LMFDB - L-function 2-4608-12.11-c1-0-50

Dirichlet series

L(s) = 1

− 3.86i·5^-s + 2.44i·7^-s + 5.46·11^-s − 2·13^-s + 6.31i·17^-s − 4.89i·19^-s − 7.46·23^-s − 9.92·25^-s − 1.03i·29^-s − 8.10i·31^-s + 9.46·35^-s + 0.535·37^-s − 9.14i·41^-s − 2.82i·43^-s − 0.535·47^-s + ⋯

L(s) = 1

− 1.72i·5^-s + 0.925i·7^-s + 1.64·11^-s − 0.554·13^-s + 1.53i·17^-s − 1.12i·19^-s − 1.55·23^-s − 1.98·25^-s − 0.192i·29^-s − 1.45i·31^-s + 1.59·35^-s + 0.0881·37^-s − 1.42i·41^-s − 0.431i·43^-s − 0.0781·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4608$ = $2^{9} \cdot 3^{2}$
Sign:	$-0.577 + 0.816i$
Analytic conductor:	$36.7950$
Root analytic conductor:	$6.06589$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4608} (4607, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4608,\ (\ :1/2),\ -0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$1.425577254$
$L(\frac12)$	$\approx$	$1.425577254$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 3.86iT - 5T^{2} $
	7	$ 1 - 2.44iT - 7T^{2} $
	11	$ 1 - 5.46T + 11T^{2} $
	13	$ 1 + 2T + 13T^{2} $
	17	$ 1 - 6.31iT - 17T^{2} $
	19	$ 1 + 4.89iT - 19T^{2} $
	23	$ 1 + 7.46T + 23T^{2} $
	29	$ 1 + 1.03iT - 29T^{2} $
	31	$ 1 + 8.10iT - 31T^{2} $
	37	$ 1 - 0.535T + 37T^{2} $
	41	$ 1 + 9.14iT - 41T^{2} $
	43	$ 1 + 2.82iT - 43T^{2} $
	47	$ 1 + 0.535T + 47T^{2} $
	53	$ 1 - 3.10iT - 53T^{2} $
	59	$ 1 - 6.92T + 59T^{2} $
	61	$ 1 + 11.4T + 61T^{2} $
	67	$ 1 + 13.3iT - 67T^{2} $
	71	$ 1 - 10.3T + 71T^{2} $
	73	$ 1 - 2T + 73T^{2} $
	79	$ 1 - 1.69iT - 79T^{2} $
	83	$ 1 + 6.53T + 83T^{2} $
	89	$ 1 + 1.41iT - 89T^{2} $
	97	$ 1 - 6.92T + 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

−8.333529720993723531381322962543, −7.50810773401113630285391136446, −6.34291927073337813234853426991, −5.88789932727723607978869937357, −5.12401670393467489300997216780, −4.21833940890494343050004462268, −3.86600021733841828665965406267, −2.24263966252081181739439754305, −1.61015289082083848118800461316, −0.39986872147877934319741598242, 1.26082439128030447426627861743, 2.38225502175604994893684056929, 3.34118880527192506039179513738, 3.84097489174161563077730888957, 4.70934245813650584462135966771, 5.94438847885873480554388822103, 6.55001561329794908816631948705, 7.10121757772562017628123623221, 7.55094923407555772454010229189, 8.451673885577523250166127507924

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.425577254\)
\(L(\frac12)\)	\(\approx\)	\(1.425577254\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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