L-function 1-3040-3040.2219-r0-0-0

• Label: 1-3040-3040.2219-r0-0-0
• Degree: $1$
• Conductor: $3040$
• Sign: $0.0113 - 0.999i$
• 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

+ (0.422 − 0.906i)3^-s + (0.866 − 0.5i)7^-s + (−0.642 − 0.766i)9^-s + (0.965 − 0.258i)11^-s + (0.906 − 0.422i)13^-s + (0.766 + 0.642i)17^-s + (−0.0871 − 0.996i)21^-s + (0.984 − 0.173i)23^-s + (−0.965 + 0.258i)27^-s + (−0.0871 + 0.996i)29^-s + (−0.5 − 0.866i)31^-s + (0.173 − 0.984i)33^-s + (−0.707 − 0.707i)37^-s − i·39^-s + (−0.342 − 0.939i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0113 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3040$    =    $2^{5} \cdot 5 \cdot 19$
Sign:	$0.0113 - 0.999i$
Analytic conductor:	$14.1177$
Root analytic conductor:	$14.1177$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3040} (2219, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3040,\ (0:\ ),\ 0.0113 - 0.999i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.867046782 - 1.888309680i$
$L(1)$	$\approx$	$1.385561239 - 0.6459444930i$

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	$1 + (0.422 - 0.906i)T$
	7	$1 + (0.866 - 0.5i)T$
	11	$1 + (0.965 - 0.258i)T$
	13	$1 + (0.906 - 0.422i)T$
	17	$1 + (0.766 + 0.642i)T$
	23	$1 + (0.984 - 0.173i)T$
	29	$1 + (-0.0871 + 0.996i)T$
	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

−19.26511647979459878533737574130, −18.58091288530988558448049277558, −17.77579614351608894004270626684, −16.98634228708912912175257933716, −16.43079078087169865298215946451, −15.60891806041100574814374072945, −15.031182146756003930775195539097, −14.3323456073818108862083012262, −13.93630335986117895845317913077, −12.97241837471634936912887594136, −11.84232028816885072204397398501, −11.45972460089521449253423504269, −10.77270117117780104737939272882, −9.79051390842772141462142792366, −9.242127109106968323079977531138, −8.56990505509244123686403798597, −7.9681443811084039374430536067, −6.96597883909753170400438582489, −6.04274939767477023293907276360, −5.13744831133096087611396574688, −4.63523691585187927909698248042, −3.7076102416717865289624008435, −3.07681543542498606265058414486, −1.99317064519129095767831654708, −1.21328985830844826986103634764, 0.89047132311139894650657769556, 1.37981384062206793716624281534, 2.25440077648065726535319381146, 3.520270781733525847330426367309, 3.782084538863046297794672605123, 5.09901712539463588613538172423, 5.88172003589413907269061724822, 6.65842625485161410037520267403, 7.402041689226055525108378829763, 8.040742842841826481420328444992, 8.757818848278064236826141360667, 9.293456764762808367533609684710, 10.604389224049269771611778785856, 11.04662984254607018100660177066, 11.88415578445611577790801287884, 12.58761009028599869858460506118, 13.26768544020898966958458023377, 14.03854386493979955013747207013, 14.51077259898262264378141505804, 15.080395874086979139321986910935, 16.16001989452895468715248100608, 17.06528274628365236491986887689, 17.44036990189724819357045982526, 18.19815277027202803353565704923, 18.96691699450604117603639703362

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