L-function 1-3040-3040.1237-r0-0-0

• Label: 1-3040-3040.1237-r0-0-0
• Degree: $1$
• Conductor: $3040$
• Sign: $-0.727 + 0.686i$
• 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

+ (−0.573 + 0.819i)3^-s + (0.5 − 0.866i)7^-s + (−0.342 − 0.939i)9^-s + (0.258 − 0.965i)11^-s + (−0.819 + 0.573i)13^-s + (−0.342 + 0.939i)17^-s + (0.422 + 0.906i)21^-s + (−0.766 − 0.642i)23^-s + (0.965 + 0.258i)27^-s + (−0.422 + 0.906i)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

$\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3040$    =    $2^{5} \cdot 5 \cdot 19$
Sign:	$-0.727 + 0.686i$
Analytic conductor:	$14.1177$
Root analytic conductor:	$14.1177$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3040} (1237, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3040,\ (0:\ ),\ -0.727 + 0.686i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.1890796287 + 0.4756203018i$
$L(1)$	$\approx$	$0.7458281282 + 0.1297536769i$

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	$1 + (-0.573 + 0.819i)T$
	7	$1 + (0.5 - 0.866i)T$
	11	$1 + (0.258 - 0.965i)T$
	13	$1 + (-0.819 + 0.573i)T$
	17	$1 + (-0.342 + 0.939i)T$
	23	$1 + (-0.766 - 0.642i)T$
	29	$1 + (-0.422 + 0.906i)T$
	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

−18.68146146023835248098635742521, −17.8646791654897666579039555517, −17.64341948869635119626958534830, −16.95726839607162166122795278390, −15.848604968491464407196639354901, −15.41816115907319136435230339765, −14.44474363259762669861543902084, −13.924653078108996400169285179514, −12.883682434141354078439570856829, −12.43279568830151793207360644262, −11.68903827571115096973251790594, −11.362498174856485499924882982275, −10.17212560910922242618406193423, −9.61240250889735698279057937164, −8.58331377150828238237243309454, −7.88622545271469576293928179838, −7.18480083341199077605575410518, −6.56662541318221068041356771565, −5.48371875745825405639981961127, −5.170090824875778126758998314922, −4.27312526033119627782568950876, −2.89822691153425888469085481793, −2.16391720052688130061951820965, −1.55053809536564146410713938060, −0.18569392015937749427808515034, 0.98133126432505941862906808691, 2.041105482415105016415685402712, 3.290532168019750200763827615313, 3.9928428057068567347932684797, 4.587260167860610183229729912185, 5.37102357967271872896087904587, 6.27019817337013837669178886558, 6.84104073341193694994077444950, 7.90008230856148286293369508190, 8.6457374027263650393122036654, 9.38386370759479707281976551771, 10.381973087586216254850472546234, 10.582921095334964482489941487119, 11.54080673583843967898334819549, 11.95637510898181591426030707237, 12.97550185779431333756083016417, 13.872748044645229397736561688048, 14.47175674264590174146399636775, 15.057804704902910805349731734369, 15.993727770763036381967051067077, 16.599523934756311144971593201272, 17.16309777897108998989583840243, 17.56383991759388361621978970922, 18.63364188202940261075912173359, 19.29693852264783128625061361551

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