LMFDB - L-function 2-12e3-24.11-c1-0-9

Dirichlet series

L(s) = 1

− 0.717·5^-s − 3.24i·7^-s + 6.63i·11^-s − 4.48·25^-s + 10.3·29^-s + 9.24i·31^-s + 2.32i·35^-s − 3.51·49^-s + 13.9·53^-s − 4.75i·55^-s + 10.3i·59^-s + 15.4·73^-s + 21.5·77^-s − 10i·79^-s + 3.76i·83^-s + ⋯

L(s) = 1

− 0.320·5^-s − 1.22i·7^-s + 1.99i·11^-s − 0.897·25^-s + 1.92·29^-s + 1.66i·31^-s + 0.393i·35^-s − 0.502·49^-s + 1.92·53^-s − 0.641i·55^-s + 1.35i·59^-s + 1.81·73^-s + 2.45·77^-s − 1.12i·79^-s + 0.412i·83^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$13.7981$
Root analytic conductor:	$3.71458$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (863, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :1/2),\ 0.707 - 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.431598579$
$L(\frac12)$	$\approx$	$1.431598579$
$L(\frac{3}{2})$		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 + 0.717T + 5T^{2} $
	7	$ 1 + 3.24iT - 7T^{2} $
	11	$ 1 - 6.63iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 + 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 - 10.3T + 29T^{2} $
	31	$ 1 - 9.24iT - 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 13.9T + 53T^{2} $
	59	$ 1 - 10.3iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 - 15.4T + 73T^{2} $
	79	$ 1 + 10iT - 79T^{2} $
	83	$ 1 - 3.76iT - 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 - 15.9T + 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

−9.570704126416580213476908303396, −8.571045295729156210621498758434, −7.64215869888907269749586952591, −7.12702623648396879913834348963, −6.49497905242817717946819666031, −5.07664288559174279888922935253, −4.43358977024251219111801174750, −3.69678586003659980215384721049, −2.36755874411560810190576121228, −1.12788810739386009682150581441, 0.63187572186566204374713609688, 2.31433029288576652432558176454, 3.16516056386780110122146432996, 4.12133887309820322501786195390, 5.37466724590196880119416843500, 5.90671153567800835495738842722, 6.64478318485121883264483921060, 7.987751680392894716562917285315, 8.367504771301243244416691272247, 9.070513443699818644261370257846

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.431598579\)
\(L(\frac12)\)	\(\approx\)	\(1.431598579\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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