LMFDB - L-function 2-3888-12.11-c1-0-21

Dirichlet series

L(s) = 1

− 1.73i·7^-s − 5·13^-s + 5.19i·19^-s + 5·25^-s − 1.73i·31^-s − 37^-s + 12.1i·43^-s + 4·49^-s + 14·61^-s + 3.46i·67^-s + 10·73^-s + 5.19i·79^-s + 8.66i·91^-s + 19·97^-s − 3.46i·103^-s + ⋯

L(s) = 1

− 0.654i·7^-s − 1.38·13^-s + 1.19i·19^-s + 25^-s − 0.311i·31^-s − 0.164·37^-s + 1.84i·43^-s + 0.571·49^-s + 1.79·61^-s + 0.423i·67^-s + 1.17·73^-s + 0.584i·79^-s + 0.907i·91^-s + 1.92·97^-s − 0.341i·103^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3888$ = $2^{4} \cdot 3^{5}$
Sign:	$0.866 - 0.5i$
Analytic conductor:	$31.0458$
Root analytic conductor:	$5.57187$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3888} (3887, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3888,\ (\ :1/2),\ 0.866 - 0.5i)$

Particular Values

$L(1)$	$\approx$	$1.536613077$
$L(\frac12)$	$\approx$	$1.536613077$
$L(\frac{3}{2})$		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 - 5T^{2} $
	7	$ 1 + 1.73iT - 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 + 5T + 13T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 - 5.19iT - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 1.73iT - 31T^{2} $
	37	$ 1 + T + 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 - 12.1iT - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 14T + 61T^{2} $
	67	$ 1 - 3.46iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 - 10T + 73T^{2} $
	79	$ 1 - 5.19iT - 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 - 19T + 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

−8.416963170923571541082869928268, −7.75518421533519040946674359356, −7.13668568638155373234807499260, −6.42733369651222158320728151636, −5.48699053123705789383172323760, −4.74115199057845646833395970396, −3.98941359928894195860114952354, −3.04854431429324880386763263848, −2.08926203324091003929469616784, −0.873410222619377677133362177247, 0.56888979240443523485451004707, 2.12866043508923671343335719466, 2.71409612348653969035337592244, 3.75224571957846151204971727794, 4.95142969122764262924888752986, 5.14471324328776300412958484157, 6.25723759574105145418968551534, 7.04533586242340999746666343824, 7.52575193544606295970146718479, 8.665539042875274873056750348447

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.536613077\)
\(L(\frac12)\)	\(\approx\)	\(1.536613077\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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