LMFDB - L-function 2-28e2-28.27-c3-0-8

Dirichlet series

L(s) = 1

+ 4.76·3^-s − 16.8i·5^-s − 4.26·9^-s + 40.7i·11^-s + 56.7i·13^-s − 80.5i·15^-s + 122. i·17^-s − 75.4·19^-s − 106. i·23^-s − 160.·25^-s − 149.·27^-s − 146.·29^-s − 42.5·31^-s + 194. i·33^-s + 80.9·37^-s + ⋯

L(s) = 1

+ 0.917·3^-s − 1.51i·5^-s − 0.157·9^-s + 1.11i·11^-s + 1.21i·13^-s − 1.38i·15^-s + 1.75i·17^-s − 0.911·19^-s − 0.962i·23^-s − 1.28·25^-s − 1.06·27^-s − 0.939·29^-s − 0.246·31^-s + 1.02i·33^-s + 0.359·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$-0.188 - 0.981i$
Analytic conductor:	$46.2574$
Root analytic conductor:	$6.80128$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{784} (783, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 784,\ (\ :3/2),\ -0.188 - 0.981i)$

Particular Values

$L(2)$	$\approx$	$1.279906011$
$L(\frac12)$	$\approx$	$1.279906011$
$L(\frac{5}{2})$		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 $
good	3	$ 1 - 4.76T + 27T^{2} $
	5	$ 1 + 16.8iT - 125T^{2} $
	11	$ 1 - 40.7iT - 1.33e3T^{2} $
	13	$ 1 - 56.7iT - 2.19e3T^{2} $
	17	$ 1 - 122. iT - 4.91e3T^{2} $
	19	$ 1 + 75.4T + 6.85e3T^{2} $
	23	$ 1 + 106. iT - 1.21e4T^{2} $
	29	$ 1 + 146.T + 2.43e4T^{2} $
	31	$ 1 + 42.5T + 2.97e4T^{2} $
	37	$ 1 - 80.9T + 5.06e4T^{2} $
	41	$ 1 + 53.8iT - 6.89e4T^{2} $
	43	$ 1 - 341. iT - 7.95e4T^{2} $
	47	$ 1 + 4.12T + 1.03e5T^{2} $
	53	$ 1 - 279.T + 1.48e5T^{2} $
	59	$ 1 + 174.T + 2.05e5T^{2} $
	61	$ 1 - 467. iT - 2.26e5T^{2} $
	67	$ 1 + 753. iT - 3.00e5T^{2} $
	71	$ 1 - 669. iT - 3.57e5T^{2} $
	73	$ 1 - 835. iT - 3.89e5T^{2} $
	79	$ 1 - 1.10e3iT - 4.93e5T^{2} $
	83	$ 1 - 552.T + 5.71e5T^{2} $
	89	$ 1 + 122. iT - 7.04e5T^{2} $
	97	$ 1 + 291. iT - 9.12e5T^{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

−9.803833226507617573124654832421, −9.111180859058438254937467423021, −8.529964532586875851209490075393, −7.933222696033878389899772222826, −6.71394618604154203450762497512, −5.64510746384462313664310514899, −4.41239304537976798242763796314, −4.01470769417659742656898872462, −2.27025786604992749361983132143, −1.52618401748374637524819004310, 0.27793092304822410343652743435, 2.30682785699420011841301618709, 3.09098661291527423159254289493, 3.59568286404829101741115058799, 5.35482013679874747891587133742, 6.19026675032044126529929080630, 7.29731338078941002560777938587, 7.83102414560140434541637814780, 8.819130524435925428886690117091, 9.598913253126976840112047326795

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.279906011\)
\(L(\frac12)\)	\(\approx\)	\(1.279906011\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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