L-function 1-300-300.227-r1-0-0

• Label: 1-300-300.227-r1-0-0
• Degree: $1$
• Conductor: $300$
• Sign: $-0.481 - 0.876i$
• Analytic cond.: $32.2394$
• Root an. cond.: $32.2394$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− i·7^-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 + (−0.951 − 0.309i)23^-s + (−0.809 − 0.587i)29^-s + (0.809 − 0.587i)31^-s + (−0.951 + 0.309i)37^-s + (−0.309 − 0.951i)41^-s − i·43^-s + (0.587 − 0.809i)47^-s − 49^-s + (0.587 − 0.809i)53^-s + (−0.309 − 0.951i)59^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$300$    =    $2^{2} \cdot 3 \cdot 5^{2}$
Sign:	$-0.481 - 0.876i$
Analytic conductor:	$32.2394$
Root analytic conductor:	$32.2394$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{300} (227, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 300,\ (1:\ ),\ -0.481 - 0.876i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.7287349768 - 1.232223551i$
$L(1)$	$\approx$	$0.9918923711 - 0.3033339400i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	5	$1$
good	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	$1 + (0.809 - 0.587i)T$
	37	$1 + (-0.951 + 0.309i)T$

Imaginary part of the first few zeros on the critical line

−25.529874322748590609203721998902, −24.65660949564244369897319137419, −23.59215771068355355284885392738, −22.76310028379643518941561877378, −21.83874921740595259226296519866, −20.99231980102785227953684658087, −20.09089289650605384572746410463, −18.99376486997275328623408244513, −18.2434070762270591973394848871, −17.38796381544082580073647527500, −16.16127961321185845406996503937, −15.423206853575468975049666955946, −14.49748140753941866904480277882, −13.44608337850967733507871827239, −12.31229608227481125379731427153, −11.68358240374398512123709599821, −10.4545840271834610101345610676, −9.32320377791901151636869602633, −8.598477250215704909922861848138, −7.3123930146541023780913195613, −6.249263758831091230374464214243, −5.20306979334760439097267454879, −4.02166891898858807312595784202, −2.6596875478104846832711175271, −1.49968878264335317125558423657, 0.43255215067138580390200831944, 1.69700042672693662672776545161, 3.47371478448214193458317766406, 4.11165578074757123100971357866, 5.74040073014098977228741826452, 6.51196554912334982984063234768, 7.900070595354452002405857955685, 8.56131966570616164568276272575, 10.040841146439055734780463553841, 10.719375211831826437560516384101, 11.73083526288977193859543009387, 12.97375551491311973036434690033, 13.76033765840412299073321908920, 14.60945625296325636863719938569, 15.81803144255728583458092170496, 16.73001317944069673408874302230, 17.38017237526214586830746626442, 18.67874833011451171937616919809, 19.337245280291960238877596861442, 20.47622781307879110892938126175, 21.1051660772030277322165794800, 22.223130612093949018391664993926, 23.1685029510597095458103962612, 23.864715372902598706430686841766, 24.78044708017380053143247628432

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