LMFDB - L-function 2-4608-8.5-c1-0-61

Dirichlet series

L(s) = 1

− 4.44i·5^-s + 4.87·7^-s + 3.46i·11^-s − 14.7·25^-s − 5.34i·29^-s + 10.5·31^-s − 21.7i·35^-s + 16.7·49^-s − 12.4i·53^-s + 15.4·55^-s − 11.3i·59^-s + 9.79·73^-s + 16.8i·77^-s + 3.32·79^-s + 17.3i·83^-s + ⋯

L(s) = 1

− 1.98i·5^-s + 1.84·7^-s + 1.04i·11^-s − 2.95·25^-s − 0.993i·29^-s + 1.89·31^-s − 3.66i·35^-s + 2.39·49^-s − 1.71i·53^-s + 2.07·55^-s − 1.47i·59^-s + 1.14·73^-s + 1.92i·77^-s + 0.373·79^-s + 1.90i·83^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$4608$ = $2^{9} \cdot 3^{2}$
Sign:	$i$
Analytic conductor:	$36.7950$
Root analytic conductor:	$6.06589$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4608} (2305, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4608,\ (\ :1/2),\ i)$

Particular Values

$L(1)$	$\approx$	$2.443930346$
$L(\frac12)$	$\approx$	$2.443930346$
$L(\frac{3}{2})$		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 + 4.44iT - 5T^{2} $
	7	$ 1 - 4.87T + 7T^{2} $
	11	$ 1 - 3.46iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 + 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 + 5.34iT - 29T^{2} $
	31	$ 1 - 10.5T + 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 + 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 + 12.4iT - 53T^{2} $
	59	$ 1 + 11.3iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 - 9.79T + 73T^{2} $
	79	$ 1 - 3.32T + 79T^{2} $
	83	$ 1 - 17.3iT - 83T^{2} $
	89	$ 1 + 89T^{2} $
	97	$ 1 + 2T + 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

−8.128714733570077948338519043057, −7.86436488513688297041881714964, −6.73631085784849839402364409430, −5.58981533536018241561040654705, −5.01992451278308531370222713881, −4.55814975762840166369673453920, −4.03430363287114868279018682855, −2.23230526572612488752597462767, −1.60543150107650345779559159516, −0.73631347300844482772161964543, 1.23052105477763498007348545102, 2.35490793655286834571939994950, 2.99007989615512283408241159818, 3.89610146715657798291923493257, 4.77562161526074657671257074320, 5.72113267740907932269962054296, 6.32532514716642567060818660114, 7.14128879706872064772708053039, 7.75395494495936581025272451927, 8.283632503842856099209274300388

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.443930346\)
\(L(\frac12)\)	\(\approx\)	\(2.443930346\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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