LMFDB - L-function 2-1815-5.4-c1-0-65

Dirichlet series

L(s) = 1

− 0.288i·2^-s + i·3^-s + 1.91·4^-s + (−0.407 + 2.19i)5^-s + 0.288·6^-s − 3.66i·7^-s − 1.13i·8^-s − 9^-s + (0.635 + 0.117i)10^-s + 1.91i·12^-s − 4.82i·13^-s − 1.05·14^-s + (−2.19 − 0.407i)15^-s + 3.50·16^-s + 3.84i·17^-s + 0.288i·18^-s + ⋯

L(s) = 1

− 0.204i·2^-s + 0.577i·3^-s + 0.958·4^-s + (−0.182 + 0.983i)5^-s + 0.117·6^-s − 1.38i·7^-s − 0.399i·8^-s − 0.333·9^-s + (0.200 + 0.0371i)10^-s + 0.553i·12^-s − 1.33i·13^-s − 0.283·14^-s + (−0.567 − 0.105i)15^-s + 0.876·16^-s + 0.931i·17^-s + 0.0680i·18^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1815$ = $3 \cdot 5 \cdot 11^{2}$
Sign:	$0.983 + 0.182i$
Analytic conductor:	$14.4928$
Root analytic conductor:	$3.80694$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1815} (364, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1815,\ (\ :1/2),\ 0.983 + 0.182i)$

Particular Values

$L(1)$	$\approx$	$2.205677669$
$L(\frac12)$	$\approx$	$2.205677669$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 - iT $
	5	$ 1 + (0.407 - 2.19i)T $
	11	$ 1 $
good	2	$ 1 + 0.288iT - 2T^{2} $
	7	$ 1 + 3.66iT - 7T^{2} $
	13	$ 1 + 4.82iT - 13T^{2} $
	17	$ 1 - 3.84iT - 17T^{2} $
	19	$ 1 - 6.96T + 19T^{2} $
	23	$ 1 - 1.20iT - 23T^{2} $
	29	$ 1 + 4.94T + 29T^{2} $
	31	$ 1 - 7.36T + 31T^{2} $
	37	$ 1 + 2.38iT - 37T^{2} $
	41	$ 1 - 4.30T + 41T^{2} $
	43	$ 1 + 0.772iT - 43T^{2} $
	47	$ 1 + 6.47iT - 47T^{2} $
	53	$ 1 + 10.7iT - 53T^{2} $
	59	$ 1 - 14.4T + 59T^{2} $
	61	$ 1 + 1.83T + 61T^{2} $
	67	$ 1 - 6.10iT - 67T^{2} $
	71	$ 1 + 3.29T + 71T^{2} $
	73	$ 1 + 0.191iT - 73T^{2} $
	79	$ 1 - 3.15T + 79T^{2} $
	83	$ 1 + 1.53iT - 83T^{2} $
	89	$ 1 - 3.04T + 89T^{2} $
	97	$ 1 - 13.7iT - 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

−9.687952290817158510245835935833, −8.162172311639927945603499500460, −7.58013639608597322283537794338, −7.00186964324766124896815345010, −6.10351035937111089567844940389, −5.26280477043297122621414573520, −3.75945337494639223144083083353, −3.51097777950906810759715355396, −2.45964808895943751299157815485, −0.929053592976226875646730022630, 1.22264650866365560691551685666, 2.21587230567589963558775127632, 3.04998016487840279317892210308, 4.55695757270829461605259114132, 5.47211524580735234403548430980, 6.02228640910562912909379695632, 6.97957942377244897574495206556, 7.65636572138969700462581214858, 8.453254466056979766231239032397, 9.206816484704027043318213048067

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.205677669\)
\(L(\frac12)\)	\(\approx\)	\(2.205677669\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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