L-function 1-273-273.257-r0-0-0

• Label: 1-273-273.257-r0-0-0
• Degree: $1$
• Conductor: $273$
• Sign: $0.927 + 0.374i$
• Analytic cond.: $1.26780$
• Root an. cond.: $1.26780$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ 2^-s + 4^-s + (0.5 + 0.866i)5^-s + 8^-s + (0.5 + 0.866i)10^-s + (−0.5 − 0.866i)11^-s + 16^-s + 17^-s + (−0.5 + 0.866i)19^-s + (0.5 + 0.866i)20^-s + (−0.5 − 0.866i)22^-s − 23^-s + (−0.5 + 0.866i)25^-s + (0.5 − 0.866i)29^-s + (−0.5 + 0.866i)31^-s + 32^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$273$    =    $3 \cdot 7 \cdot 13$
Sign:	$0.927 + 0.374i$
Analytic conductor:	$1.26780$
Root analytic conductor:	$1.26780$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{273} (257, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 273,\ (0:\ ),\ 0.927 + 0.374i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.412899711 + 0.4690856261i$
$L(1)$	$\approx$	$1.974956782 + 0.2282725771i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−25.53407967226194095416395172711, −24.59455251951113730397316124458, −23.7658019639427937019436261384, −23.08173449724473522509713769878, −21.887081669207775978809410430540, −21.23152117327349309479319379040, −20.35692267887874687288325552849, −19.71242879014442997361933930165, −18.22652803898206997238574077969, −17.12790833395704030711434565822, −16.297200575215240674580167741756, −15.37367641139748903016522708838, −14.375769688522254731481911943333, −13.39933699438816453800336559494, −12.63284671413707962715637468720, −11.91315308913042173846699237760, −10.55836705851273950980576023523, −9.64496542943816392607517558898, −8.24067558884500497527961933480, −7.14514325035651326371196587205, −5.9101454143574864846547981372, −5.04458941050903129935097575068, −4.157045135444236194831868344145, −2.66003053655980546698228287846, −1.533321521405612395296018474228, 1.82687399004739372206766210842, 2.98263686421173500008818484705, 3.87261286085133974853536228782, 5.465735478564525367735596987498, 6.06475335094636524115340795234, 7.19791909295807911609061583850, 8.260687722126872890785474954269, 10.06448271404097554068994536792, 10.656030688468376769397678486621, 11.76215661410033397215342977996, 12.727606914419680789903893179673, 13.96298992612011698682471287402, 14.23261084466025287583925168551, 15.43197804425441438606480588419, 16.299977478612533703981020188369, 17.36654494330226881440316003551, 18.65098519683341697309050778088, 19.31932557706000505887934992097, 20.705186827523612767201567517120, 21.36006173180843685050487565864, 22.11539826420894403376302801031, 23.02661883102827812065755947957, 23.75233569978345764567661871490, 24.82317665637523888485973243854, 25.63037115409518782711705477057

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