LMFDB - L-function 2-1792-8.3-c2-0-3

Dirichlet series

L(s) = 1

− 2.55·3^-s − 9.86i·5^-s − 2.64i·7^-s − 2.47·9^-s − 13.1·11^-s − 5.86i·13^-s + 25.2i·15^-s − 0.570·17^-s − 15.6·19^-s + 6.75i·21^-s + 16.4i·23^-s − 72.3·25^-s + 29.3·27^-s − 29.7i·29^-s + 54.8i·31^-s + ⋯

L(s) = 1

− 0.851·3^-s − 1.97i·5^-s − 0.377i·7^-s − 0.274·9^-s − 1.19·11^-s − 0.451i·13^-s + 1.68i·15^-s − 0.0335·17^-s − 0.824·19^-s + 0.321i·21^-s + 0.716i·23^-s − 2.89·25^-s + 1.08·27^-s − 1.02i·29^-s + 1.76i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1792$ = $2^{8} \cdot 7$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$48.8284$
Root analytic conductor:	$6.98773$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1792} (127, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1792,\ (\ :1),\ 0.707 - 0.707i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.1591462991$
$L(\frac12)$	$\approx$	$0.1591462991$
$L(2)$		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 + 2.64iT $
good	3	$ 1 + 2.55T + 9T^{2} $
	5	$ 1 + 9.86iT - 25T^{2} $
	11	$ 1 + 13.1T + 121T^{2} $
	13	$ 1 + 5.86iT - 169T^{2} $
	17	$ 1 + 0.570T + 289T^{2} $
	19	$ 1 + 15.6T + 361T^{2} $
	23	$ 1 - 16.4iT - 529T^{2} $
	29	$ 1 + 29.7iT - 841T^{2} $
	31	$ 1 - 54.8iT - 961T^{2} $
	37	$ 1 - 42.0iT - 1.36e3T^{2} $
	41	$ 1 + 0.773T + 1.68e3T^{2} $
	43	$ 1 + 41.7T + 1.84e3T^{2} $
	47	$ 1 + 58.4iT - 2.20e3T^{2} $
	53	$ 1 + 5.65iT - 2.80e3T^{2} $
	59	$ 1 + 42.6T + 3.48e3T^{2} $
	61	$ 1 + 95.9iT - 3.72e3T^{2} $
	67	$ 1 - 69.8T + 4.48e3T^{2} $
	71	$ 1 + 92.0iT - 5.04e3T^{2} $
	73	$ 1 + 9.97T + 5.32e3T^{2} $
	79	$ 1 - 20.1iT - 6.24e3T^{2} $
	83	$ 1 - 151.T + 6.88e3T^{2} $
	89	$ 1 + 5.79T + 7.92e3T^{2} $
	97	$ 1 - 103.T + 9.40e3T^{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

−9.119639924527825943457932361549, −8.242771774846597189846476831953, −7.993690427572302441621555348980, −6.63948621346319581961593610755, −5.70795842623471733436094692557, −5.05238773457060479527331432169, −4.69882532030620100817011758005, −3.40579603816687196859250585006, −1.86861760978213009302467036206, −0.68805362026780745509060067157, 0.06935936660550567175887235959, 2.27106362247830295978068703043, 2.77781002852378292833513458342, 3.91674954307099309046137590575, 5.10882146202701959117744760525, 6.03227861696028648022886884620, 6.40725232722158575759606704484, 7.29420728635497371603287164728, 7.994035456234957080294012934800, 9.059622837654926986029771131072

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.1591462991\)
\(L(\frac12)\)	\(\approx\)	\(0.1591462991\)
\(L(2)\)		not available
\(L(1)\)		not available

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