LMFDB - L-function 2-1216-152.75-c1-0-3

Dirichlet series

L(s) = 1

− 3.40i·3^-s + 4.62i·7^-s − 8.56·9^-s − 0.223·13^-s − 3.69·17^-s + 4.35i·19^-s + 15.7·21^-s + 6.49i·23^-s + 5·25^-s + 18.9i·27^-s − 6.57·29^-s + 8.71·37^-s + 0.761i·39^-s + 6i·47^-s − 14.4·49^-s + ⋯

L(s) = 1

− 1.96i·3^-s + 1.74i·7^-s − 2.85·9^-s − 0.0620·13^-s − 0.896·17^-s + 0.999i·19^-s + 3.43·21^-s + 1.35i·23^-s + 25^-s + 3.64i·27^-s − 1.22·29^-s + 1.43·37^-s + 0.121i·39^-s + 0.875i·47^-s − 2.06·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1216$ = $2^{6} \cdot 19$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$9.70980$
Root analytic conductor:	$3.11605$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1216} (607, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1216,\ (\ :1/2),\ 0.707 - 0.707i)$

Particular Values

$L(1)$	$\approx$	$0.8591246126$
$L(\frac12)$	$\approx$	$0.8591246126$
$L(\frac{3}{2})$		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 $
	19	$ 1 - 4.35iT $
good	3	$ 1 + 3.40iT - 3T^{2} $
	5	$ 1 - 5T^{2} $
	7	$ 1 - 4.62iT - 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 + 0.223T + 13T^{2} $
	17	$ 1 + 3.69T + 17T^{2} $
	23	$ 1 - 6.49iT - 23T^{2} $
	29	$ 1 + 6.57T + 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 - 8.71T + 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 - 6iT - 47T^{2} $
	53	$ 1 + 13.8T + 53T^{2} $
	59	$ 1 - 2.95iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 10.6iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 1.82T + 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 - 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

−9.402970185345317384614524803689, −8.898347554196132577145659278944, −8.030179454810842379354939931149, −7.49474376054674785617231408714, −6.35685515594104635607487324737, −5.97081154396229275917817415892, −5.12552490668844645305537566303, −3.19374490205225294880471124287, −2.30676200482846665774037759052, −1.52248119877635404466962452961, 0.35649047348992046243818960061, 2.72759647693396293446416429936, 3.74049200610976929900937415185, 4.47233484491098425370201172495, 4.88553479086233434933661687792, 6.20707242935184470864951439469, 7.12344838332619782226158857662, 8.258107895054660166040459255119, 9.041291154394641287401591917419, 9.758983556537289394013249798144

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8591246126\)
\(L(\frac12)\)	\(\approx\)	\(0.8591246126\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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