LMFDB - L-function 2-756-21.20-c3-0-21

Dirichlet series

L(s) = 1

− 0.0103·5^-s + (16.5 + 8.35i)7^-s − 60.1i·11^-s + 31.7i·13^-s + 91.2·17^-s + 12.6i·19^-s + 86.0i·23^-s − 124.·25^-s − 122. i·29^-s − 297. i·31^-s + (−0.170 − 0.0863i)35^-s + 159.·37^-s + 182.·41^-s − 95.7·43^-s + 43.8·47^-s + ⋯

L(s) = 1

− 0.000923·5^-s + (0.892 + 0.451i)7^-s − 1.64i·11^-s + 0.676i·13^-s + 1.30·17^-s + 0.152i·19^-s + 0.780i·23^-s − 0.999·25^-s − 0.781i·29^-s − 1.72i·31^-s + (−0.000824 − 0.000416i)35^-s + 0.706·37^-s + 0.693·41^-s − 0.339·43^-s + 0.136·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$756$ = $2^{2} \cdot 3^{3} \cdot 7$
Sign:	$0.892 + 0.451i$
Analytic conductor:	$44.6054$
Root analytic conductor:	$6.67873$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{756} (377, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 756,\ (\ :3/2),\ 0.892 + 0.451i)$

Particular Values

$L(2)$	$\approx$	$2.303707738$
$L(\frac12)$	$\approx$	$2.303707738$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	7	$ 1 + (-16.5 - 8.35i)T $
good	5	$ 1 + 0.0103T + 125T^{2} $
	11	$ 1 + 60.1iT - 1.33e3T^{2} $
	13	$ 1 - 31.7iT - 2.19e3T^{2} $
	17	$ 1 - 91.2T + 4.91e3T^{2} $
	19	$ 1 - 12.6iT - 6.85e3T^{2} $
	23	$ 1 - 86.0iT - 1.21e4T^{2} $
	29	$ 1 + 122. iT - 2.43e4T^{2} $
	31	$ 1 + 297. iT - 2.97e4T^{2} $
	37	$ 1 - 159.T + 5.06e4T^{2} $
	41	$ 1 - 182.T + 6.89e4T^{2} $
	43	$ 1 + 95.7T + 7.95e4T^{2} $
	47	$ 1 - 43.8T + 1.03e5T^{2} $
	53	$ 1 - 17.6iT - 1.48e5T^{2} $
	59	$ 1 - 317.T + 2.05e5T^{2} $
	61	$ 1 - 802. iT - 2.26e5T^{2} $
	67	$ 1 + 383.T + 3.00e5T^{2} $
	71	$ 1 + 230. iT - 3.57e5T^{2} $
	73	$ 1 + 847. iT - 3.89e5T^{2} $
	79	$ 1 - 411.T + 4.93e5T^{2} $
	83	$ 1 - 1.23e3T + 5.71e5T^{2} $
	89	$ 1 - 1.54e3T + 7.04e5T^{2} $
	97	$ 1 + 1.58e3iT - 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

−9.791448655727965731903936888149, −9.020786777821194519373604737128, −8.054824872361239835202102894993, −7.60468538185179765094161870685, −5.96738239811878680827185885865, −5.71849981215616663874676916116, −4.36495226810999486674679254391, −3.35010490705191859076322155437, −2.06160900694409199505477177784, −0.77481263774223699862861326896, 1.04590672830656873244839151110, 2.16605087907073793124379399827, 3.56578873119663151060073593537, 4.69287261106190091004482272362, 5.31842810605081875786013017050, 6.65650273987977170088564060054, 7.56268904153926579945232034240, 8.067871030148309658030391892426, 9.239931995902898494743495112326, 10.17974953582296012643236188979

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 $\F_{q}$
Belyi maps

\(L(2)\)	\(\approx\)	\(2.303707738\)
\(L(\frac12)\)	\(\approx\)	\(2.303707738\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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