LMFDB - L-function 2-1368-57.56-c1-0-3

Dirichlet series

L(s) = 1

+ 3.42i·5^-s − 0.394·7^-s + 2.33i·11^-s − 3.36i·13^-s + 0.494i·17^-s + (−0.300 + 4.34i)19^-s + 3.79i·23^-s − 6.75·25^-s + 2.14·29^-s + 7.04i·31^-s − 1.35i·35^-s − 7.30i·37^-s + 3.81·41^-s − 8.45·43^-s + 2.43i·47^-s + ⋯

L(s) = 1

+ 1.53i·5^-s − 0.148·7^-s + 0.703i·11^-s − 0.933i·13^-s + 0.119i·17^-s + (−0.0690 + 0.997i)19^-s + 0.790i·23^-s − 1.35·25^-s + 0.399·29^-s + 1.26i·31^-s − 0.228i·35^-s − 1.20i·37^-s + 0.595·41^-s − 1.28·43^-s + 0.355i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1368$ = $2^{3} \cdot 3^{2} \cdot 19$
Sign:	$-0.774 - 0.632i$
Analytic conductor:	$10.9235$
Root analytic conductor:	$3.30507$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1368} (1025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1368,\ (\ :1/2),\ -0.774 - 0.632i)$

Particular Values

$L(1)$	$\approx$	$1.140516768$
$L(\frac12)$	$\approx$	$1.140516768$
$L(\frac{3}{2})$		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 $
	19	$ 1 + (0.300 - 4.34i)T $
good	5	$ 1 - 3.42iT - 5T^{2} $
	7	$ 1 + 0.394T + 7T^{2} $
	11	$ 1 - 2.33iT - 11T^{2} $
	13	$ 1 + 3.36iT - 13T^{2} $
	17	$ 1 - 0.494iT - 17T^{2} $
	23	$ 1 - 3.79iT - 23T^{2} $
	29	$ 1 - 2.14T + 29T^{2} $
	31	$ 1 - 7.04iT - 31T^{2} $
	37	$ 1 + 7.30iT - 37T^{2} $
	41	$ 1 - 3.81T + 41T^{2} $
	43	$ 1 + 8.45T + 43T^{2} $
	47	$ 1 - 2.43iT - 47T^{2} $
	53	$ 1 + 10.9T + 53T^{2} $
	59	$ 1 + 6.33T + 59T^{2} $
	61	$ 1 + 10.5T + 61T^{2} $
	67	$ 1 + 5.01iT - 67T^{2} $
	71	$ 1 + 0.759T + 71T^{2} $
	73	$ 1 - 3.06T + 73T^{2} $
	79	$ 1 - 6.40iT - 79T^{2} $
	83	$ 1 - 1.54iT - 83T^{2} $
	89	$ 1 - 13.2T + 89T^{2} $
	97	$ 1 - 1.71iT - 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

−10.07257318963960417808022285823, −9.290927171786668046591132527939, −8.014853193372043872938792792799, −7.50402538830599437657095336123, −6.62764086616359990063499210287, −5.97092694305641797645491404834, −4.89706109367886177040675666894, −3.58294764907386195423519916388, −3.00215949537470110838355759445, −1.78008067403451847915735729923, 0.45784250157850717202435449387, 1.71198597640306219034714639486, 3.10068011865215176237631267127, 4.45664716607557111623625230441, 4.80316146520882097105184845859, 5.95893218030682022212211291573, 6.69823941218086259108411350576, 7.904698395178155961273791593061, 8.566481784169703150317078684386, 9.190449275708081851262088035230

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.140516768\)
\(L(\frac12)\)	\(\approx\)	\(1.140516768\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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