LMFDB - L-function 2-1344-56.27-c3-0-26

Dirichlet series

L(s) = 1

− 3i·3^-s − 12.5·5^-s + (18.2 + 2.88i)7^-s − 9·9^-s − 55.4·11^-s − 92.1·13^-s + 37.5i·15^-s + 118. i·17^-s − 155. i·19^-s + (8.66 − 54.8i)21^-s − 125. i·23^-s + 31.6·25^-s + 27i·27^-s − 131. i·29^-s − 66.0·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 1.11·5^-s + (0.987 + 0.155i)7^-s − 0.333·9^-s − 1.51·11^-s − 1.96·13^-s + 0.646i·15^-s + 1.68i·17^-s − 1.87i·19^-s + (0.0900 − 0.570i)21^-s − 1.13i·23^-s + 0.253·25^-s + 0.192i·27^-s − 0.842i·29^-s − 0.382·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.994 - 0.105i$
Analytic conductor:	$79.2985$
Root analytic conductor:	$8.90497$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1344} (223, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1344,\ (\ :3/2),\ 0.994 - 0.105i)$

Particular Values

$L(2)$	$\approx$	$0.7571711871$
$L(\frac12)$	$\approx$	$0.7571711871$
$L(\frac{5}{2})$		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 + 3iT $
	7	$ 1 + (-18.2 - 2.88i)T $
good	5	$ 1 + 12.5T + 125T^{2} $
	11	$ 1 + 55.4T + 1.33e3T^{2} $
	13	$ 1 + 92.1T + 2.19e3T^{2} $
	17	$ 1 - 118. iT - 4.91e3T^{2} $
	19	$ 1 + 155. iT - 6.85e3T^{2} $
	23	$ 1 + 125. iT - 1.21e4T^{2} $
	29	$ 1 + 131. iT - 2.43e4T^{2} $
	31	$ 1 + 66.0T + 2.97e4T^{2} $
	37	$ 1 - 147. iT - 5.06e4T^{2} $
	41	$ 1 - 20.3iT - 6.89e4T^{2} $
	43	$ 1 - 355.T + 7.95e4T^{2} $
	47	$ 1 + 79.5T + 1.03e5T^{2} $
	53	$ 1 - 463. iT - 1.48e5T^{2} $
	59	$ 1 - 580. iT - 2.05e5T^{2} $
	61	$ 1 + 587.T + 2.26e5T^{2} $
	67	$ 1 - 496.T + 3.00e5T^{2} $
	71	$ 1 - 232. iT - 3.57e5T^{2} $
	73	$ 1 + 551. iT - 3.89e5T^{2} $
	79	$ 1 + 437. iT - 4.93e5T^{2} $
	83	$ 1 + 191. iT - 5.71e5T^{2} $
	89	$ 1 - 93.7iT - 7.04e5T^{2} $
	97	$ 1 + 758. iT - 9.12e5T^{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

−8.993960145558176529323336942247, −8.179629873200502679738610911156, −7.66162668186647178741278487224, −7.17772352996039387526761866865, −5.91069678656593408348006105481, −4.82499317800247835744893223900, −4.41777165562835168934620458531, −2.79724789653328446116858249633, −2.17977500724904667067686464367, −0.52098516549736571303265261302, 0.32582595686868828942059894215, 2.11228489695047678092771357525, 3.15718430083546439008185411379, 4.16775137505383423639644773319, 5.15165557850592681157888096745, 5.32758639595747662703783516085, 7.25336453983057757759223394186, 7.66110217047978057539272056836, 8.121026739611262398030357472933, 9.370445282025195712466277711407

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.7571711871\)
\(L(\frac12)\)	\(\approx\)	\(0.7571711871\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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