LMFDB - L-function 2-1008-84.83-c2-0-12

Dirichlet series

L(s) = 1

− 0.718·5^-s + (−6.52 + 2.54i)7^-s − 13.1·11^-s − 12.0i·13^-s + 17.0·17^-s + 6.05·19^-s + 10.8·23^-s − 24.4·25^-s + 8.20i·29^-s + 36.6·31^-s + (4.68 − 1.82i)35^-s + 45.0·37^-s + 62.4·41^-s + 12.0i·43^-s + 87.3i·47^-s + ⋯

L(s) = 1

− 0.143·5^-s + (−0.931 + 0.362i)7^-s − 1.19·11^-s − 0.926i·13^-s + 1.00·17^-s + 0.318·19^-s + 0.473·23^-s − 0.979·25^-s + 0.282i·29^-s + 1.18·31^-s + (0.133 − 0.0521i)35^-s + 1.21·37^-s + 1.52·41^-s + 0.279i·43^-s + 1.85i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1008$ = $2^{4} \cdot 3^{2} \cdot 7$
Sign:	$0.961 - 0.275i$
Analytic conductor:	$27.4660$
Root analytic conductor:	$5.24080$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1008} (1007, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1008,\ (\ :1),\ 0.961 - 0.275i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.384670923$
$L(\frac12)$	$\approx$	$1.384670923$
$L(2)$		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 $
	7	$ 1 + (6.52 - 2.54i)T $
good	5	$ 1 + 0.718T + 25T^{2} $
	11	$ 1 + 13.1T + 121T^{2} $
	13	$ 1 + 12.0iT - 169T^{2} $
	17	$ 1 - 17.0T + 289T^{2} $
	19	$ 1 - 6.05T + 361T^{2} $
	23	$ 1 - 10.8T + 529T^{2} $
	29	$ 1 - 8.20iT - 841T^{2} $
	31	$ 1 - 36.6T + 961T^{2} $
	37	$ 1 - 45.0T + 1.36e3T^{2} $
	41	$ 1 - 62.4T + 1.68e3T^{2} $
	43	$ 1 - 12.0iT - 1.84e3T^{2} $
	47	$ 1 - 87.3iT - 2.20e3T^{2} $
	53	$ 1 - 74.7iT - 2.80e3T^{2} $
	59	$ 1 - 22.0iT - 3.48e3T^{2} $
	61	$ 1 + 57.2iT - 3.72e3T^{2} $
	67	$ 1 + 47.6iT - 4.48e3T^{2} $
	71	$ 1 + 66.4T + 5.04e3T^{2} $
	73	$ 1 + 36.0iT - 5.32e3T^{2} $
	79	$ 1 - 46.6iT - 6.24e3T^{2} $
	83	$ 1 - 38.2iT - 6.88e3T^{2} $
	89	$ 1 - 126.T + 7.92e3T^{2} $
	97	$ 1 + 126. 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

−9.854908782381582631198906549290, −9.106599197761128629003186593654, −7.84677233345055933615847341222, −7.63493826677682917178082400564, −6.17282309289161490183830230536, −5.66826307815467759548116527421, −4.58520201256068598176021879215, −3.22808766836003100536358197877, −2.65061200265493941541506823069, −0.77507488383799599536848990584, 0.65841020326894464945722380786, 2.36017085129514335254145534791, 3.38437037956125004330984950966, 4.36680384178587798654652332291, 5.48165495566295515223272870351, 6.32423429113545043905154816643, 7.30918240639323934825271522154, 7.921080468012269838126563253407, 8.992066835255995563545910318154, 9.915819729782387393539214197606

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

\(L(\frac{3}{2})\)	\(\approx\)	\(1.384670923\)
\(L(\frac12)\)	\(\approx\)	\(1.384670923\)
\(L(2)\)		not available
\(L(1)\)		not available

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