L-function 1-847-847.164-r0-0-0

• Label: 1-847-847.164-r0-0-0
• Degree: $1$
• Conductor: $847$
• Sign: $-0.647 - 0.762i$
• Analytic cond.: $3.93345$
• Root an. cond.: $3.93345$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.928 − 0.371i)2^-s + (0.5 + 0.866i)3^-s + (0.723 + 0.690i)4^-s + (−0.580 − 0.814i)5^-s + (−0.142 − 0.989i)6^-s + (−0.415 − 0.909i)8^-s + (−0.5 + 0.866i)9^-s + (0.235 + 0.971i)10^-s + (−0.235 + 0.971i)12^-s + (−0.959 + 0.281i)13^-s + (0.415 − 0.909i)15^-s + (0.0475 + 0.998i)16^-s + (−0.327 + 0.945i)17^-s + (0.786 − 0.618i)18^-s + (−0.327 − 0.945i)19^-s + (0.142 − 0.989i)20^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.647 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$847$    =    $7 \cdot 11^{2}$
Sign:	$-0.647 - 0.762i$
Analytic conductor:	$3.93345$
Root analytic conductor:	$3.93345$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{847} (164, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 847,\ (0:\ ),\ -0.647 - 0.762i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.1078188370 - 0.2330263997i$
$L(1)$	$\approx$	$0.5770590097 + 0.01113182222i$

Euler product

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

	$p$	$F_p(T)$
bad	7	$1$
	11	$1$
good	2	$1 + (-0.928 - 0.371i)T$
	3	$1 + (0.5 + 0.866i)T$
	5	$1 + (-0.580 - 0.814i)T$
	13	$1 + (-0.959 + 0.281i)T$
	17	$1 + (-0.327 + 0.945i)T$
	19	$1 + (-0.327 - 0.945i)T$
	23	$1 + (0.0475 + 0.998i)T$
	29	$1 + (0.654 - 0.755i)T$
	31	$1 + (-0.235 - 0.971i)T$

Imaginary part of the first few zeros on the critical line

−22.72952358236653308426058947526, −21.49839629569770508008903455387, −20.13510064344482895343323355832, −19.99639460403083234168702340707, −19.02280561997161481881872889528, −18.393658018272314127585834862200, −17.96291701534502687736397111332, −16.89661060241913593311649705385, −16.05771119718610625223904091649, −14.95593419367336148764629310141, −14.62971520391875712700488734438, −13.77582598502410813637806042672, −12.382679168930848642446097776783, −11.85716654811056410580317145748, −10.806481487030205140284420375569, −10.02431703623266045173897781036, −9.00431678696464853045354106725, −8.13756239600870207520887919876, −7.49771537344972611990865648838, −6.80307129586462702079765375704, −6.114820541635589968373687970068, −4.70439408032115353175941596803, −3.055863355475874145975508001974, −2.54458336498652798569037580229, −1.28226332067308417038924850594, 0.1478142974155387147168078399, 1.78921131731641096846332621, 2.73912364452291746019122152716, 3.90225320820250883127291839562, 4.50422234103927148569846586125, 5.753122642640357747220414219997, 7.23419036815125010949135403785, 7.93780240641522586729219143311, 8.81001450445389592410092090906, 9.31859291182684599568219048040, 10.1641345898354708579655702005, 11.08744028846659031055385007622, 11.79250205502490622493326037601, 12.734611455405207590298742409069, 13.61729137079524981513858670156, 15.05106852701194676702560690046, 15.43793930942507598519964287097, 16.29846991183088361490040864445, 17.09057541271000548382598038471, 17.52414422125967627815212717807, 19.12527083577754936327263016249, 19.44153071458233002595485598239, 20.11942260145236491547539496109, 20.84413661996493174298736021556, 21.63402628512394576044475157622

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