LMFDB - L-function 2-3744-104.77-c1-0-51

Dirichlet series

L(s) = 1

+ 3.60·13^-s − 7.21·19^-s − 5·25^-s − 10.1i·29^-s + 7.21·37^-s − 2.82i·41^-s − 5.65i·47^-s + 7·49^-s − 10.1i·53^-s − 7.21·67^-s + 11.3i·71^-s − 2·79^-s + 14.1i·89^-s − 10.1i·101^-s + 10·103^-s + ⋯

L(s) = 1

+ 1.00·13^-s − 1.65·19^-s − 25^-s − 1.89i·29^-s + 1.18·37^-s − 0.441i·41^-s − 0.825i·47^-s + 49^-s − 1.40i·53^-s − 0.880·67^-s + 1.34i·71^-s − 0.225·79^-s + 1.49i·89^-s − 1.01i·101^-s + 0.985·103^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$3744$ = $2^{5} \cdot 3^{2} \cdot 13$
Sign:	$i$
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:	$(2,\ 3744,\ (\ :1/2),\ i)$

Particular Values

$L(1)$	$\approx$	$1.317895032$
$L(\frac12)$	$\approx$	$1.317895032$
$L(\frac{3}{2})$		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 - 3.60T $
good	5	$ 1 + 5T^{2} $
	7	$ 1 - 7T^{2} $
	11	$ 1 + 11T^{2} $
	17	$ 1 + 17T^{2} $
	19	$ 1 + 7.21T + 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 + 10.1iT - 29T^{2} $
	31	$ 1 - 31T^{2} $
	37	$ 1 - 7.21T + 37T^{2} $
	41	$ 1 + 2.82iT - 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 + 5.65iT - 47T^{2} $
	53	$ 1 + 10.1iT - 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 7.21T + 67T^{2} $
	71	$ 1 - 11.3iT - 71T^{2} $
	73	$ 1 - 73T^{2} $
	79	$ 1 + 2T + 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 14.1iT - 89T^{2} $
	97	$ 1 - 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

−8.320153206702214563662025298860, −7.74631524567482147413955450366, −6.71840690623019257861507404865, −6.12448496282336692352671385850, −5.48313622277366309796749764492, −4.20848283635574320689847299014, −3.95171665692815117336618055898, −2.63724801806009038893821277940, −1.80904024960721747286926365724, −0.40413327189704634228886609407, 1.17980227569556980694895937376, 2.21530496932102058297521428219, 3.27387449890024005885656502151, 4.10988291133818551480504168582, 4.81928076659010143758786599636, 5.97787557604180808054822971011, 6.26516189366389970844456633476, 7.28781432842321830906017538655, 7.972853041055862444447977043934, 8.809286916247570003123764541244

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.317895032\)
\(L(\frac12)\)	\(\approx\)	\(1.317895032\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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