LMFDB - L-function 2-4140-69.68-c1-0-18

Dirichlet series

L(s) = 1

− 5^-s + 4.35i·7^-s − 0.0533·11^-s − 4.90·13^-s − 1.18·17^-s − 5.30i·19^-s + (1.90 − 4.40i)23^-s + 25^-s + 3.23i·29^-s − 6.80·31^-s − 4.35i·35^-s − 4.86i·37^-s − 3.15i·41^-s + 10.5i·43^-s − 11.6i·47^-s + ⋯

L(s) = 1

− 0.447·5^-s + 1.64i·7^-s − 0.0160·11^-s − 1.36·13^-s − 0.286·17^-s − 1.21i·19^-s + (0.397 − 0.917i)23^-s + 0.200·25^-s + 0.600i·29^-s − 1.22·31^-s − 0.736i·35^-s − 0.799i·37^-s − 0.493i·41^-s + 1.60i·43^-s − 1.70i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4140$ = $2^{2} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$0.519 + 0.854i$
Analytic conductor:	$33.0580$
Root analytic conductor:	$5.74961$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4140} (1241, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4140,\ (\ :1/2),\ 0.519 + 0.854i)$

Particular Values

$L(1)$	$\approx$	$0.9134990207$
$L(\frac12)$	$\approx$	$0.9134990207$
$L(\frac{3}{2})$		not available
$L(1)$		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 + T $
	23	$ 1 + (-1.90 + 4.40i)T $
good	7	$ 1 - 4.35iT - 7T^{2} $
	11	$ 1 + 0.0533T + 11T^{2} $
	13	$ 1 + 4.90T + 13T^{2} $
	17	$ 1 + 1.18T + 17T^{2} $
	19	$ 1 + 5.30iT - 19T^{2} $
	29	$ 1 - 3.23iT - 29T^{2} $
	31	$ 1 + 6.80T + 31T^{2} $
	37	$ 1 + 4.86iT - 37T^{2} $
	41	$ 1 + 3.15iT - 41T^{2} $
	43	$ 1 - 10.5iT - 43T^{2} $
	47	$ 1 + 11.6iT - 47T^{2} $
	53	$ 1 - 6.20T + 53T^{2} $
	59	$ 1 - 4.11iT - 59T^{2} $
	61	$ 1 + 0.265iT - 61T^{2} $
	67	$ 1 + 12.6iT - 67T^{2} $
	71	$ 1 - 1.13iT - 71T^{2} $
	73	$ 1 - 9.67T + 73T^{2} $
	79	$ 1 + 3.25iT - 79T^{2} $
	83	$ 1 - 17.8T + 83T^{2} $
	89	$ 1 - 12.1T + 89T^{2} $
	97	$ 1 + 6.81iT - 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−8.429947882934777214699723788821, −7.52787044554240114518349608774, −6.89841807821369594878160414740, −6.08869377224799413601060437925, −5.05531298942279646226063895101, −4.89664511717913155350364716619, −3.58508321179614325258510489976, −2.59446817222188024987921735952, −2.13176970346816614049255719564, −0.31566262551497935031940796783, 0.893601690928651512912498094475, 2.05428558299146821695267913899, 3.32943064420049753652362526469, 3.94551802356460563488105890893, 4.66556635287750886138928842581, 5.47457785313247019822507715515, 6.52720530924386447950714951377, 7.35254644135624705911310398893, 7.54510979108427125331937609015, 8.341334035098418015406619975611

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9134990207\)
\(L(\frac12)\)	\(\approx\)	\(0.9134990207\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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