LMFDB - L-function 2-1332-37.36-c1-0-12

Dirichlet series

L(s) = 1

− 0.540i·5^-s − 1.23·7^-s − 2.47·11^-s + 4.57i·13^-s − 3.36i·17^-s − 1.74i·19^-s − 8.61i·23^-s + 4.70·25^-s − 7.94i·29^-s − 6.32i·31^-s + 0.667i·35^-s + (−2.23 + 5.65i)37^-s − 2·41^-s − 2.82i·43^-s − 4·47^-s + ⋯

L(s) = 1

− 0.241i·5^-s − 0.467·7^-s − 0.745·11^-s + 1.26i·13^-s − 0.817i·17^-s − 0.401i·19^-s − 1.79i·23^-s + 0.941·25^-s − 1.47i·29^-s − 1.13i·31^-s + 0.112i·35^-s + (−0.367 + 0.929i)37^-s − 0.312·41^-s − 0.431i·43^-s − 0.583·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1332$ = $2^{2} \cdot 3^{2} \cdot 37$
Sign:	$-0.367 + 0.929i$
Analytic conductor:	$10.6360$
Root analytic conductor:	$3.26129$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1332} (73, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1332,\ (\ :1/2),\ -0.367 + 0.929i)$

Particular Values

$L(1)$	$\approx$	$0.9046813766$
$L(\frac12)$	$\approx$	$0.9046813766$
$L(\frac{3}{2})$		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 $
	37	$ 1 + (2.23 - 5.65i)T $
good	5	$ 1 + 0.540iT - 5T^{2} $
	7	$ 1 + 1.23T + 7T^{2} $
	11	$ 1 + 2.47T + 11T^{2} $
	13	$ 1 - 4.57iT - 13T^{2} $
	17	$ 1 + 3.36iT - 17T^{2} $
	19	$ 1 + 1.74iT - 19T^{2} $
	23	$ 1 + 8.61iT - 23T^{2} $
	29	$ 1 + 7.94iT - 29T^{2} $
	31	$ 1 + 6.32iT - 31T^{2} $
	41	$ 1 + 2T + 41T^{2} $
	43	$ 1 + 2.82iT - 43T^{2} $
	47	$ 1 + 4T + 47T^{2} $
	53	$ 1 + 10.9T + 53T^{2} $
	59	$ 1 + 2.28iT - 59T^{2} $
	61	$ 1 + 5.65iT - 61T^{2} $
	67	$ 1 + 7.70T + 67T^{2} $
	71	$ 1 - 8.94T + 71T^{2} $
	73	$ 1 - 3.23T + 73T^{2} $
	79	$ 1 + 9.56iT - 79T^{2} $
	83	$ 1 + 1.52T + 83T^{2} $
	89	$ 1 - 8.61iT - 89T^{2} $
	97	$ 1 + 13.7iT - 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

−9.424473388955722941125594506929, −8.603283315752581888196057736665, −7.81067960927221170089615026581, −6.76097314755542205701691500724, −6.26339167351768923780838560713, −4.94224435793901610201267791082, −4.41653472149188786936337094710, −3.06837160504201529636609647518, −2.12963057233581368700800745375, −0.36975534593719310342534207823, 1.47475117229597855117306622548, 3.01875549311061907785076861559, 3.51454618197324643720917790078, 5.02245290452896366274559258139, 5.62269594312928240543189568791, 6.61430630827477067777931173623, 7.49584420544267775016767728764, 8.184170442651570584646751209999, 9.069128291146366418843371892644, 10.00531556628751875658733949915

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.9046813766\)
\(L(\frac12)\)	\(\approx\)	\(0.9046813766\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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