LMFDB - L-function 2-48e2-24.5-c2-0-7

Dirichlet series

L(s) = 1

− 4.87·5^-s − 11.7·7^-s − 9.75·11^-s + 22.6i·13^-s + 22.1i·17^-s + 17.7i·19^-s − 14.1i·23^-s − 1.20·25^-s − 20.0·29^-s − 39.7·31^-s + 57.5·35^-s − 2.40i·37^-s + 64.3i·41^-s + 3.19i·43^-s + 41.8i·47^-s + ⋯

L(s) = 1

− 0.975·5^-s − 1.68·7^-s − 0.886·11^-s + 1.74i·13^-s + 1.30i·17^-s + 0.936i·19^-s − 0.614i·23^-s − 0.0480·25^-s − 0.689·29^-s − 1.28·31^-s + 1.64·35^-s − 0.0649i·37^-s + 1.56i·41^-s + 0.0742i·43^-s + 0.890i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$0.169 + 0.985i$
Analytic conductor:	$62.7794$
Root analytic conductor:	$7.92334$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2304} (2177, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2304,\ (\ :1),\ 0.169 + 0.985i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.1114509448$
$L(\frac12)$	$\approx$	$0.1114509448$
$L(2)$		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 + 4.87T + 25T^{2} $
	7	$ 1 + 11.7T + 49T^{2} $
	11	$ 1 + 9.75T + 121T^{2} $
	13	$ 1 - 22.6iT - 169T^{2} $
	17	$ 1 - 22.1iT - 289T^{2} $
	19	$ 1 - 17.7iT - 361T^{2} $
	23	$ 1 + 14.1iT - 529T^{2} $
	29	$ 1 + 20.0T + 841T^{2} $
	31	$ 1 + 39.7T + 961T^{2} $
	37	$ 1 + 2.40iT - 1.36e3T^{2} $
	41	$ 1 - 64.3iT - 1.68e3T^{2} $
	43	$ 1 - 3.19iT - 1.84e3T^{2} $
	47	$ 1 - 41.8iT - 2.20e3T^{2} $
	53	$ 1 + 55.5T + 2.80e3T^{2} $
	59	$ 1 - 111.T + 3.48e3T^{2} $
	61	$ 1 - 10.8iT - 3.72e3T^{2} $
	67	$ 1 - 18.2iT - 4.48e3T^{2} $
	71	$ 1 + 34.7iT - 5.04e3T^{2} $
	73	$ 1 + 87.5T + 5.32e3T^{2} $
	79	$ 1 + 151.T + 6.24e3T^{2} $
	83	$ 1 - 61.8T + 6.88e3T^{2} $
	89	$ 1 + 72.9iT - 7.92e3T^{2} $
	97	$ 1 + 87.3T + 9.40e3T^{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

−9.463614168118140405321683551822, −8.638943834324290413669111600164, −7.86800076596756586325532307822, −7.08138008830288771973417884505, −6.37234392521189484921301045786, −5.69898198017146377196219017218, −4.30096506393193447655690601026, −3.83913046840544296800399662133, −2.96169336834097354255313136891, −1.73740621299698850228428281813, 0.06107455031956434836682323431, 0.43898281667443287308237988768, 2.58463627242419835605285113592, 3.22769802460301738053473178148, 3.88811290613004399552663469790, 5.26864875330780580287984135003, 5.65579289531694526948189081137, 6.99681224094899572631447124918, 7.31779830245384828334347530220, 8.125227650667676466735386688453

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.1114509448\)
\(L(\frac12)\)	\(\approx\)	\(0.1114509448\)
\(L(2)\)		not available
\(L(1)\)		not available

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