LMFDB - L-function 2-675-5.4-c3-0-5

Dirichlet series

L(s) = 1

+ 5.20i·2^-s − 19.0·4^-s + 24.4i·7^-s − 57.4i·8^-s − 28.9·11^-s + 65.3i·13^-s − 126.·14^-s + 146.·16^-s + 68.1i·17^-s − 104.·19^-s − 150. i·22^-s + 154. i·23^-s − 340.·26^-s − 464. i·28^-s + 205.·29^-s + ⋯

L(s) = 1

+ 1.83i·2^-s − 2.38·4^-s + 1.31i·7^-s − 2.53i·8^-s − 0.794·11^-s + 1.39i·13^-s − 2.42·14^-s + 2.28·16^-s + 0.972i·17^-s − 1.26·19^-s − 1.46i·22^-s + 1.40i·23^-s − 2.56·26^-s − 3.13i·28^-s + 1.31·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$675$ = $3^{3} \cdot 5^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$39.8262$
Root analytic conductor:	$6.31080$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{675} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 675,\ (\ :3/2),\ 0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$0.5939780254$
$L(\frac12)$	$\approx$	$0.5939780254$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 $
good	2	$ 1 - 5.20iT - 8T^{2} $
	7	$ 1 - 24.4iT - 343T^{2} $
	11	$ 1 + 28.9T + 1.33e3T^{2} $
	13	$ 1 - 65.3iT - 2.19e3T^{2} $
	17	$ 1 - 68.1iT - 4.91e3T^{2} $
	19	$ 1 + 104.T + 6.85e3T^{2} $
	23	$ 1 - 154. iT - 1.21e4T^{2} $
	29	$ 1 - 205.T + 2.43e4T^{2} $
	31	$ 1 + 18.2T + 2.97e4T^{2} $
	37	$ 1 + 337. iT - 5.06e4T^{2} $
	41	$ 1 + 195.T + 6.89e4T^{2} $
	43	$ 1 + 334. iT - 7.95e4T^{2} $
	47	$ 1 + 5.00iT - 1.03e5T^{2} $
	53	$ 1 - 319. iT - 1.48e5T^{2} $
	59	$ 1 + 430.T + 2.05e5T^{2} $
	61	$ 1 - 594.T + 2.26e5T^{2} $
	67	$ 1 - 195. iT - 3.00e5T^{2} $
	71	$ 1 - 425.T + 3.57e5T^{2} $
	73	$ 1 + 929. iT - 3.89e5T^{2} $
	79	$ 1 + 24.4T + 4.93e5T^{2} $
	83	$ 1 - 545. iT - 5.71e5T^{2} $
	89	$ 1 - 84.1T + 7.04e5T^{2} $
	97	$ 1 - 827. iT - 9.12e5T^{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

−10.69082772970058549305236857753, −9.526827462673495767294920467716, −8.807196798815046496242127271612, −8.317180492262126466353327027812, −7.30742308634950985235320591166, −6.39809765747283885470274749351, −5.75450931492486003774020820337, −4.92251823871788678271149017096, −3.89088142332429302203860439398, −2.09991871873655084352031644837, 0.20604686625198852874512881354, 0.969093584635709347704566490879, 2.50360962358946449610175176023, 3.25187007387353048281582502364, 4.42341650003283301003544025171, 5.01968882865059807055323890372, 6.62651635333911371211059839358, 7.974419144598456965370932301748, 8.547819293126021313861241693393, 9.923788457087067410563906709865

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.5939780254\)
\(L(\frac12)\)	\(\approx\)	\(0.5939780254\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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