L-function 1-740-740.43-r1-0-0

• Label: 1-740-740.43-r1-0-0
• Degree: $1$
• Conductor: $740$
• Sign: $-0.988 - 0.148i$
• Analytic cond.: $79.5240$
• Root an. cond.: $79.5240$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ i·3^-s + i·7^-s − 9^-s + 11^-s − 13^-s + 17^-s + i·19^-s − 21^-s + 23^-s − i·27^-s + i·29^-s + i·31^-s + i·33^-s − i·39^-s − 41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$740$    =    $2^{2} \cdot 5 \cdot 37$
Sign:	$-0.988 - 0.148i$
Analytic conductor:	$79.5240$
Root analytic conductor:	$79.5240$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{740} (43, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 740,\ (1:\ ),\ -0.988 - 0.148i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-0.1120803925 + 1.499131158i$
$L(1)$	$\approx$	$0.8449369730 + 0.5941933785i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−22.08096244932013913755742591011, −20.87111314471836546747659466662, −20.068312614906752998655590152665, −19.357552570226933761471118528815, −18.87309804130248876691211767648, −17.53691681152068273776539206738, −17.19353146068479221207533403936, −16.51876677897857674094247338373, −15.01785450857239328167945335639, −14.39290396123406606499323064484, −13.55499255806185664315613194646, −12.8902465376128137559539909903, −11.86734788531659039783152395247, −11.32245683794823164531230194807, −10.1200214390597804513617447609, −9.25370860239648972315461230891, −8.163089642182598414982297259, −7.235853075887480810382473611297, −6.83518528053911272263253504565, −5.68664418961849509301350604802, −4.57909364074746036323800640367, −3.451246214291847415930509428234, −2.364766734538010662175305438164, −1.17039870108390695481332124644, −0.37132901116501828285734675346, 1.41355424003937333577206220836, 2.793638084679504348389025939233, 3.53580153892092372919932111530, 4.73025195048452004963062333808, 5.43179447593760368575207717938, 6.337530967180191402494059313776, 7.58833760200925073951856026243, 8.726274269484627934072115875763, 9.29179880019005010830775140388, 10.09662808333633405580122244203, 11.01922017406355517348129772036, 12.10354975641396418516119079641, 12.39262887608183060228678124850, 14.085926601881266117263976785502, 14.630271854515294060713623059981, 15.234666957134883086715332265, 16.28083893778206818432329536087, 16.84549586220580662760683984440, 17.67865374915843596750012278008, 18.831877528853778115609898662577, 19.477482372192116231108055509569, 20.41160748762880996561482148248, 21.25977398138091182369890238419, 21.85587044701500452074294285920, 22.51015040574524961711972647015

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