L-function 1-2793-2793.2081-r0-0-0

• Label: 1-2793-2793.2081-r0-0-0
• Degree: $1$
• Conductor: $2793$
• Sign: $-0.425 - 0.904i$
• Analytic cond.: $12.9706$
• Root an. cond.: $12.9706$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.980 − 0.198i)2^-s + (0.921 − 0.388i)4^-s + (0.583 − 0.811i)5^-s + (0.826 − 0.563i)8^-s + (0.411 − 0.911i)10^-s + (0.988 + 0.149i)11^-s + (−0.878 − 0.478i)13^-s + (0.698 − 0.715i)16^-s + (−0.921 − 0.388i)17^-s + (0.222 − 0.974i)20^-s + (0.998 − 0.0498i)22^-s + (0.124 − 0.992i)23^-s + (−0.318 − 0.947i)25^-s + (−0.955 − 0.294i)26^-s + (0.542 − 0.840i)29^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2793$    =    $3 \cdot 7^{2} \cdot 19$
Sign:	$-0.425 - 0.904i$
Analytic conductor:	$12.9706$
Root analytic conductor:	$12.9706$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2793} (2081, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2793,\ (0:\ ),\ -0.425 - 0.904i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.835892567 - 2.893083407i$
$L(1)$	$\approx$	$1.857134178 - 0.9133383995i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	7	$1$
	19	$1$
good	2	$1 + (0.980 - 0.198i)T$
	5	$1 + (0.583 - 0.811i)T$
	11	$1 + (0.988 + 0.149i)T$
	13	$1 + (-0.878 - 0.478i)T$
	17	$1 + (-0.921 - 0.388i)T$
	23	$1 + (0.124 - 0.992i)T$
	29	$1 + (0.542 - 0.840i)T$
	31	$1 - T$
	37	$1 + (-0.955 + 0.294i)T$

Imaginary part of the first few zeros on the critical line

−19.63561364580688756869195448019, −18.89070821889699879886683481793, −17.76361271798172106006991322280, −17.31996445183525166322355228734, −16.59604700353587963966930479316, −15.77046469266368128674464718445, −14.86754336318289520149688283806, −14.59523383852351029886646133060, −13.80972564544417913239675731463, −13.28994488455199719832559926999, −12.3521262277883137752645126919, −11.68376756915529601213248223953, −11.02517601284157128833899923198, −10.3145686253933571398755240571, −9.39979618321588026705449922732, −8.62027914812998411350164738207, −7.38705296537989676776452802935, −6.90218341839464790243147455512, −6.34332633468831410344522925425, −5.473150120983403706556959001613, −4.74778828127495643855423412136, −3.73816532093405150005879676270, −3.211121958543969092911697002983, −2.09993487730434020859528594590, −1.65234557725429515011076608076, 0.65476687516510098820056677481, 1.769744832825668068008648823394, 2.3497656634548874144698076864, 3.378825322238102414559522623624, 4.36186702266025603360104324358, 4.871308174308353073244162029185, 5.57310991348800448449884504472, 6.53574877666167801025346271399, 6.95648859051644941487978373503, 8.14264656306366889550322166401, 8.95121322690256904812870636957, 9.82617812107376227968891217919, 10.340814735276561886406774856545, 11.43302725154337797761103827610, 12.00648259294101747029935012758, 12.64538681267623952041014667881, 13.32412595274793556875805888941, 13.90179040933545936510984199789, 14.73853866887183078539914288390, 15.21658726200295318435512703982, 16.23341247759763672290249767458, 16.74773175624691958066656346775, 17.42155984114205248828978406538, 18.23763249836792455525994757432, 19.35185590110061592924151528327

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