L-function 2-1440-8.5-c1-0-17

• Label: 2-1440-8.5-c1-0-17
• Degree: $2$
• Conductor: $1440$
• Sign: $-0.707 + 0.707i$
• Analytic cond.: $11.4984$
• Root an. cond.: $3.39093$
• Motivic weight: $1$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− i·5^-s − 2·7^-s − 4i·11^-s + 6·17^-s + 4i·19^-s − 4·23^-s − 25^-s − 6i·29^-s − 10·31^-s + 2i·35^-s − 4i·37^-s − 10·41^-s − 4i·43^-s − 4·47^-s − 3·49^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}$

Invariants

Degree:	$2$
Conductor:	$1440$    =    $2^{5} \cdot 3^{2} \cdot 5$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$11.4984$
Root analytic conductor:	$3.39093$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1440} (721, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1440,\ (\ :1/2),\ -0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$0.8479143869$
$L(\frac{3}{2})$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	5	$1 + iT$
good	7	$1 + 2T + 7T^{2}$
	11	$1 + 4iT - 11T^{2}$
	13	$1 - 13T^{2}$
	17	$1 - 6T + 17T^{2}$
	19	$1 - 4iT - 19T^{2}$
	23	$1 + 4T + 23T^{2}$
	29	$1 + 6iT - 29T^{2}$
	31	$1 + 10T + 31T^{2}$
	37	$1 + 4iT - 37T^{2}$

Imaginary part of the first few zeros on the critical line

−9.354094253682609085681039595783, −8.289424434526591841867225924358, −7.86282324019680072946234411252, −6.67565302581894547538290429185, −5.81800693973327795464505440020, −5.30241617411372266717401701421, −3.77902288694413270098235946716, −3.36046617832412111145231734701, −1.81808407337671702396876871288, −0.33172000155736559573941842711, 1.63036692113692049996276112208, 2.90385070972141966395465254389, 3.69490182960377994125609730582, 4.85440100223836599974660812731, 5.72337206167562305029026070245, 6.76630783323006265674513399981, 7.24466066830627603973922874964, 8.138988632522686263166671829296, 9.255104980425820087005714032841, 9.821829431708892956964254494006

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