LMFDB - L-function 2-3744-104.77-c1-0-27

Dirichlet series

L(s) = 1

− 1.66·5^-s + 3.57i·7^-s + 1.02·11^-s + (1.87 − 3.07i)13^-s + 5.05·17^-s − 1.16·19^-s + 8.17·23^-s − 2.23·25^-s + 4.29i·29^-s − 7.98i·31^-s − 5.93i·35^-s − 9.83·37^-s − 1.62i·41^-s + 2.35i·43^-s + 7.73i·47^-s + ⋯

L(s) = 1

− 0.743·5^-s + 1.35i·7^-s + 0.309·11^-s + (0.520 − 0.853i)13^-s + 1.22·17^-s − 0.266·19^-s + 1.70·23^-s − 0.447·25^-s + 0.798i·29^-s − 1.43i·31^-s − 1.00i·35^-s − 1.61·37^-s − 0.253i·41^-s + 0.358i·43^-s + 1.12i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3744$ = $2^{5} \cdot 3^{2} \cdot 13$
Sign:	$0.759 - 0.650i$
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:	$(2,\ 3744,\ (\ :1/2),\ 0.759 - 0.650i)$

Particular Values

$L(1)$	$\approx$	$1.690673051$
$L(\frac12)$	$\approx$	$1.690673051$
$L(\frac{3}{2})$		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 + (-1.87 + 3.07i)T $
good	5	$ 1 + 1.66T + 5T^{2} $
	7	$ 1 - 3.57iT - 7T^{2} $
	11	$ 1 - 1.02T + 11T^{2} $
	17	$ 1 - 5.05T + 17T^{2} $
	19	$ 1 + 1.16T + 19T^{2} $
	23	$ 1 - 8.17T + 23T^{2} $
	29	$ 1 - 4.29iT - 29T^{2} $
	31	$ 1 + 7.98iT - 31T^{2} $
	37	$ 1 + 9.83T + 37T^{2} $
	41	$ 1 + 1.62iT - 41T^{2} $
	43	$ 1 - 2.35iT - 43T^{2} $
	47	$ 1 - 7.73iT - 47T^{2} $
	53	$ 1 + 11.2iT - 53T^{2} $
	59	$ 1 - 9.73T + 59T^{2} $
	61	$ 1 + 11.4iT - 61T^{2} $
	67	$ 1 - 7.23T + 67T^{2} $
	71	$ 1 - 10.5iT - 71T^{2} $
	73	$ 1 - 4.41iT - 73T^{2} $
	79	$ 1 - 6.94T + 79T^{2} $
	83	$ 1 - 2.29T + 83T^{2} $
	89	$ 1 - 9.02iT - 89T^{2} $
	97	$ 1 - 11.5iT - 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

−8.472842583956739678712124616853, −8.039997590049777508319800050052, −7.18702148410809813138001599009, −6.32982835448984770882712997154, −5.46815941106871076065201859963, −5.06458438880574488804810503495, −3.73574412915091951344471676120, −3.23010645435965284113120057786, −2.19386539093542436478710640580, −0.884477197281367230673619617988, 0.69990825632321635383724216256, 1.63021477994219260485274585972, 3.22011799096765414869080217929, 3.76324636030395756338200969433, 4.46405386486623401654670002054, 5.31718556628422382440566790034, 6.40749520936154516617623724622, 7.17673055269094092452710094106, 7.44345637594362224723428756054, 8.479806431177178489767937178650

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.690673051\)
\(L(\frac12)\)	\(\approx\)	\(1.690673051\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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