LMFDB - L-function 2-3744-8.5-c1-0-28

Dirichlet series

L(s) = 1

− 0.592i·5^-s − 3.54·7^-s + 2.11i·11^-s − i·13^-s − 1.91·17^-s + 2.26i·19^-s − 1.43·23^-s + 4.64·25^-s − 0.214i·29^-s − 1.97·31^-s + 2.10i·35^-s − 3.57i·37^-s − 0.377·41^-s + 0.319i·43^-s + 9.40·47^-s + ⋯

L(s) = 1

− 0.264i·5^-s − 1.34·7^-s + 0.637i·11^-s − 0.277i·13^-s − 0.463·17^-s + 0.518i·19^-s − 0.299·23^-s + 0.929·25^-s − 0.0399i·29^-s − 0.355·31^-s + 0.355i·35^-s − 0.587i·37^-s − 0.0589·41^-s + 0.0486i·43^-s + 1.37·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3744$ = $2^{5} \cdot 3^{2} \cdot 13$
Sign:	$0.763 + 0.645i$
Analytic conductor:	$29.8959$
Root analytic conductor:	$5.46772$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3744} (1873, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3744,\ (\ :1/2),\ 0.763 + 0.645i)$

Particular Values

$L(1)$	$\approx$	$1.218262987$
$L(\frac12)$	$\approx$	$1.218262987$
$L(\frac{3}{2})$		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 + iT $
good	5	$ 1 + 0.592iT - 5T^{2} $
	7	$ 1 + 3.54T + 7T^{2} $
	11	$ 1 - 2.11iT - 11T^{2} $
	17	$ 1 + 1.91T + 17T^{2} $
	19	$ 1 - 2.26iT - 19T^{2} $
	23	$ 1 + 1.43T + 23T^{2} $
	29	$ 1 + 0.214iT - 29T^{2} $
	31	$ 1 + 1.97T + 31T^{2} $
	37	$ 1 + 3.57iT - 37T^{2} $
	41	$ 1 + 0.377T + 41T^{2} $
	43	$ 1 - 0.319iT - 43T^{2} $
	47	$ 1 - 9.40T + 47T^{2} $
	53	$ 1 + 9.40iT - 53T^{2} $
	59	$ 1 - 2.37iT - 59T^{2} $
	61	$ 1 - 10.7iT - 61T^{2} $
	67	$ 1 + 6.49iT - 67T^{2} $
	71	$ 1 - 11.0T + 71T^{2} $
	73	$ 1 + 3.57T + 73T^{2} $
	79	$ 1 + 4.55T + 79T^{2} $
	83	$ 1 + 11.3iT - 83T^{2} $
	89	$ 1 - 13.2T + 89T^{2} $
	97	$ 1 - 10.7T + 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

−8.590076450721900095786551936980, −7.56573029219509882177223081100, −6.96328658007001145481532801154, −6.22752785131258743463874276754, −5.52758112653838570936860148435, −4.57101578870649245497671745451, −3.75266310760009580430077772796, −2.93397168849234229296313183357, −1.93946223746560294761572744775, −0.50331913039915313525533151143, 0.77291279307250827121090614609, 2.31816974988674510610027321892, 3.11233636152363740009145201238, 3.81217780312528192505641464161, 4.80099075538725038233524914718, 5.77096868666143806399347314917, 6.48375384300074894731088333796, 6.92055068168873206000867447245, 7.81459001636716789002298772471, 8.800623520884090510989938261672

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.218262987\)
\(L(\frac12)\)	\(\approx\)	\(1.218262987\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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