LMFDB - L-function 2-84e2-21.20-c1-0-52

Dirichlet series

L(s) = 1

− 3.37·5^-s + 0.828i·11^-s + 3.37i·13^-s − 1.39·17^-s − 6.75i·19^-s + 2i·23^-s + 6.41·25^-s + 4.82i·29^-s + 6.75i·31^-s − 2.58·37^-s − 8.15·41^-s + 12.4·43^-s + 6.75·47^-s + 7.07i·53^-s − 2.79i·55^-s + ⋯

L(s) = 1

− 1.51·5^-s + 0.249i·11^-s + 0.937i·13^-s − 0.339·17^-s − 1.55i·19^-s + 0.417i·23^-s + 1.28·25^-s + 0.896i·29^-s + 1.21i·31^-s − 0.425·37^-s − 1.27·41^-s + 1.90·43^-s + 0.985·47^-s + 0.971i·53^-s − 0.377i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$-0.716 + 0.698i$
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:	$(2,\ 7056,\ (\ :1/2),\ -0.716 + 0.698i)$

Particular Values

$L(1)$	$\approx$	$0.1765134530$
$L(\frac12)$	$\approx$	$0.1765134530$
$L(\frac{3}{2})$		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 + 3.37T + 5T^{2} $
	11	$ 1 - 0.828iT - 11T^{2} $
	13	$ 1 - 3.37iT - 13T^{2} $
	17	$ 1 + 1.39T + 17T^{2} $
	19	$ 1 + 6.75iT - 19T^{2} $
	23	$ 1 - 2iT - 23T^{2} $
	29	$ 1 - 4.82iT - 29T^{2} $
	31	$ 1 - 6.75iT - 31T^{2} $
	37	$ 1 + 2.58T + 37T^{2} $
	41	$ 1 + 8.15T + 41T^{2} $
	43	$ 1 - 12.4T + 43T^{2} $
	47	$ 1 - 6.75T + 47T^{2} $
	53	$ 1 - 7.07iT - 53T^{2} $
	59	$ 1 - 6.75T + 59T^{2} $
	61	$ 1 + 8.15iT - 61T^{2} $
	67	$ 1 + 8.48T + 67T^{2} $
	71	$ 1 - 4.82iT - 71T^{2} $
	73	$ 1 + 1.39iT - 73T^{2} $
	79	$ 1 - 9.65T + 79T^{2} $
	83	$ 1 + 13.5T + 83T^{2} $
	89	$ 1 + 6.17T + 89T^{2} $
	97	$ 1 - 1.39iT - 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

−7.46030339165219011867953745039, −7.12647919504995425818612047648, −6.57772722754232507757624799593, −5.39276507201394412616678059919, −4.65290829692977063682745093839, −4.13797751775257666169675402226, −3.35198298586698080503832491603, −2.51824631695324832158276186123, −1.29191832746272161862674408278, −0.05743720043484823956650769982, 0.882724409831705366553607951473, 2.26776952378465629430712287428, 3.22179582374106985687948057846, 3.94427879919725795316477955228, 4.36151740946110335605477083651, 5.48263038293373786777099821456, 6.01221287356759935072251979740, 6.99870509291527148129901174437, 7.65161473611829646923869581534, 8.133341375337232911795274007996

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.1765134530\)
\(L(\frac12)\)	\(\approx\)	\(0.1765134530\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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