L-function 1-200-200.27-r0-0-0

• Label: 1-200-200.27-r0-0-0
• Degree: $1$
• Conductor: $200$
• Sign: $0.684 - 0.728i$
• Analytic cond.: $0.928796$
• Root an. cond.: $0.928796$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.587 + 0.809i)3^-s − i·7^-s + (−0.309 − 0.951i)9^-s + (0.309 − 0.951i)11^-s + (−0.951 + 0.309i)13^-s + (−0.587 − 0.809i)17^-s + (0.809 − 0.587i)19^-s + (0.809 + 0.587i)21^-s + (0.951 + 0.309i)23^-s + (0.951 + 0.309i)27^-s + (−0.809 − 0.587i)29^-s + (0.809 − 0.587i)31^-s + (0.587 + 0.809i)33^-s + (0.951 − 0.309i)37^-s + (0.309 − 0.951i)39^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$200$    =    $2^{3} \cdot 5^{2}$
Sign:	$0.684 - 0.728i$
Analytic conductor:	$0.928796$
Root analytic conductor:	$0.928796$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{200} (27, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 200,\ (0:\ ),\ 0.684 - 0.728i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.7494990598 - 0.3243372055i$
$L(1)$	$\approx$	$0.8304687260 - 0.05743465635i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
good	3	$1 + (-0.587 + 0.809i)T$
	7	$1 - iT$
	11	$1 + (0.309 - 0.951i)T$
	13	$1 + (-0.951 + 0.309i)T$
	17	$1 + (-0.587 - 0.809i)T$
	19	$1 + (0.809 - 0.587i)T$
	23	$1 + (0.951 + 0.309i)T$
	29	$1 + (-0.809 - 0.587i)T$
	31	$1 + (0.809 - 0.587i)T$

Imaginary part of the first few zeros on the critical line

−27.220753979245981813624124957183, −25.86947565462636305029648735202, −24.79356563396706024480838298355, −24.504484297599097546783600395652, −23.13245844161621521497110594598, −22.429793971995722475995796820308, −21.60318486494653518063936667084, −20.1589787287478950250124291358, −19.28077882439917848992380892609, −18.31335754843015045024538776094, −17.56144414896297970029363177573, −16.65338319274024167295953434091, −15.32148952885278189258944612009, −14.481152781961091831652415114204, −13.00115413750133161410627481738, −12.36101168603037293187498118467, −11.54539423657381225593249160949, −10.26278505597679500183634368067, −9.02271866689015038873717254629, −7.78357263155248175133889126777, −6.797536899328404393123585512978, −5.68839262501768797180832832204, −4.70496030233045628749839575609, −2.73228611678974902283581031659, −1.597147006499628074525741942721, 0.70751026559172448316068740264, 2.980425301631732533537491786115, 4.20866057971495749200689386749, 5.13025346000769720643410861394, 6.45119287315562127665108592309, 7.518688935377878261610904424650, 9.141419169909876477444601542895, 9.87610290403246612855779242831, 11.14511858991378970304063186473, 11.60342948753042657295983336959, 13.20535193655885946098793175189, 14.17197981948480795442401370237, 15.26611026332142210721941145644, 16.370522956943999830993579764628, 16.97127888670888521277495771374, 17.88179166315855743351899053760, 19.28454650537807388243891505326, 20.23702843946161169934120029223, 21.15825871007768280687653485886, 22.14212870494611552212872407264, 22.81759496673155289037827148339, 23.90729170720960452209067281919, 24.69637075171613885630720629988, 26.31863692532461111183515286323, 26.78966059438783086385612382274

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