LMFDB - L-function 2-1344-28.27-c3-0-5

Dirichlet series

L(s) = 1

− 3·3^-s − 19.8i·5^-s + (−13.1 − 13.0i)7^-s + 9·9^-s − 11.9i·11^-s + 82.5i·13^-s + 59.4i·15^-s − 100. i·17^-s + 104.·19^-s + (39.5 + 39.0i)21^-s + 144. i·23^-s − 268.·25^-s − 27·27^-s − 92.0·29^-s − 195.·31^-s + ⋯

L(s) = 1

− 0.577·3^-s − 1.77i·5^-s + (−0.711 − 0.702i)7^-s + 0.333·9^-s − 0.327i·11^-s + 1.76i·13^-s + 1.02i·15^-s − 1.43i·17^-s + 1.25·19^-s + (0.410 + 0.405i)21^-s + 1.31i·23^-s − 2.14·25^-s − 0.192·27^-s − 0.589·29^-s − 1.13·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.702 - 0.711i$
Analytic conductor:	$79.2985$
Root analytic conductor:	$8.90497$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1344} (895, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1344,\ (\ :3/2),\ 0.702 - 0.711i)$

Particular Values

$L(2)$	$\approx$	$0.5706307957$
$L(\frac12)$	$\approx$	$0.5706307957$
$L(\frac{5}{2})$		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 + 3T $
	7	$ 1 + (13.1 + 13.0i)T $
good	5	$ 1 + 19.8iT - 125T^{2} $
	11	$ 1 + 11.9iT - 1.33e3T^{2} $
	13	$ 1 - 82.5iT - 2.19e3T^{2} $
	17	$ 1 + 100. iT - 4.91e3T^{2} $
	19	$ 1 - 104.T + 6.85e3T^{2} $
	23	$ 1 - 144. iT - 1.21e4T^{2} $
	29	$ 1 + 92.0T + 2.43e4T^{2} $
	31	$ 1 + 195.T + 2.97e4T^{2} $
	37	$ 1 + 281.T + 5.06e4T^{2} $
	41	$ 1 + 435. iT - 6.89e4T^{2} $
	43	$ 1 - 362. iT - 7.95e4T^{2} $
	47	$ 1 - 35.6T + 1.03e5T^{2} $
	53	$ 1 - 68.5T + 1.48e5T^{2} $
	59	$ 1 + 460.T + 2.05e5T^{2} $
	61	$ 1 - 443. iT - 2.26e5T^{2} $
	67	$ 1 - 867. iT - 3.00e5T^{2} $
	71	$ 1 - 577. iT - 3.57e5T^{2} $
	73	$ 1 + 590. iT - 3.89e5T^{2} $
	79	$ 1 - 282. iT - 4.93e5T^{2} $
	83	$ 1 - 816.T + 5.71e5T^{2} $
	89	$ 1 + 849. iT - 7.04e5T^{2} $
	97	$ 1 - 711. 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

−9.325979019843825148728720498724, −8.880422388649725703214803644579, −7.46179726472015190834587284389, −7.08655241010410695950432284239, −5.78987532784121924412983574789, −5.17346216491933888728340537223, −4.34283777853686306204653542349, −3.51468942100515890147499212769, −1.69496827488069616593851499834, −0.838811623866897148648674645137, 0.18180387572110853844887656059, 2.00880990714061614677892385717, 3.10813119932690651530320452046, 3.58258700911157985264570926437, 5.21871175790499482343602032366, 5.97636399786886804763701493833, 6.53463388704680621808882730102, 7.39148985481016016141945582300, 8.129638066443840358664927389023, 9.362101598951327802409439790499

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.5706307957\)
\(L(\frac12)\)	\(\approx\)	\(0.5706307957\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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