LMFDB - L-function 2-3344-76.75-c1-0-23

Dirichlet series

L(s) = 1

− 1.71·3^-s − 3.66·5^-s + 2.65i·7^-s − 0.0489·9^-s + i·11^-s + 4.00i·13^-s + 6.30·15^-s + 5.00·17^-s + (−1.66 + 4.02i)19^-s − 4.55i·21^-s + 0.204i·23^-s + 8.45·25^-s + 5.23·27^-s − 6.62i·29^-s + 8.91·31^-s + ⋯

L(s) = 1

− 0.991·3^-s − 1.64·5^-s + 1.00i·7^-s − 0.0163·9^-s + 0.301i·11^-s + 1.11i·13^-s + 1.62·15^-s + 1.21·17^-s + (−0.383 + 0.923i)19^-s − 0.993i·21^-s + 0.0425i·23^-s + 1.69·25^-s + 1.00·27^-s − 1.23i·29^-s + 1.60·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3344$ = $2^{4} \cdot 11 \cdot 19$
Sign:	$-0.923 - 0.383i$
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:	$(2,\ 3344,\ (\ :1/2),\ -0.923 - 0.383i)$

Particular Values

$L(1)$	$\approx$	$0.6165399719$
$L(\frac12)$	$\approx$	$0.6165399719$
$L(\frac{3}{2})$		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 + (1.66 - 4.02i)T $
good	3	$ 1 + 1.71T + 3T^{2} $
	5	$ 1 + 3.66T + 5T^{2} $
	7	$ 1 - 2.65iT - 7T^{2} $
	13	$ 1 - 4.00iT - 13T^{2} $
	17	$ 1 - 5.00T + 17T^{2} $
	23	$ 1 - 0.204iT - 23T^{2} $
	29	$ 1 + 6.62iT - 29T^{2} $
	31	$ 1 - 8.91T + 31T^{2} $
	37	$ 1 - 11.2iT - 37T^{2} $
	41	$ 1 - 1.16iT - 41T^{2} $
	43	$ 1 - 8.59iT - 43T^{2} $
	47	$ 1 + 7.02iT - 47T^{2} $
	53	$ 1 - 5.64iT - 53T^{2} $
	59	$ 1 - 10.7T + 59T^{2} $
	61	$ 1 + 15.0T + 61T^{2} $
	67	$ 1 - 9.02T + 67T^{2} $
	71	$ 1 + 3.63T + 71T^{2} $
	73	$ 1 + 7.45T + 73T^{2} $
	79	$ 1 - 11.2T + 79T^{2} $
	83	$ 1 - 3.09iT - 83T^{2} $
	89	$ 1 + 1.40iT - 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

−8.722082355990793748880804507771, −8.153904359953613316945372526512, −7.58250763335421907195045534892, −6.48038107333125945286464908160, −6.10298301544039391125389281844, −5.03149552343734781697247695397, −4.46851990243790842357214279899, −3.57291506986695765100366755652, −2.58000123356814089512799312589, −1.08119396517185032054297630290, 0.36668755039988498007632750496, 0.843574944962185250687413891635, 2.95714477965041190216796453064, 3.62098783107181814591369976967, 4.46670546482133459578169884241, 5.19028527956554310876093341084, 5.98405681812081874029981960756, 7.02172630222757741259107995984, 7.42365316611551491062914606629, 8.189692586399415400624103891446

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6165399719\)
\(L(\frac12)\)	\(\approx\)	\(0.6165399719\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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