LMFDB - L-function 2-4032-21.20-c1-0-19

Dirichlet series

L(s) = 1

− 1.53·5^-s + (−0.414 − 2.61i)7^-s + 0.585i·11^-s + 2.16i·13^-s + 5.86·17^-s + 5.22i·19^-s + 2.24i·23^-s − 2.65·25^-s − 5.41i·29^-s − 4.32i·31^-s + (0.634 + 4i)35^-s − 4·37^-s + 8.92·41^-s − 10.4·43^-s − 7.39·47^-s + ⋯

L(s) = 1

− 0.684·5^-s + (−0.156 − 0.987i)7^-s + 0.176i·11^-s + 0.600i·13^-s + 1.42·17^-s + 1.19i·19^-s + 0.467i·23^-s − 0.531·25^-s − 1.00i·29^-s − 0.777i·31^-s + (0.107 + 0.676i)35^-s − 0.657·37^-s + 1.39·41^-s − 1.59·43^-s − 1.07·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.716 - 0.698i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (3905, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ 0.716 - 0.698i)$

Particular Values

$L(1)$	$\approx$	$1.308861554$
$L(\frac12)$	$\approx$	$1.308861554$
$L(\frac{3}{2})$		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 + (0.414 + 2.61i)T $
good	5	$ 1 + 1.53T + 5T^{2} $
	11	$ 1 - 0.585iT - 11T^{2} $
	13	$ 1 - 2.16iT - 13T^{2} $
	17	$ 1 - 5.86T + 17T^{2} $
	19	$ 1 - 5.22iT - 19T^{2} $
	23	$ 1 - 2.24iT - 23T^{2} $
	29	$ 1 + 5.41iT - 29T^{2} $
	31	$ 1 + 4.32iT - 31T^{2} $
	37	$ 1 + 4T + 37T^{2} $
	41	$ 1 - 8.92T + 41T^{2} $
	43	$ 1 + 10.4T + 43T^{2} $
	47	$ 1 + 7.39T + 47T^{2} $
	53	$ 1 - 5.41iT - 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 9.65T + 67T^{2} $
	71	$ 1 - 4.58iT - 71T^{2} $
	73	$ 1 - 12.6iT - 73T^{2} $
	79	$ 1 + 2.34T + 79T^{2} $
	83	$ 1 - 13.5T + 83T^{2} $
	89	$ 1 - 5.86T + 89T^{2} $
	97	$ 1 - 8.28iT - 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

−8.196677387348136904346694041122, −7.83441453615444005522665845499, −7.24861586764558549770999076659, −6.36339399386998302722153218369, −5.61661829001715308458759127347, −4.60340717691068196560443940232, −3.81752430250803032654201941659, −3.42008753661285430358930416752, −1.98425817551950842106514686391, −0.896785538381928438628286090175, 0.48406261256322927099567108117, 1.87563634017008976878223143952, 3.11026620412325203562019634312, 3.41613028720953489337755072120, 4.77419806515143777286503356686, 5.27574865651968225930048849095, 6.12331350174901187440851653557, 6.91807202501067355325949687442, 7.73248449915986865991890706342, 8.345936860356127618257182853315

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.308861554\)
\(L(\frac12)\)	\(\approx\)	\(1.308861554\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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