LMFDB - L-function 2-3344-76.75-c1-0-57

Dirichlet series

L(s) = 1

− 2.23·3^-s + 1.13·5^-s + 3.35i·7^-s + 1.99·9^-s − i·11^-s − 1.30i·13^-s − 2.53·15^-s + 5.93·17^-s + (−3.18 + 2.97i)19^-s − 7.50i·21^-s − 7.29i·23^-s − 3.71·25^-s + 2.24·27^-s − 7.16i·29^-s − 8.09·31^-s + ⋯

L(s) = 1

− 1.29·3^-s + 0.507·5^-s + 1.26i·7^-s + 0.664·9^-s − 0.301i·11^-s − 0.362i·13^-s − 0.654·15^-s + 1.43·17^-s + (−0.730 + 0.682i)19^-s − 1.63i·21^-s − 1.52i·23^-s − 0.742·25^-s + 0.432·27^-s − 1.33i·29^-s − 1.45·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3344$ = $2^{4} \cdot 11 \cdot 19$
Sign:	$0.682 + 0.730i$
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:	$(2,\ 3344,\ (\ :1/2),\ 0.682 + 0.730i)$

Particular Values

$L(1)$	$\approx$	$0.9279841852$
$L(\frac12)$	$\approx$	$0.9279841852$
$L(\frac{3}{2})$		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 + (3.18 - 2.97i)T $
good	3	$ 1 + 2.23T + 3T^{2} $
	5	$ 1 - 1.13T + 5T^{2} $
	7	$ 1 - 3.35iT - 7T^{2} $
	13	$ 1 + 1.30iT - 13T^{2} $
	17	$ 1 - 5.93T + 17T^{2} $
	23	$ 1 + 7.29iT - 23T^{2} $
	29	$ 1 + 7.16iT - 29T^{2} $
	31	$ 1 + 8.09T + 31T^{2} $
	37	$ 1 + 0.144iT - 37T^{2} $
	41	$ 1 - 5.48iT - 41T^{2} $
	43	$ 1 + 5.70iT - 43T^{2} $
	47	$ 1 - 8.39iT - 47T^{2} $
	53	$ 1 + 10.0iT - 53T^{2} $
	59	$ 1 - 13.0T + 59T^{2} $
	61	$ 1 + 1.98T + 61T^{2} $
	67	$ 1 + 11.7T + 67T^{2} $
	71	$ 1 - 7.66T + 71T^{2} $
	73	$ 1 - 6.53T + 73T^{2} $
	79	$ 1 + 0.728T + 79T^{2} $
	83	$ 1 - 4.79iT - 83T^{2} $
	89	$ 1 + 5.21iT - 89T^{2} $
	97	$ 1 - 1.14iT - 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

−8.452118964459696001116213766461, −7.901693975250704862695076668379, −6.70209074694715898879129418562, −5.93024357738023136334387404396, −5.73934688086815466573932067839, −5.05094054603983274099100537804, −3.95183502211580307013348675746, −2.76990212138156742374839015039, −1.83749953798591232302151753894, −0.43010835880835987769170416459, 0.897220567316445398366969406619, 1.82476160651314140557793539100, 3.38805685757363133565311627196, 4.14139271365139463903562791876, 5.17335095996638016937591794801, 5.57006236063831653021565522476, 6.44769619465866239135151072480, 7.20919156689816598760106030584, 7.59610971473606751860135935903, 8.857317336987880442141121170182

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9279841852\)
\(L(\frac12)\)	\(\approx\)	\(0.9279841852\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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