LMFDB - L-function 2-3344-76.75-c1-0-32

Dirichlet series

L(s) = 1

− 0.924·3^-s − 2.57·5^-s − 0.765i·7^-s − 2.14·9^-s − i·11^-s + 2.60i·13^-s + 2.37·15^-s − 3.74·17^-s + (−2.28 + 3.71i)19^-s + 0.707i·21^-s + 3.60i·23^-s + 1.62·25^-s + 4.75·27^-s − 3.77i·29^-s − 7.30·31^-s + ⋯

L(s) = 1

− 0.533·3^-s − 1.15·5^-s − 0.289i·7^-s − 0.714·9^-s − 0.301i·11^-s + 0.723i·13^-s + 0.614·15^-s − 0.909·17^-s + (−0.523 + 0.852i)19^-s + 0.154i·21^-s + 0.751i·23^-s + 0.324·25^-s + 0.915·27^-s − 0.700i·29^-s − 1.31·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3344$ = $2^{4} \cdot 11 \cdot 19$
Sign:	$0.852 + 0.523i$
Analytic conductor:	$26.7019$
Root analytic conductor:	$5.16739$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3344} (1519, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3344,\ (\ :1/2),\ 0.852 + 0.523i)$

Particular Values

$L(1)$	$\approx$	$0.5253206878$
$L(\frac12)$	$\approx$	$0.5253206878$
$L(\frac{3}{2})$		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 $
	11	$ 1 + iT $
	19	$ 1 + (2.28 - 3.71i)T $
good	3	$ 1 + 0.924T + 3T^{2} $
	5	$ 1 + 2.57T + 5T^{2} $
	7	$ 1 + 0.765iT - 7T^{2} $
	13	$ 1 - 2.60iT - 13T^{2} $
	17	$ 1 + 3.74T + 17T^{2} $
	23	$ 1 - 3.60iT - 23T^{2} $
	29	$ 1 + 3.77iT - 29T^{2} $
	31	$ 1 + 7.30T + 31T^{2} $
	37	$ 1 + 8.15iT - 37T^{2} $
	41	$ 1 - 5.25iT - 41T^{2} $
	43	$ 1 - 0.574iT - 43T^{2} $
	47	$ 1 - 1.77iT - 47T^{2} $
	53	$ 1 - 11.7iT - 53T^{2} $
	59	$ 1 + 4.11T + 59T^{2} $
	61	$ 1 - 9.17T + 61T^{2} $
	67	$ 1 + 1.66T + 67T^{2} $
	71	$ 1 + 11.0T + 71T^{2} $
	73	$ 1 + 2.32T + 73T^{2} $
	79	$ 1 + 9.04T + 79T^{2} $
	83	$ 1 + 17.2iT - 83T^{2} $
	89	$ 1 + 3.39iT - 89T^{2} $
	97	$ 1 + 16.0iT - 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

−8.659050860247445335903061578735, −7.63829522138716510539973014834, −7.26183901021654836712637888893, −6.16019203649107621919181461050, −5.71141600853937094207157966533, −4.49924303681747925944494229880, −4.03327581423825042579773822177, −3.11892789994205926789507531148, −1.86564694285609070149222683538, −0.34321049233859647861384294791, 0.52463167412438525872350850506, 2.23197038144054632292774325667, 3.16535970366401328644248056228, 4.08088802276195606091217548135, 4.91588017969952775457822436932, 5.55272203634273226961979814654, 6.57364965194209528467086598794, 7.09205594966955671145527276860, 8.049067405804314399187938095158, 8.616837208534316032310091395588

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5253206878\)
\(L(\frac12)\)	\(\approx\)	\(0.5253206878\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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