LMFDB - L-function 2-3744-104.77-c1-0-7

Dirichlet series

L(s) = 1

− 2.68·5^-s − 4.15i·7^-s − 4.35·11^-s + (−3.53 + 0.726i)13^-s + 5.87·17^-s − 5.71·19^-s − 3.62·23^-s + 2.23·25^-s − 3.08i·29^-s − 9.28i·31^-s + 11.1i·35^-s + 2.69·37^-s + 11.1i·41^-s − 3.80i·43^-s + 4.91i·47^-s + ⋯

L(s) = 1

− 1.20·5^-s − 1.56i·7^-s − 1.31·11^-s + (−0.979 + 0.201i)13^-s + 1.42·17^-s − 1.31·19^-s − 0.756·23^-s + 0.447·25^-s − 0.573i·29^-s − 1.66i·31^-s + 1.88i·35^-s + 0.443·37^-s + 1.73i·41^-s − 0.580i·43^-s + 0.716i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3744$ = $2^{5} \cdot 3^{2} \cdot 13$
Sign:	$0.412 - 0.910i$
Analytic conductor:	$29.8959$
Root analytic conductor:	$5.46772$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3744} (1585, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3744,\ (\ :1/2),\ 0.412 - 0.910i)$

Particular Values

$L(1)$	$\approx$	$0.3556791898$
$L(\frac12)$	$\approx$	$0.3556791898$
$L(\frac{3}{2})$		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 $
	13	$ 1 + (3.53 - 0.726i)T $
good	5	$ 1 + 2.68T + 5T^{2} $
	7	$ 1 + 4.15iT - 7T^{2} $
	11	$ 1 + 4.35T + 11T^{2} $
	17	$ 1 - 5.87T + 17T^{2} $
	19	$ 1 + 5.71T + 19T^{2} $
	23	$ 1 + 3.62T + 23T^{2} $
	29	$ 1 + 3.08iT - 29T^{2} $
	31	$ 1 + 9.28iT - 31T^{2} $
	37	$ 1 - 2.69T + 37T^{2} $
	41	$ 1 - 11.1iT - 41T^{2} $
	43	$ 1 + 3.80iT - 43T^{2} $
	47	$ 1 - 4.91iT - 47T^{2} $
	53	$ 1 - 1.17iT - 53T^{2} $
	59	$ 1 + 2.29T + 59T^{2} $
	61	$ 1 + 7.05iT - 61T^{2} $
	67	$ 1 - 10.0T + 67T^{2} $
	71	$ 1 - 2.08iT - 71T^{2} $
	73	$ 1 - 13.4iT - 73T^{2} $
	79	$ 1 + 10.9T + 79T^{2} $
	83	$ 1 - 9.73T + 83T^{2} $
	89	$ 1 + 12.1iT - 89T^{2} $
	97	$ 1 - 5.13iT - 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

−8.097741990206328030471054527602, −7.917349994399650941034795652875, −7.48233320793511863517737212459, −6.60277786097768879483792371087, −5.61672662099302481456741345783, −4.52331117268904467151173174388, −4.19214357069463754439411070754, −3.31344560160848534353515429987, −2.27061737156624567550888979575, −0.70602798363387383534832986836, 0.15436511485936360270871583126, 2.02760830744348645074468119266, 2.82422346458096772464314264421, 3.58927663806281652252839198672, 4.73790557457294990129462561288, 5.32143544516393940414266226174, 5.98056543857813524977605405312, 7.11738237606814922921942523509, 7.75224204941441804568791888057, 8.358388856442276777409127836577

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3556791898\)
\(L(\frac12)\)	\(\approx\)	\(0.3556791898\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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