LMFDB - L-function 2-825-5.4-c3-0-87

Dirichlet series

L(s) = 1

− i·2^-s + 3i·3^-s + 7·4^-s + 3·6^-s − 36i·7^-s − 15i·8^-s − 9·9^-s + 11·11^-s + 21i·12^-s + 2i·13^-s − 36·14^-s + 41·16^-s − 66i·17^-s + 9i·18^-s − 140·19^-s + ⋯

L(s) = 1

− 0.353i·2^-s + 0.577i·3^-s + 0.875·4^-s + 0.204·6^-s − 1.94i·7^-s − 0.662i·8^-s − 0.333·9^-s + 0.301·11^-s + 0.505i·12^-s + 0.0426i·13^-s − 0.687·14^-s + 0.640·16^-s − 0.941i·17^-s + 0.117i·18^-s − 1.69·19^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$825$ = $3 \cdot 5^{2} \cdot 11$
Sign:	$-0.894 + 0.447i$
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:	$(2,\ 825,\ (\ :3/2),\ -0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$1.491951105$
$L(\frac12)$	$\approx$	$1.491951105$
$L(\frac{5}{2})$		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 + iT - 8T^{2} $
	7	$ 1 + 36iT - 343T^{2} $
	13	$ 1 - 2iT - 2.19e3T^{2} $
	17	$ 1 + 66iT - 4.91e3T^{2} $
	19	$ 1 + 140T + 6.85e3T^{2} $
	23	$ 1 + 68iT - 1.21e4T^{2} $
	29	$ 1 + 150T + 2.43e4T^{2} $
	31	$ 1 + 128T + 2.97e4T^{2} $
	37	$ 1 - 314iT - 5.06e4T^{2} $
	41	$ 1 + 118T + 6.89e4T^{2} $
	43	$ 1 - 172iT - 7.95e4T^{2} $
	47	$ 1 - 324iT - 1.03e5T^{2} $
	53	$ 1 - 82iT - 1.48e5T^{2} $
	59	$ 1 - 740T + 2.05e5T^{2} $
	61	$ 1 - 122T + 2.26e5T^{2} $
	67	$ 1 - 124iT - 3.00e5T^{2} $
	71	$ 1 + 988T + 3.57e5T^{2} $
	73	$ 1 - 2iT - 3.89e5T^{2} $
	79	$ 1 + 1.10e3T + 4.93e5T^{2} $
	83	$ 1 + 868iT - 5.71e5T^{2} $
	89	$ 1 - 470T + 7.04e5T^{2} $
	97	$ 1 + 1.18e3iT - 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

−9.933079419289534288369554854077, −8.697097780186471783881046135554, −7.57009251904567912956849743678, −6.94354992980385881650851553884, −6.14731655937689618778257540320, −4.63082293161128468797669799173, −3.96163909793507138156104933663, −2.99155539733255144283625123537, −1.60228214691283004127041661452, −0.34370911471092556443256149036, 1.90092904012246334966189776388, 2.25616368766992110717609679843, 3.63024217310991587771042707689, 5.38237693049968110083686631504, 5.90648351793450641598580544304, 6.61288404838827388917823716989, 7.58528124903231350113213555905, 8.580385133159884591494865182172, 8.918334055496485553458204346740, 10.27974751694400137312139795956

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.491951105\)
\(L(\frac12)\)	\(\approx\)	\(1.491951105\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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