LMFDB - L-function 2-60e2-15.14-c2-0-17

Dirichlet series

L(s) = 1

− 13.4i·7^-s + 17.6i·11^-s − 7.48i·13^-s + 16.9·17^-s − 10.9·19^-s − 21.9·23^-s + 47.3i·29^-s + 16.9·31^-s + 5.53i·37^-s + 66.3i·41^-s − 38.9i·43^-s − 32.5·47^-s − 132.·49^-s − 11.2·53^-s − 31.8i·59^-s + ⋯

L(s) = 1

− 1.92i·7^-s + 1.60i·11^-s − 0.575i·13^-s + 0.998·17^-s − 0.577·19^-s − 0.952·23^-s + 1.63i·29^-s + 0.547·31^-s + 0.149i·37^-s + 1.61i·41^-s − 0.906i·43^-s − 0.692·47^-s − 2.71·49^-s − 0.212·53^-s − 0.540i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3600$ = $2^{4} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.472 - 0.881i$
Analytic conductor:	$98.0928$
Root analytic conductor:	$9.90418$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3600} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3600,\ (\ :1),\ 0.472 - 0.881i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.389392821$
$L(\frac12)$	$\approx$	$1.389392821$
$L(2)$		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 $
	5	$ 1 $
good	7	$ 1 + 13.4iT - 49T^{2} $
	11	$ 1 - 17.6iT - 121T^{2} $
	13	$ 1 + 7.48iT - 169T^{2} $
	17	$ 1 - 16.9T + 289T^{2} $
	19	$ 1 + 10.9T + 361T^{2} $
	23	$ 1 + 21.9T + 529T^{2} $
	29	$ 1 - 47.3iT - 841T^{2} $
	31	$ 1 - 16.9T + 961T^{2} $
	37	$ 1 - 5.53iT - 1.36e3T^{2} $
	41	$ 1 - 66.3iT - 1.68e3T^{2} $
	43	$ 1 + 38.9iT - 1.84e3T^{2} $
	47	$ 1 + 32.5T + 2.20e3T^{2} $
	53	$ 1 + 11.2T + 2.80e3T^{2} $
	59	$ 1 + 31.8iT - 3.48e3T^{2} $
	61	$ 1 - 46.9T + 3.72e3T^{2} $
	67	$ 1 + 76iT - 4.48e3T^{2} $
	71	$ 1 - 77.7iT - 5.04e3T^{2} $
	73	$ 1 - 94.9iT - 5.32e3T^{2} $
	79	$ 1 + 6.92T + 6.24e3T^{2} $
	83	$ 1 + 62.1T + 6.88e3T^{2} $
	89	$ 1 - 62.2iT - 7.92e3T^{2} $
	97	$ 1 - 124. iT - 9.40e3T^{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

−8.175695440376451061418014801397, −7.79609389246694275421226874716, −6.98402649435342330836534655429, −6.60717889073541760019800369465, −5.33876968746641801978628591625, −4.58643152978600387645766590017, −3.95461049263755515477943565001, −3.12916673354415999668791424580, −1.79200230048540397080831256950, −0.948191148567082882339114593744, 0.33595367282213386666758391833, 1.80636890573119695140440947985, 2.63143291552246246274566639318, 3.40894990417658219465447380376, 4.43681362628719667559394103088, 5.56543332697218958133885926086, 5.88824174652394140737811575874, 6.46483996441856554266808967403, 7.77598663822500161893825272039, 8.383713863026903886209126811768

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.389392821\)
\(L(\frac12)\)	\(\approx\)	\(1.389392821\)
\(L(2)\)		not available
\(L(1)\)		not available

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