LMFDB - L-function 2-84e2-21.20-c1-0-49

Dirichlet series

L(s) = 1

− 0.717·5^-s + 3i·11^-s + 2.44i·13^-s − 5.91·17^-s + 5.91i·19^-s + 4.24i·23^-s − 4.48·25^-s − 7.24i·29^-s − 9.08i·31^-s − 0.242·37^-s − 11.8·41^-s + 0.242·43^-s + 5.91·47^-s − 7.24i·53^-s − 2.15i·55^-s + ⋯

L(s) = 1

− 0.320·5^-s + 0.904i·11^-s + 0.679i·13^-s − 1.43·17^-s + 1.35i·19^-s + 0.884i·23^-s − 0.897·25^-s − 1.34i·29^-s − 1.63i·31^-s − 0.0398·37^-s − 1.84·41^-s + 0.0370·43^-s + 0.862·47^-s − 0.994i·53^-s − 0.290i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$-0.0980 + 0.995i$
Analytic conductor:	$56.3424$
Root analytic conductor:	$7.50616$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7056} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7056,\ (\ :1/2),\ -0.0980 + 0.995i)$

Particular Values

$L(1)$	$\approx$	$0.5788890273$
$L(\frac12)$	$\approx$	$0.5788890273$
$L(\frac{3}{2})$		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 $
	7	$ 1 $
good	5	$ 1 + 0.717T + 5T^{2} $
	11	$ 1 - 3iT - 11T^{2} $
	13	$ 1 - 2.44iT - 13T^{2} $
	17	$ 1 + 5.91T + 17T^{2} $
	19	$ 1 - 5.91iT - 19T^{2} $
	23	$ 1 - 4.24iT - 23T^{2} $
	29	$ 1 + 7.24iT - 29T^{2} $
	31	$ 1 + 9.08iT - 31T^{2} $
	37	$ 1 + 0.242T + 37T^{2} $
	41	$ 1 + 11.8T + 41T^{2} $
	43	$ 1 - 0.242T + 43T^{2} $
	47	$ 1 - 5.91T + 47T^{2} $
	53	$ 1 + 7.24iT - 53T^{2} $
	59	$ 1 + 8.06T + 59T^{2} $
	61	$ 1 - 1.01iT - 61T^{2} $
	67	$ 1 - 10T + 67T^{2} $
	71	$ 1 + 1.75iT - 71T^{2} $
	73	$ 1 - 1.43iT - 73T^{2} $
	79	$ 1 - 2.75T + 79T^{2} $
	83	$ 1 - 6.63T + 83T^{2} $
	89	$ 1 - 10.3T + 89T^{2} $
	97	$ 1 + 13.5iT - 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

−7.76453589063073225818866371570, −7.11768070555094674380552672389, −6.35777073938391810427100196228, −5.75381179956928472447128937431, −4.75071989638265962076183092630, −4.12930401395614195573500873948, −3.57934170893770712271425596789, −2.19171934430473069091317529684, −1.82001496355857464923156134461, −0.16502663567503521367594274302, 0.870513774538379316294797767107, 2.14603077676094581677458177926, 3.03357264331842822635424749992, 3.65911018945373437650200803292, 4.72862739998885051506780539449, 5.12023758372681768627474554002, 6.15760656576693218003371549796, 6.73892045751274329347174134811, 7.34680536411697831375336145697, 8.281589201091906526897274268462

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5788890273\)
\(L(\frac12)\)	\(\approx\)	\(0.5788890273\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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