L-function 1-6384-6384.1661-r0-0-0

• Label: 1-6384-6384.1661-r0-0-0
• Degree: $1$
• Conductor: $6384$
• Sign: $0.0303 + 0.999i$
• Analytic cond.: $29.6471$
• Root an. cond.: $29.6471$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.866 − 0.5i)5^-s + (−0.866 − 0.5i)11^-s + (0.866 + 0.5i)13^-s − 17^-s + 23^-s + (0.5 + 0.866i)25^-s + (−0.866 − 0.5i)29^-s + (0.5 − 0.866i)31^-s + (0.866 − 0.5i)37^-s + (0.5 + 0.866i)41^-s + (−0.866 + 0.5i)43^-s − 47^-s + (−0.866 + 0.5i)53^-s + (0.5 + 0.866i)55^-s − i·59^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0303 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$6384$    =    $2^{4} \cdot 3 \cdot 7 \cdot 19$
Sign:	$0.0303 + 0.999i$
Analytic conductor:	$29.6471$
Root analytic conductor:	$29.6471$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6384} (1661, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 6384,\ (0:\ ),\ 0.0303 + 0.999i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3881241853 + 0.4000933284i$
$L(1)$	$\approx$	$0.7758478940 - 0.05234576474i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	7	$1$
	19	$1$
good	5	$1 + (-0.866 - 0.5i)T$
	11	$1 + (-0.866 - 0.5i)T$
	13	$1 + (0.866 + 0.5i)T$
	17	$1 - T$
	23	$1 + T$
	29	$1 + (-0.866 - 0.5i)T$
	31	$1 + (0.5 - 0.866i)T$
	37	$1 + (0.866 - 0.5i)T$
	41	$1 + (0.5 + 0.866i)T$

Imaginary part of the first few zeros on the critical line

−17.61271480754093334907414461768, −16.67477537680935665083698179159, −16.003299134379012389986219871206, −15.346835505181484561499579378214, −15.13966316744803550574674631760, −14.257368656345227548119882169284, −13.383359608590662492882224332146, −12.922695722898052246152214965487, −12.23570208211723611881708112028, −11.34775790339973376342930147457, −10.90163856759028801472492218101, −10.41625501143107453882878759186, −9.49753416989401049164440679346, −8.642351001071361830841394031493, −8.14418467471177689667288054522, −7.38490830054826467856351311530, −6.82298993843668164801962398475, −6.11021261179316271385348400203, −5.09773359508332904556513298675, −4.60433893750968244226675617273, −3.65219939529606262411279455923, −3.10070909045657489000202170695, −2.33707863467245588394826067081, −1.32683657241413128598888957961, −0.17907338980214307335082347490, 0.80511452820026174405847259753, 1.71830010070338999702842470851, 2.74616720450181885183777881626, 3.406733481880296381231372874717, 4.32841051449545947656116601243, 4.68037699789548931398514863129, 5.678530039714430004150941612119, 6.31145920155069505121032812945, 7.17933034561042330992643894414, 7.88827783614777023762959561749, 8.42252018737306394692929650679, 9.05956270845546374807371817251, 9.73122383309391355844254029399, 10.79384107485130464046325735177, 11.33098257400300249526740302560, 11.57584818344890805869991540070, 12.77253217449168049027293411912, 13.14517594355992599441671444362, 13.609409248653399202126725091317, 14.75981023148546360541915940970, 15.17103657252704609183073148063, 16.001384912131070419321017845575, 16.257833432691237051190630869930, 17.00784312054505267876603027109, 17.79663211729599128509926734031

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