LMFDB - L-function 2-3648-76.75-c1-0-47

Dirichlet series

L(s) = 1

− 3^-s + 3.37·5^-s − 0.792i·7^-s + 9^-s + 0.792i·11^-s − 5.04i·13^-s − 3.37·15^-s + 5.37·17^-s + (−4 − 1.73i)19^-s + 0.792i·21^-s + 8.51i·23^-s + 6.37·25^-s − 27^-s + 10.0i·29^-s + 8.74·31^-s + ⋯

L(s) = 1

− 0.577·3^-s + 1.50·5^-s − 0.299i·7^-s + 0.333·9^-s + 0.238i·11^-s − 1.40i·13^-s − 0.870·15^-s + 1.30·17^-s + (−0.917 − 0.397i)19^-s + 0.172i·21^-s + 1.77i·23^-s + 1.27·25^-s − 0.192·27^-s + 1.87i·29^-s + 1.57·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3648$ = $2^{6} \cdot 3 \cdot 19$
Sign:	$0.917 + 0.397i$
Analytic conductor:	$29.1294$
Root analytic conductor:	$5.39716$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3648} (2431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3648,\ (\ :1/2),\ 0.917 + 0.397i)$

Particular Values

$L(1)$	$\approx$	$2.210237544$
$L(\frac12)$	$\approx$	$2.210237544$
$L(\frac{3}{2})$		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 + T $
	19	$ 1 + (4 + 1.73i)T $
good	5	$ 1 - 3.37T + 5T^{2} $
	7	$ 1 + 0.792iT - 7T^{2} $
	11	$ 1 - 0.792iT - 11T^{2} $
	13	$ 1 + 5.04iT - 13T^{2} $
	17	$ 1 - 5.37T + 17T^{2} $
	23	$ 1 - 8.51iT - 23T^{2} $
	29	$ 1 - 10.0iT - 29T^{2} $
	31	$ 1 - 8.74T + 31T^{2} $
	37	$ 1 + 5.04iT - 37T^{2} $
	41	$ 1 + 6.92iT - 41T^{2} $
	43	$ 1 + 9.30iT - 43T^{2} $
	47	$ 1 - 4.25iT - 47T^{2} $
	53	$ 1 + 3.16iT - 53T^{2} $
	59	$ 1 - 4T + 59T^{2} $
	61	$ 1 + 5.37T + 61T^{2} $
	67	$ 1 - 9.48T + 67T^{2} $
	71	$ 1 + 4T + 71T^{2} $
	73	$ 1 + 4.11T + 73T^{2} $
	79	$ 1 - 4.74T + 79T^{2} $
	83	$ 1 + 3.46iT - 83T^{2} $
	89	$ 1 - 13.2iT - 89T^{2} $
	97	$ 1 + 13.2iT - 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

−8.577527904031370130113469060572, −7.58601820658398912642592641671, −6.97693208161627766721231542599, −6.06192087274228902801985120585, −5.44190516491237406829524029159, −5.13716559796510178774713217236, −3.79487663212530819106619672532, −2.87368551646971438685952505441, −1.78948282612848675309691070043, −0.865714006245449104706532923834, 1.02553839776104416761535986845, 2.05777015518477270953930011399, 2.76740086842857847388025583367, 4.24376562135573241620136209342, 4.80014771447471940370307214340, 5.86863286485460344604228139262, 6.25394714005523595873950215914, 6.68905726128040525639204665694, 7.977388390632402905403291484453, 8.606051117068531373226975707321

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.210237544\)
\(L(\frac12)\)	\(\approx\)	\(2.210237544\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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