LMFDB - L-function 2-1792-56.27-c1-0-52

Dirichlet series

L(s) = 1

− 3.09i·3^-s + 3.09·5^-s + (2.44 − i)7^-s − 6.58·9^-s + 5.58·11^-s − 1.80·13^-s − 9.58i·15^-s + 1.29i·17^-s + 4.38i·19^-s + (−3.09 − 7.58i)21^-s − 3.58i·23^-s + 4.58·25^-s + 11.0i·27^-s − 6i·29^-s + 1.29·31^-s + ⋯

L(s) = 1

− 1.78i·3^-s + 1.38·5^-s + (0.925 − 0.377i)7^-s − 2.19·9^-s + 1.68·11^-s − 0.500·13^-s − 2.47i·15^-s + 0.313i·17^-s + 1.00i·19^-s + (−0.675 − 1.65i)21^-s − 0.747i·23^-s + 0.916·25^-s + 2.13i·27^-s − 1.11i·29^-s + 0.232·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1792$ = $2^{8} \cdot 7$
Sign:	$-0.387 + 0.921i$
Analytic conductor:	$14.3091$
Root analytic conductor:	$3.78274$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1792} (895, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1792,\ (\ :1/2),\ -0.387 + 0.921i)$

Particular Values

$L(1)$	$\approx$	$2.592355295$
$L(\frac12)$	$\approx$	$2.592355295$
$L(\frac{3}{2})$		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 $
	7	$ 1 + (-2.44 + i)T $
good	3	$ 1 + 3.09iT - 3T^{2} $
	5	$ 1 - 3.09T + 5T^{2} $
	11	$ 1 - 5.58T + 11T^{2} $
	13	$ 1 + 1.80T + 13T^{2} $
	17	$ 1 - 1.29iT - 17T^{2} $
	19	$ 1 - 4.38iT - 19T^{2} $
	23	$ 1 + 3.58iT - 23T^{2} $
	29	$ 1 + 6iT - 29T^{2} $
	31	$ 1 - 1.29T + 31T^{2} $
	37	$ 1 - 2iT - 37T^{2} $
	41	$ 1 + 6.19iT - 41T^{2} $
	43	$ 1 + 5.58T + 43T^{2} $
	47	$ 1 - 1.29T + 47T^{2} $
	53	$ 1 - 9.16iT - 53T^{2} $
	59	$ 1 - 1.80iT - 59T^{2} $
	61	$ 1 + 6.70T + 61T^{2} $
	67	$ 1 - 13.5T + 67T^{2} $
	71	$ 1 + 9.16iT - 71T^{2} $
	73	$ 1 - 7.48iT - 73T^{2} $
	79	$ 1 - 2iT - 79T^{2} $
	83	$ 1 + 3.09iT - 83T^{2} $
	89	$ 1 - 7.48iT - 89T^{2} $
	97	$ 1 - 3.87iT - 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

−8.856712504080112973741987859318, −8.149838078417558793768808664755, −7.38284702856024214573921663818, −6.50426157153057462237778202845, −6.17258867099920791897434495711, −5.26514334881032094342806001640, −3.99563824989670693396342935338, −2.44345315210792354828808247144, −1.73462730846083983670954231484, −1.08500392096782468753226132629, 1.58111784207210820536979757737, 2.74731856434269940526238662491, 3.79781647014352589463826707893, 4.80991911922536724086867348107, 5.19124182852509179099749414303, 6.06172671302831641266540882513, 6.96644501461390730944044309982, 8.425345722716515032318972895109, 9.068763042794672171804127039307, 9.544377879917647502138680352325

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.592355295\)
\(L(\frac12)\)	\(\approx\)	\(2.592355295\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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