L-function 1-1472-1472.229-r1-0-0

• Label: 1-1472-1472.229-r1-0-0
• Degree: $1$
• Conductor: $1472$
• Sign: $-0.995 - 0.0980i$
• Analytic cond.: $158.188$
• Root an. cond.: $158.188$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.382 − 0.923i)3^-s + (0.923 + 0.382i)5^-s + (0.707 + 0.707i)7^-s + (−0.707 + 0.707i)9^-s + (−0.382 + 0.923i)11^-s + (−0.923 + 0.382i)13^-s − i·15^-s − i·17^-s + (−0.923 + 0.382i)19^-s + (0.382 − 0.923i)21^-s + (0.707 + 0.707i)25^-s + (0.923 + 0.382i)27^-s + (0.382 + 0.923i)29^-s − 31^-s + 33^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1472$    =    $2^{6} \cdot 23$
Sign:	$-0.995 - 0.0980i$
Analytic conductor:	$158.188$
Root analytic conductor:	$158.188$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1472} (229, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1472,\ (1:\ ),\ -0.995 - 0.0980i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.02283845520 + 0.4648874355i$
$L(1)$	$\approx$	$0.9194964356 + 0.08537392023i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−20.3201523039212997431240237134, −19.5647348391014363297504263589, −18.3664218339535677283841926941, −17.53459335928161277855074223173, −17.17030283636118995993513611244, −16.42739501756628818319695833284, −15.73551413219801511229927950182, −14.66481423304659249253942315679, −14.17270331634143838897444459165, −13.35272413567800470362813735527, −12.483130227920050147979774338782, −11.38692747070939436110663038525, −10.85879928904245190197591585601, −10.074676462186098595010554311086, −9.43902384674799899108390924702, −8.58711273901169158201325119931, −7.7146961154773845000707939726, −6.54759477407648719342732666362, −5.7060806652244473982193724285, −4.91832387629215200752166148766, −4.47097944561654185463754673706, −3.19709155986031735653916875475, −2.31538305486104618542891074694, −0.9541794929671753883152083415, −0.09374587100710976254812818618, 1.71126699312696083010647550173, 1.90601046817269251763498320119, 2.78531637258987443364618276139, 4.37850797204201810182828207878, 5.374103052848471940058171641413, 5.81890104059041386351189454089, 6.889493214488791761179396031109, 7.388474254706020984039169119692, 8.466094286965919946890629799691, 9.158308814101096881849975760009, 10.38640779012982185868217348869, 10.761770566035169946322696165843, 11.94754311390091813389129962420, 12.51665622014219500369315768297, 13.056867736131312013779733293971, 14.27521378814795409803796129442, 14.52925382731349795279808584012, 15.41945548962643786317873397056, 16.722543268886140338427905559010, 17.40119001336108954092383341884, 17.76088208209420214346772721235, 18.58617644801517866601627595942, 19.102842067496640487183455190231, 20.057539268323852514243568616866, 20.997213658824409776773188846231

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