LMFDB - L-function 2-2016-28.27-c1-0-30

Dirichlet series

L(s) = 1

− 1.69i·5^-s + (2.13 + 1.56i)7^-s − 0.794i·11^-s − 1.87i·13^-s − 4.34i·17^-s − 2.39·19^-s − 3.62i·23^-s + 2.12·25^-s − 4.41·29^-s + 1.87·31^-s + (2.64 − 3.62i)35^-s − 3.12·37^-s − 4.34i·41^-s + 0.876i·43^-s + 12.0·47^-s + ⋯

L(s) = 1

− 0.758i·5^-s + (0.807 + 0.590i)7^-s − 0.239i·11^-s − 0.519i·13^-s − 1.05i·17^-s − 0.550·19^-s − 0.755i·23^-s + 0.424·25^-s − 0.820·29^-s + 0.336·31^-s + (0.447 − 0.612i)35^-s − 0.513·37^-s − 0.678i·41^-s + 0.133i·43^-s + 1.76·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2016$ = $2^{5} \cdot 3^{2} \cdot 7$
Sign:	$0.153 + 0.988i$
Analytic conductor:	$16.0978$
Root analytic conductor:	$4.01221$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2016} (1567, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2016,\ (\ :1/2),\ 0.153 + 0.988i)$

Particular Values

$L(1)$	$\approx$	$1.683872309$
$L(\frac12)$	$\approx$	$1.683872309$
$L(\frac{3}{2})$		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 + (-2.13 - 1.56i)T $
good	5	$ 1 + 1.69iT - 5T^{2} $
	11	$ 1 + 0.794iT - 11T^{2} $
	13	$ 1 + 1.87iT - 13T^{2} $
	17	$ 1 + 4.34iT - 17T^{2} $
	19	$ 1 + 2.39T + 19T^{2} $
	23	$ 1 + 3.62iT - 23T^{2} $
	29	$ 1 + 4.41T + 29T^{2} $
	31	$ 1 - 1.87T + 31T^{2} $
	37	$ 1 + 3.12T + 37T^{2} $
	41	$ 1 + 4.34iT - 41T^{2} $
	43	$ 1 - 0.876iT - 43T^{2} $
	47	$ 1 - 12.0T + 47T^{2} $
	53	$ 1 + 10.0T + 53T^{2} $
	59	$ 1 - 6.78T + 59T^{2} $
	61	$ 1 + 11.4iT - 61T^{2} $
	67	$ 1 + 0.876iT - 67T^{2} $
	71	$ 1 + 5.21iT - 71T^{2} $
	73	$ 1 - 3.74iT - 73T^{2} $
	79	$ 1 + 3.12iT - 79T^{2} $
	83	$ 1 - 12.0T + 83T^{2} $
	89	$ 1 + 7.73iT - 89T^{2} $
	97	$ 1 - 13.3iT - 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

−8.904772722907853420953529868483, −8.302495134525258828868921339752, −7.58532999838501538860009175150, −6.58809615833167723221696336691, −5.55739060868485915205080421292, −5.02726358972204695537812062589, −4.22096602726023082144437335028, −2.94647284964337490103699605620, −1.93637997089235914505991448350, −0.62792567148257249281719905948, 1.38771590546920786593901380916, 2.38313236492968008597229512207, 3.64772469989198192890000860334, 4.30674472170655960707475775452, 5.31596189190524574791155836195, 6.28716425685146433104675256370, 7.03600408382667661392996422458, 7.68729086137919330719227569862, 8.477254294605855273049493025167, 9.310540064026711153407289939784

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.683872309\)
\(L(\frac12)\)	\(\approx\)	\(1.683872309\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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