L-function 1-728-728.627-r1-0-0

• Label: 1-728-728.627-r1-0-0
• Degree: $1$
• Conductor: $728$
• Sign: $0.949 - 0.313i$
• Analytic cond.: $78.2344$
• Root an. cond.: $78.2344$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ 3^-s + (0.5 − 0.866i)5^-s + 9^-s + 11^-s + (0.5 − 0.866i)15^-s + (−0.5 + 0.866i)17^-s + 19^-s + (0.5 + 0.866i)23^-s + (−0.5 − 0.866i)25^-s + 27^-s + (0.5 − 0.866i)29^-s + (0.5 + 0.866i)31^-s + 33^-s + (0.5 + 0.866i)37^-s + (−0.5 + 0.866i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$728$    =    $2^{3} \cdot 7 \cdot 13$
Sign:	$0.949 - 0.313i$
Analytic conductor:	$78.2344$
Root analytic conductor:	$78.2344$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{728} (627, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 728,\ (1:\ ),\ 0.949 - 0.313i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$4.132345428 - 0.6642529918i$
$L(1)$	$\approx$	$1.895518411 - 0.2008129936i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−22.29054443068276131506515810303, −21.5996038260292367818951668415, −20.64785344708540052096257130662, −19.99056503030891244246917512134, −19.135296837828864162646719156769, −18.39813512901564611799292693184, −17.73737393237501344792804530933, −16.598510123727063453799459328391, −15.64937374968051452466225738874, −14.795202990683810995534331573089, −14.14090959391754946935549756168, −13.637087599231907276512552220959, −12.57290759205840898212248227452, −11.49693980345364036047443486610, −10.5897055922249629831397635472, −9.54769348370674301662890789002, −9.15549986306049931196503371137, −7.96406598803877596587686421922, −7.00828189134731875011306096074, −6.453297301305768878679401004571, −5.050807308491987311024771749266, −3.91319835375859133969876751498, −2.99810407164146274450221646152, −2.234811168482889098095890390251, −1.037949062367644411465999224478, 1.073761128736046900062890964389, 1.757269777851103122339865911511, 3.00908017389289188865414666634, 4.02653210235416116933276302246, 4.86935570857464899319884692278, 6.06944222243564519116888097102, 7.0468174631225349582505778965, 8.15444222517794560615608391887, 8.83447631354467650587895888680, 9.535245478898973764357925547597, 10.279332324869489432579029960799, 11.70068388754443883784275281309, 12.441430971130005835163371146549, 13.60076420591438629906994450092, 13.70921917518267433404055628950, 14.95663034169509286613975279628, 15.58535702924486723302017801597, 16.643665860659053857046184896834, 17.342214674499793556008204055, 18.25781992986216835878091205932, 19.3386741327195409785050178888, 19.93020992448212295485726825307, 20.50695797975080073600860404412, 21.56355974317543344113636608530, 21.85690148475645920907005172458

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