L-function 1-3040-3040.1637-r0-0-0

• Label: 1-3040-3040.1637-r0-0-0
• Degree: $1$
• Conductor: $3040$
• Sign: $0.951 + 0.309i$
• 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

+ (−0.996 − 0.0871i)3^-s + (0.5 − 0.866i)7^-s + (0.984 + 0.173i)9^-s + (−0.258 + 0.965i)11^-s + (0.0871 + 0.996i)13^-s + (0.984 − 0.173i)17^-s + (−0.573 + 0.819i)21^-s + (0.939 − 0.342i)23^-s + (−0.965 − 0.258i)27^-s + (0.573 + 0.819i)29^-s + (0.5 − 0.866i)31^-s + (0.342 − 0.939i)33^-s + (0.707 − 0.707i)37^-s − i·39^-s + (0.642 + 0.766i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3040$    =    $2^{5} \cdot 5 \cdot 19$
Sign:	$0.951 + 0.309i$
Analytic conductor:	$14.1177$
Root analytic conductor:	$14.1177$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3040} (1637, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3040,\ (0:\ ),\ 0.951 + 0.309i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.366099911 + 0.2164285033i$
$L(1)$	$\approx$	$0.9257882290 + 0.01229853447i$

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	$1 + (-0.996 - 0.0871i)T$
	7	$1 + (0.5 - 0.866i)T$
	11	$1 + (-0.258 + 0.965i)T$
	13	$1 + (0.0871 + 0.996i)T$
	17	$1 + (0.984 - 0.173i)T$
	23	$1 + (0.939 - 0.342i)T$
	29	$1 + (0.573 + 0.819i)T$
	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

−19.02680163095747854324688596359, −18.14233314227445078116436725603, −17.66089888034394831844418443378, −17.00934216624985379918551254785, −16.133087485902146096136741312331, −15.679141890453800962087159572663, −14.94836830977733197996484883152, −14.1686071327916488270423356390, −13.13420400827212249264803319690, −12.63512352991348162073398086828, −11.80483164902412427955892083730, −11.31342599417061309613210408370, −10.57479333594950563832395999311, −9.93261812220869052619211543645, −8.99577984117938705449957804467, −8.14109304288807794201827234000, −7.597469101569938241688568819673, −6.4105612772778734882348487229, −5.84059023451579814879810005047, −5.26382432296447145000471087696, −4.61232372883949888612519016708, −3.4025289167061343479680178929, −2.73974019450844701338513734260, −1.44753939737080820287795877044, −0.68246798303738429474031534006, 0.87690297121139850235908165631, 1.522116786436144236516878014653, 2.5838237367202577286249510366, 3.89551374776936370004149494595, 4.5424468611080483546635041786, 5.06322437454883626174845627890, 6.040435834372923741479480671120, 6.87966758462807243142040806211, 7.36488810186744361782584112682, 8.076770476545707254747010620509, 9.33011387357203984251030817908, 9.91038671232563662403458695601, 10.70390688192792266805479379702, 11.22889939052089829242925536996, 11.98476210520671432072192723660, 12.65553213316524331285109052282, 13.32068922759384808528413571599, 14.26611174798216624475084599088, 14.76698673123732657884288960601, 15.805274806140453918189427137787, 16.42598681307070893827802821499, 17.062707128867347638880175652, 17.52338772607333415994218373681, 18.35131062472431237754824386859, 18.846325135473928961218721760992

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