LMFDB - L-function 2-825-5.4-c3-0-41

Dirichlet series

L(s) = 1

− 3.98i·2^-s + 3i·3^-s − 7.89·4^-s + 11.9·6^-s + 12.5i·7^-s − 0.411i·8^-s − 9·9^-s − 11·11^-s − 23.6i·12^-s − 36.0i·13^-s + 50.0·14^-s − 64.8·16^-s + 39.7i·17^-s + 35.8i·18^-s + 148.·19^-s + ⋯

L(s) = 1

− 1.40i·2^-s + 0.577i·3^-s − 0.987·4^-s + 0.813·6^-s + 0.678i·7^-s − 0.0181i·8^-s − 0.333·9^-s − 0.301·11^-s − 0.569i·12^-s − 0.769i·13^-s + 0.956·14^-s − 1.01·16^-s + 0.566i·17^-s + 0.469i·18^-s + 1.78·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$825$ = $3 \cdot 5^{2} \cdot 11$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$48.6765$
Root analytic conductor:	$6.97686$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{825} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 825,\ (\ :3/2),\ 0.447 + 0.894i)$

Particular Values

$L(2)$	$\approx$	$1.886820050$
$L(\frac12)$	$\approx$	$1.886820050$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 - 3iT $
	5	$ 1 $
	11	$ 1 + 11T $
good	2	$ 1 + 3.98iT - 8T^{2} $
	7	$ 1 - 12.5iT - 343T^{2} $
	13	$ 1 + 36.0iT - 2.19e3T^{2} $
	17	$ 1 - 39.7iT - 4.91e3T^{2} $
	19	$ 1 - 148.T + 6.85e3T^{2} $
	23	$ 1 - 35.0iT - 1.21e4T^{2} $
	29	$ 1 + 88.2T + 2.43e4T^{2} $
	31	$ 1 + 166.T + 2.97e4T^{2} $
	37	$ 1 - 85.2iT - 5.06e4T^{2} $
	41	$ 1 - 329.T + 6.89e4T^{2} $
	43	$ 1 - 278. iT - 7.95e4T^{2} $
	47	$ 1 + 272. iT - 1.03e5T^{2} $
	53	$ 1 + 223. iT - 1.48e5T^{2} $
	59	$ 1 - 467.T + 2.05e5T^{2} $
	61	$ 1 - 752.T + 2.26e5T^{2} $
	67	$ 1 - 733. iT - 3.00e5T^{2} $
	71	$ 1 - 537.T + 3.57e5T^{2} $
	73	$ 1 + 397. iT - 3.89e5T^{2} $
	79	$ 1 - 1.07e3T + 4.93e5T^{2} $
	83	$ 1 + 683. iT - 5.71e5T^{2} $
	89	$ 1 - 166.T + 7.04e5T^{2} $
	97	$ 1 + 694. 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

−9.807993966665738649969142595518, −9.282749176372629978862726625527, −8.300741559875148001821090908743, −7.24273256107432825296812635116, −5.78552982553888537594691408165, −5.09886496719466505690602612810, −3.81542662959316823262937669299, −3.11903028185857041871829819008, −2.13763025337107342144291924396, −0.792406293486479651168939452229, 0.73512580128861731747000591073, 2.29742819765406902197052617667, 3.79764942528097243540096891081, 5.02752633210828710084753412813, 5.72704114743985802645264387260, 6.77803104210051312211384405987, 7.35184909413400720651458229847, 7.83234414258475663451590834541, 8.954805418902562309625107368875, 9.603250651142975161983090216394

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.886820050\)
\(L(\frac12)\)	\(\approx\)	\(1.886820050\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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