LMFDB - L-function 2-84e2-21.20-c1-0-42

Dirichlet series

L(s) = 1

− 0.765·5^-s − 4i·11^-s − 4.90i·13^-s + 5.54·17^-s + 7.39i·19^-s + 5.65i·23^-s − 4.41·25^-s − 1.65i·29^-s + 4.32i·31^-s + 8.24·37^-s − 1.84·41^-s + 1.65·43^-s + 7.39·47^-s + 8.24i·53^-s + 3.06i·55^-s + ⋯

L(s) = 1

− 0.342·5^-s − 1.20i·11^-s − 1.36i·13^-s + 1.34·17^-s + 1.69i·19^-s + 1.17i·23^-s − 0.882·25^-s − 0.307i·29^-s + 0.777i·31^-s + 1.35·37^-s − 0.288·41^-s + 0.252·43^-s + 1.07·47^-s + 1.13i·53^-s + 0.412i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$0.896 + 0.442i$
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:	$(2,\ 7056,\ (\ :1/2),\ 0.896 + 0.442i)$

Particular Values

$L(1)$	$\approx$	$1.830968073$
$L(\frac12)$	$\approx$	$1.830968073$
$L(\frac{3}{2})$		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 + 0.765T + 5T^{2} $
	11	$ 1 + 4iT - 11T^{2} $
	13	$ 1 + 4.90iT - 13T^{2} $
	17	$ 1 - 5.54T + 17T^{2} $
	19	$ 1 - 7.39iT - 19T^{2} $
	23	$ 1 - 5.65iT - 23T^{2} $
	29	$ 1 + 1.65iT - 29T^{2} $
	31	$ 1 - 4.32iT - 31T^{2} $
	37	$ 1 - 8.24T + 37T^{2} $
	41	$ 1 + 1.84T + 41T^{2} $
	43	$ 1 - 1.65T + 43T^{2} $
	47	$ 1 - 7.39T + 47T^{2} $
	53	$ 1 - 8.24iT - 53T^{2} $
	59	$ 1 + 4.32T + 59T^{2} $
	61	$ 1 + 14.0iT - 61T^{2} $
	67	$ 1 + 4T + 67T^{2} $
	71	$ 1 - 13.6iT - 71T^{2} $
	73	$ 1 + 3.82iT - 73T^{2} $
	79	$ 1 + 5.65T + 79T^{2} $
	83	$ 1 - 13.5T + 83T^{2} $
	89	$ 1 - 18.6T + 89T^{2} $
	97	$ 1 - 3.82iT - 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

−7.80099855834027553199057546975, −7.60490882387467740775226618840, −6.25965737392668519040588690686, −5.70156286397414497427841211643, −5.37621996972929561676659615594, −4.08555775239379111345141509786, −3.43672586283860806773988914004, −2.95199772779435542567308858386, −1.55612585802299381375467355866, −0.65885264545920876554780655613, 0.75516726219929843711675585950, 1.98942698999685003047311575715, 2.64727009609113206772103362046, 3.78227097088213649236235442946, 4.46921218347468516664707099791, 4.93223116410704515803310288475, 6.00763858246002575748322245526, 6.66450568068588297909034292825, 7.38869960460721529446886915103, 7.75930331053075352735104644438

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.830968073\)
\(L(\frac12)\)	\(\approx\)	\(1.830968073\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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