LMFDB - L-function 2-5292-21.20-c1-0-18

Dirichlet series

L(s) = 1

− 2.15·5^-s + 1.33i·11^-s + 2.35i·13^-s + 7.20·17^-s − 1.19i·19^-s + 2.40i·23^-s − 0.335·25^-s − 6.14i·29^-s − 8.21i·31^-s − 5.14·37^-s + 4.47·41^-s − 0.664·43^-s − 2.15·47^-s + 12.7i·53^-s − 2.88i·55^-s + ⋯

L(s) = 1

− 0.965·5^-s + 0.402i·11^-s + 0.651i·13^-s + 1.74·17^-s − 0.274i·19^-s + 0.501i·23^-s − 0.0670·25^-s − 1.14i·29^-s − 1.47i·31^-s − 0.846·37^-s + 0.698·41^-s − 0.101·43^-s − 0.315·47^-s + 1.75i·53^-s − 0.388i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5292$ = $2^{2} \cdot 3^{3} \cdot 7^{2}$
Sign:	$0.755 - 0.654i$
Analytic conductor:	$42.2568$
Root analytic conductor:	$6.50052$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5292} (2645, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5292,\ (\ :1/2),\ 0.755 - 0.654i)$

Particular Values

$L(1)$	$\approx$	$1.418829954$
$L(\frac12)$	$\approx$	$1.418829954$
$L(\frac{3}{2})$		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 $
	7	$ 1 $
good	5	$ 1 + 2.15T + 5T^{2} $
	11	$ 1 - 1.33iT - 11T^{2} $
	13	$ 1 - 2.35iT - 13T^{2} $
	17	$ 1 - 7.20T + 17T^{2} $
	19	$ 1 + 1.19iT - 19T^{2} $
	23	$ 1 - 2.40iT - 23T^{2} $
	29	$ 1 + 6.14iT - 29T^{2} $
	31	$ 1 + 8.21iT - 31T^{2} $
	37	$ 1 + 5.14T + 37T^{2} $
	41	$ 1 - 4.47T + 41T^{2} $
	43	$ 1 + 0.664T + 43T^{2} $
	47	$ 1 + 2.15T + 47T^{2} $
	53	$ 1 - 12.7iT - 53T^{2} $
	59	$ 1 - 9.66T + 59T^{2} $
	61	$ 1 + 12.2iT - 61T^{2} $
	67	$ 1 + 3.73T + 67T^{2} $
	71	$ 1 - 11.4iT - 71T^{2} $
	73	$ 1 - 10.4iT - 73T^{2} $
	79	$ 1 + 9.07T + 79T^{2} $
	83	$ 1 + 10.7T + 83T^{2} $
	89	$ 1 - 5.19T + 89T^{2} $
	97	$ 1 + 1.23iT - 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

−8.077563248922878772576549986151, −7.60193064490426827416242372947, −7.07533365042914704153580470862, −6.04694413327712688509743920002, −5.43109361479128280516302587636, −4.38169809407606738582289766860, −3.92256022180451267772829089527, −3.05298531127967734977309485469, −1.99516843324335662204818018881, −0.793690602203814241837718011396, 0.54240346822463969154246225601, 1.61931775524765594594655994785, 3.18769167691775606262498219895, 3.33588529814102965711175073745, 4.37858901618853895650195784370, 5.28778148332534910523147468511, 5.79170836583325193913059193629, 6.88883324514891963317480388242, 7.40687853349767087851362596432, 8.229446473949153082249013378841

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

\(L(1)\)	\(\approx\)	\(1.418829954\)
\(L(\frac12)\)	\(\approx\)	\(1.418829954\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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