245328094791, 12.56976965113252878624871868995, 13.41389376477924514422881807293, 13.98329844865826860936475493663, 14.95201830865988972473598630565, 15.54703232474263706834198744642, 16.13461658379959747363627149874, 16.972781795590992326075208547693, 18.079030985921333683252261159, 18.47361718823038546191794762535, 19.284160422958225555732124605140, 20.0905770676599834572411781831, 20.34157752584007048698321145532

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