LMFDB - L-function 2-4896-8.5-c1-0-35

Dirichlet series

L(s) = 1

− 0.436i·5^-s + 1.90·7^-s + 2.45i·11^-s + 2.93i·13^-s + 17^-s − 0.713i·19^-s − 0.640·23^-s + 4.80·25^-s − 7.72i·29^-s + 6.23·31^-s − 0.832i·35^-s − 1.93i·37^-s + 1.06·41^-s + 7.65i·43^-s − 2.73·47^-s + ⋯

L(s) = 1

− 0.195i·5^-s + 0.719·7^-s + 0.740i·11^-s + 0.812i·13^-s + 0.242·17^-s − 0.163i·19^-s − 0.133·23^-s + 0.961·25^-s − 1.43i·29^-s + 1.11·31^-s − 0.140i·35^-s − 0.318i·37^-s + 0.166·41^-s + 1.16i·43^-s − 0.399·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4896$ = $2^{5} \cdot 3^{2} \cdot 17$
Sign:	$0.803 - 0.595i$
Analytic conductor:	$39.0947$
Root analytic conductor:	$6.25258$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4896} (2449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4896,\ (\ :1/2),\ 0.803 - 0.595i)$

Particular Values

$L(1)$	$\approx$	$2.136019001$
$L(\frac12)$	$\approx$	$2.136019001$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	17	$ 1 - T $
good	5	$ 1 + 0.436iT - 5T^{2} $
	7	$ 1 - 1.90T + 7T^{2} $
	11	$ 1 - 2.45iT - 11T^{2} $
	13	$ 1 - 2.93iT - 13T^{2} $
	19	$ 1 + 0.713iT - 19T^{2} $
	23	$ 1 + 0.640T + 23T^{2} $
	29	$ 1 + 7.72iT - 29T^{2} $
	31	$ 1 - 6.23T + 31T^{2} $
	37	$ 1 + 1.93iT - 37T^{2} $
	41	$ 1 - 1.06T + 41T^{2} $
	43	$ 1 - 7.65iT - 43T^{2} $
	47	$ 1 + 2.73T + 47T^{2} $
	53	$ 1 - 4.91iT - 53T^{2} $
	59	$ 1 - 6.77iT - 59T^{2} $
	61	$ 1 + 13.9iT - 61T^{2} $
	67	$ 1 - 13.2iT - 67T^{2} $
	71	$ 1 - 14.0T + 71T^{2} $
	73	$ 1 + 9.52T + 73T^{2} $
	79	$ 1 - 3.35T + 79T^{2} $
	83	$ 1 - 12.6iT - 83T^{2} $
	89	$ 1 + 0.155T + 89T^{2} $
	97	$ 1 - 3.59T + 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

−8.189420380967677030953165845685, −7.77389598741090067264379591189, −6.84865347393691776468038299611, −6.28402320578603138986503411390, −5.26814762396957130504116808337, −4.58892134331093242446433499509, −4.08473607749760493421036870563, −2.82544865179205832666966603915, −1.98562829598392909153030602082, −1.00602746223749479771409314825, 0.69698074135811298502121442647, 1.73778755686533325857891769165, 2.93351600430429026440039621664, 3.47343820466157694171680778288, 4.63056043324438943613854976773, 5.22600958516086778702676786484, 5.97206365614984770573256248814, 6.78350299479019372119907061026, 7.51194955986234430978789718682, 8.359962414926466137475623100471

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.136019001\)
\(L(\frac12)\)	\(\approx\)	\(2.136019001\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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