L-function 1-3040-3040.123-r0-0-0

• Label: 1-3040-3040.123-r0-0-0
• Degree: $1$
• Conductor: $3040$
• Sign: $-0.579 + 0.814i$
• Analytic cond.: $14.1177$
• Root an. cond.: $14.1177$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.819 − 0.573i)3^-s + (0.5 + 0.866i)7^-s + (0.342 − 0.939i)9^-s + (−0.965 + 0.258i)11^-s + (−0.573 + 0.819i)13^-s + (−0.342 − 0.939i)17^-s + (0.906 + 0.422i)21^-s + (−0.766 + 0.642i)23^-s + (−0.258 − 0.965i)27^-s + (−0.906 + 0.422i)29^-s + (0.5 + 0.866i)31^-s + (−0.642 + 0.766i)33^-s + (0.707 − 0.707i)37^-s − i·39^-s + (−0.984 − 0.173i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3040$    =    $2^{5} \cdot 5 \cdot 19$
Sign:	$-0.579 + 0.814i$
Analytic conductor:	$14.1177$
Root analytic conductor:	$14.1177$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3040} (123, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3040,\ (0:\ ),\ -0.579 + 0.814i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3646407976 + 0.7070556306i$
$L(1)$	$\approx$	$1.092730682 + 0.01526790464i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	19	$1$
good	3	$1 + (0.819 - 0.573i)T$
	7	$1 + (0.5 + 0.866i)T$
	11	$1 + (-0.965 + 0.258i)T$
	13	$1 + (-0.573 + 0.819i)T$
	17	$1 + (-0.342 - 0.939i)T$
	23	$1 + (-0.766 + 0.642i)T$
	29	$1 + (-0.906 + 0.422i)T$
	31	$1 + (0.5 + 0.866i)T$
	37	$1 + (0.707 - 0.707i)T$

Imaginary part of the first few zeros on the critical line

−18.741657046409406892462215882354, −18.28790648850405036807987280895, −17.0959232456188830052130094411, −16.88114233246022415737864627962, −15.85334053703701513040794952410, −15.13429522066864044249280381451, −14.80744739236922307147823652142, −13.81370286272700179101964841894, −13.34246911166676407942056177930, −12.71940956423692340447033925046, −11.56089239448978463494503793483, −10.74143639642261153181771432909, −10.227115936404260056436984198549, −9.745139906292880497334438069070, −8.5697117451517147205717990003, −7.98044816884991669718951080772, −7.664116702546069561435615201185, −6.53449730781423349658176009825, −5.48974416296720769556735477258, −4.73894486203884558704866576217, −4.0744492705127150829351622578, −3.27231023385823442744108818754, −2.43036185560497254961889621478, −1.638350422339934958385013392168, −0.1869691592606181237126862721, 1.43957111268028537984183165601, 2.184909918631710888524745841524, 2.70689281384722676820884413798, 3.67158481101546254736409301292, 4.75524075877643946678001163836, 5.33652792955033188202913675549, 6.404438848977652858698384212083, 7.16072181033565896830974890294, 7.84094315906587646999438502238, 8.446923392866269554191496583432, 9.33406580873370745442152424258, 9.68867167935509684282644951753, 10.8515509280516992628728547103, 11.77817454257763578557870865428, 12.173029864033313113871044562676, 13.04905372098247807857305036750, 13.67523611442641163227236929809, 14.410149612350987811566165306903, 14.97306075964882205333643302486, 15.692730097187034447417520310101, 16.30663451049570443543340619286, 17.48817237006647590614148010224, 18.10632282769215175325588588430, 18.508543272690405393988671061436, 19.22349519773175628332140581944

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