LMFDB - L-function 2-2736-57.56-c1-0-31

Dirichlet series

L(s) = 1

− 0.0959i·5^-s + 3.22·7^-s − 0.634i·11^-s − 1.00i·13^-s − 3.46i·17^-s + (−3.89 − 1.95i)19^-s − 6.51i·23^-s + 4.99·25^-s − 6.76·29^-s − 9.20i·31^-s − 0.309i·35^-s + 10.9i·37^-s − 10.2·41^-s + 0.906·43^-s − 6.82i·47^-s + ⋯

L(s) = 1

− 0.0429i·5^-s + 1.22·7^-s − 0.191i·11^-s − 0.279i·13^-s − 0.839i·17^-s + (−0.894 − 0.447i)19^-s − 1.35i·23^-s + 0.998·25^-s − 1.25·29^-s − 1.65i·31^-s − 0.0523i·35^-s + 1.80i·37^-s − 1.60·41^-s + 0.138·43^-s − 0.996i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$0.150 + 0.988i$
Analytic conductor:	$21.8470$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (1025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1/2),\ 0.150 + 0.988i)$

Particular Values

$L(1)$	$\approx$	$1.749833100$
$L(\frac12)$	$\approx$	$1.749833100$
$L(\frac{3}{2})$		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 $
	19	$ 1 + (3.89 + 1.95i)T $
good	5	$ 1 + 0.0959iT - 5T^{2} $
	7	$ 1 - 3.22T + 7T^{2} $
	11	$ 1 + 0.634iT - 11T^{2} $
	13	$ 1 + 1.00iT - 13T^{2} $
	17	$ 1 + 3.46iT - 17T^{2} $
	23	$ 1 + 6.51iT - 23T^{2} $
	29	$ 1 + 6.76T + 29T^{2} $
	31	$ 1 + 9.20iT - 31T^{2} $
	37	$ 1 - 10.9iT - 37T^{2} $
	41	$ 1 + 10.2T + 41T^{2} $
	43	$ 1 - 0.906T + 43T^{2} $
	47	$ 1 + 6.82iT - 47T^{2} $
	53	$ 1 - 4.18T + 53T^{2} $
	59	$ 1 - 11.4T + 59T^{2} $
	61	$ 1 - 7.41T + 61T^{2} $
	67	$ 1 + 4.28iT - 67T^{2} $
	71	$ 1 + 5.42T + 71T^{2} $
	73	$ 1 + 1.75T + 73T^{2} $
	79	$ 1 + 10.5iT - 79T^{2} $
	83	$ 1 - 3.62iT - 83T^{2} $
	89	$ 1 - 3.30T + 89T^{2} $
	97	$ 1 - 6.30iT - 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

−8.469375323080120015930048323267, −8.116382323452787967882045509119, −7.11636874688892593584123509186, −6.46692795097198451934335285379, −5.35225050748257949333902993479, −4.81716265755963311591472877501, −3.99696716185678129909656282925, −2.77163999627394706853714554938, −1.90065045273467797391352976594, −0.56784824311960904935874382823, 1.41741557647705570143685353113, 2.09220117641485757543390499536, 3.49285844945708422297380120312, 4.24345038458863360623188817100, 5.17079548314866064638406987443, 5.75731903160161310564122862239, 6.88536164605589375908002554203, 7.44505260377114733505987643185, 8.397983154081912896524471137421, 8.744974916557503009091802131805

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.749833100\)
\(L(\frac12)\)	\(\approx\)	\(1.749833100\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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