LMFDB - L-function 2-48e2-24.11-c1-0-29

Dirichlet series

L(s) = 1

+ 3.41·5^-s − 4.82i·7^-s − 2.82i·11^-s + 2.82i·13^-s − 5.41i·17^-s − 5.65·19^-s + 1.17·23^-s + 6.65·25^-s − 0.585·29^-s − 3.17i·31^-s − 16.4i·35^-s − 3.65i·37^-s + 2.58i·41^-s + 9.65·43^-s − 12.4·47^-s + ⋯

L(s) = 1

+ 1.52·5^-s − 1.82i·7^-s − 0.852i·11^-s + 0.784i·13^-s − 1.31i·17^-s − 1.29·19^-s + 0.244·23^-s + 1.33·25^-s − 0.108·29^-s − 0.569i·31^-s − 2.78i·35^-s − 0.601i·37^-s + 0.403i·41^-s + 1.47·43^-s − 1.82·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$-0.169 + 0.985i$
Analytic conductor:	$18.3975$
Root analytic conductor:	$4.28923$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2304} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2304,\ (\ :1/2),\ -0.169 + 0.985i)$

Particular Values

$L(1)$	$\approx$	$2.089545528$
$L(\frac12)$	$\approx$	$2.089545528$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
good	5	$ 1 - 3.41T + 5T^{2} $
	7	$ 1 + 4.82iT - 7T^{2} $
	11	$ 1 + 2.82iT - 11T^{2} $
	13	$ 1 - 2.82iT - 13T^{2} $
	17	$ 1 + 5.41iT - 17T^{2} $
	19	$ 1 + 5.65T + 19T^{2} $
	23	$ 1 - 1.17T + 23T^{2} $
	29	$ 1 + 0.585T + 29T^{2} $
	31	$ 1 + 3.17iT - 31T^{2} $
	37	$ 1 + 3.65iT - 37T^{2} $
	41	$ 1 - 2.58iT - 41T^{2} $
	43	$ 1 - 9.65T + 43T^{2} $
	47	$ 1 + 12.4T + 47T^{2} $
	53	$ 1 + 5.07T + 53T^{2} $
	59	$ 1 - 2.34iT - 59T^{2} $
	61	$ 1 - 7.65iT - 61T^{2} $
	67	$ 1 - 12T + 67T^{2} $
	71	$ 1 + 4.48T + 71T^{2} $
	73	$ 1 + 4T + 73T^{2} $
	79	$ 1 - 6.48iT - 79T^{2} $
	83	$ 1 - 5.17iT - 83T^{2} $
	89	$ 1 - 12.2iT - 89T^{2} $
	97	$ 1 - 13.6T + 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

−8.989381313536108481470657863133, −7.985181640840948455460246837401, −7.04456878691551356019486277994, −6.54326021304858408639042391064, −5.75342954271676378654892243713, −4.74069361002316984864468349053, −4.01228376974065599760273592070, −2.85406355455886356601623612481, −1.77452068044452813381761043006, −0.67116515437534502075938333588, 1.79497165810668352076306835982, 2.19088929907138344121106205761, 3.21068923856874190041442762923, 4.71622992510796016796297414721, 5.38890400458878353731772284274, 6.16905849183434429478258517947, 6.44230384417422423822698072962, 7.86491495852013596117845120668, 8.675903643959334782610057154929, 9.129045666977687303711072572400

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(2.089545528\)
\(L(\frac12)\)	\(\approx\)	\(2.089545528\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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