L-function 1-1020-1020.947-r0-0-0

• Label: 1-1020-1020.947-r0-0-0
• Degree: $1$
• Conductor: $1020$
• Sign: $-0.374 - 0.927i$
• Analytic cond.: $4.73686$
• Root an. cond.: $4.73686$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.382 − 0.923i)7^-s + (−0.382 − 0.923i)11^-s + 13^-s + (−0.707 + 0.707i)19^-s + (0.923 − 0.382i)23^-s + (−0.923 − 0.382i)29^-s + (0.382 − 0.923i)31^-s + (0.923 + 0.382i)37^-s + (−0.923 + 0.382i)41^-s + (0.707 − 0.707i)43^-s − 47^-s + (−0.707 + 0.707i)49^-s + (−0.707 − 0.707i)53^-s + (−0.707 − 0.707i)59^-s + (−0.923 + 0.382i)61^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign:	$-0.374 - 0.927i$
Analytic conductor:	\(4.73686\)
Root analytic conductor:	\(4.73686\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1020} (947, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 1020,\ (0:\ ),\ -0.374 - 0.927i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.5942445070 - 0.8810543594i\)
\(L(1)\)	\(\approx\)	\(0.9133944211 - 0.2696698277i\)

Euler product

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

	$p$	$F_p(T)$
bad	2	\( 1 \)
	3	\( 1 \)
	5	\( 1 \)
	17	\( 1 \)
good	7	\( 1 + (-0.382 - 0.923i)T \)
	11	\( 1 + (-0.382 - 0.923i)T \)
	13	\( 1 + T \)
	19	\( 1 + (-0.707 + 0.707i)T \)
	23	\( 1 + (0.923 - 0.382i)T \)
	29	\( 1 + (-0.923 - 0.382i)T \)
	31	\( 1 + (0.382 - 0.923i)T \)
	37	\( 1 + (0.923 + 0.382i)T \)
	41	\( 1 + (-0.923 + 0.382i)T \)

Imaginary part of the first few zeros on the critical line

−21.67001808353322151060223318755, −21.225371243236919048485563875667, −20.27950248669581858781054651656, −19.49783240226958406774210078151, −18.630345994959384369569176086247, −18.0798369143296455437068010145, −17.21871670952316743653814410666, −16.25735886210384145744243127173, −15.39984164098659065712723271115, −15.06674124979201714316206139638, −13.89753249115665026010765253254, −12.88992751013047720016025027679, −12.5667079814298902625088390179, −11.38962550821093595111136782788, −10.75942833994883368063358683523, −9.626786718412492573466932449871, −9.01061282842135162335773072946, −8.16633709369148210971783034287, −7.08638710904117219315421398989, −6.29956354994250512418150612223, −5.37127524862246402041232849124, −4.52099372065414932800316845288, −3.33483320076789583139384226982, −2.47126848620484247376240035819, −1.41210453744725709358857871844, 0.45896676196517870542763683343, 1.63204111736145531289296927869, 3.03107580640039378186934271106, 3.74650478586234595500296707717, 4.67601239401668876736371165397, 5.94331539538782933685233217955, 6.45840028619557552896279004271, 7.62259975176291667482968124509, 8.31187630580247477060389769925, 9.25458395897290317192371266747, 10.261567365888018503688730810354, 10.907476205781060766457430341437, 11.60475324011868629320305561402, 13.00601582028952923223805602221, 13.24980637660882841496766098518, 14.168217928978957642519646132963, 15.068294174545818476936426804069, 16.00312661211943944445205777748, 16.70136598201590650309378577024, 17.222257184612708860472786389538, 18.535283267161048402284926481676, 18.85420081948035611689845049892, 19.82142286879961518814324335094, 20.76289450065067364914091083754, 21.095849317257531669214149793

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