LMFDB - L-function 8-832e4-1.1-c1e4-0-16

Dirichlet series

L(s) = 1

− 4·5^-s + 6·9^-s + 6·13^-s + 2·17^-s + 14·25^-s + 10·29^-s + 2·37^-s + 10·41^-s − 24·45^-s + 14·49^-s + 28·53^-s + 10·61^-s − 24·65^-s + 12·73^-s + 9·81^-s − 8·85^-s − 20·89^-s − 36·97^-s + 2·101^-s − 24·109^-s − 14·113^-s + 36·117^-s + 22·121^-s − 64·125^-s + 127^-s + 131^-s + 137^-s + ⋯

L(s) = 1

− 1.78·5^-s + 2·9^-s + 1.66·13^-s + 0.485·17^-s + 14/5·25^-s + 1.85·29^-s + 0.328·37^-s + 1.56·41^-s − 3.57·45^-s + 2·49^-s + 3.84·53^-s + 1.28·61^-s − 2.97·65^-s + 1.40·73^-s + 81^-s − 0.867·85^-s − 2.11·89^-s − 3.65·97^-s + 0.199·101^-s − 2.29·109^-s − 1.31·113^-s + 3.32·117^-s + 2·121^-s − 5.72·125^-s + 0.0887·127^-s + 0.0873·131^-s + 0.0854·137^-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree:	$8$
Conductor:	$2^{24} \cdot 13^{4}$
Sign:	$1$
Analytic conductor:	$1948.05$
Root analytic conductor:	$2.57750$
Motivic weight:	$1$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 2^{24} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$

Particular Values

$L(1)$	$\approx$	$3.636013848$
$L(\frac12)$	$\approx$	$3.636013848$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}

	$p$	$\Gal(F_p)$	$F_p(T)$	Isogeny Class over $\mathbf{F}_p$
bad	2		$1$
	13	$C_2^2$	$1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4}$
good	3	$C_2$	$( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2}$	4.3.a_ag_a_bb
	5	$C_2^2$	$( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$	4.5.e_c_q_dn
	7	$C_2^2$	$( 1 - p T^{2} + p^{2} T^{4} )^{2}$	4.7.a_ao_a_fr
	11	$C_2^2$	$( 1 - p T^{2} + p^{2} T^{4} )^{2}$	4.11.a_aw_a_nz
	17	$C_2$ $\times$ $C_2^2$	$( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} )$	4.17.ac_r_dq_ahg
	19	$C_2^2$	$( 1 - p T^{2} + p^{2} T^{4} )^{2}$	4.19.a_abm_a_bpr
	23	$C_2^2$	$( 1 - p T^{2} + p^{2} T^{4} )^{2}$	4.23.a_abu_a_cjb
	29	$C_2$ $\times$ $C_2^2$	$( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} )$	4.29.ak_bd_afa_bya
	31	$C_2$	$( 1 + p T^{2} )^{4}$	4.31.a_eu_a_inu
	37	$C_2$ $\times$ $C_2^2$	$( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )$	4.37.ac_bl_ig_aqm
	41	$C_2$ $\times$ $C_2^2$	$( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} )$	4.41.ak_bp_iw_adkm
	43	$C_2^2$	$( 1 - p T^{2} + p^{2} T^{4} )^{2}$	4.43.a_adi_a_ifj
	47	$C_2$	$( 1 + p T^{2} )^{4}$	4.47.a_hg_a_tpu
	53	$C_2^2$	$( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}$	4.53.abc_so_aidc_crhr
	59	$C_2^2$	$( 1 - p T^{2} + p^{2} T^{4} )^{2}$	4.59.a_aeo_a_plr
	61	$C_2$ $\times$ $C_2^2$	$( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )$	4.61.ak_cj_bfy_amhg
	67	$C_2^2$	$( 1 - p T^{2} + p^{2} T^{4} )^{2}$	4.67.a_afe_a_txz
	71	$C_2^2$	$( 1 - p T^{2} + p^{2} T^{4} )^{2}$	4.71.a_afm_a_wjr
	73	$C_2^2$	$( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$	4.73.am_abm_aqq_zot
	79	$C_2$	$( 1 + p T^{2} )^{4}$	4.79.a_me_a_cdkg
	83	$C_2$	$( 1 + p T^{2} )^{4}$	4.83.a_mu_a_cjdu
	89	$C_2^2$	$( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}$	4.89.u_es_cyy_bxyp
	97	$C_2^2$	$( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2}$	4.97.bk_bdy_rgq_hpbf
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L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.32456289254254282086171325256, −7.11478437130649698482413304070, −6.87360197021863132651143148562, −6.79139461715931586476520847496, −6.67124480121587678737305448806, −6.25588444438717162541363592433, −5.79282805802190037060522883054, −5.77039539865653728489539145373, −5.55435006928759479084408583914, −5.12764861783047012396148875616, −4.93694287971795001527906351117, −4.52353758887044459762626346130, −4.43843951632667894305509682956, −4.03567161531092997170504951083, −3.87615301317608835419108432422, −3.85499276898535288409211173056, −3.71032316470829712170155206853, −3.16518765702227992097418965696, −2.62214148073303372759935715417, −2.51782263686722652720846050597, −2.48251574095226401889594628541, −1.44294101272403318532430877000, −1.18668516801452826381590313233, −1.10320839767098872391392769323, −0.60562780862176733974533240617, 0.60562780862176733974533240617, 1.10320839767098872391392769323, 1.18668516801452826381590313233, 1.44294101272403318532430877000, 2.48251574095226401889594628541, 2.51782263686722652720846050597, 2.62214148073303372759935715417, 3.16518765702227992097418965696, 3.71032316470829712170155206853, 3.85499276898535288409211173056, 3.87615301317608835419108432422, 4.03567161531092997170504951083, 4.43843951632667894305509682956, 4.52353758887044459762626346130, 4.93694287971795001527906351117, 5.12764861783047012396148875616, 5.55435006928759479084408583914, 5.77039539865653728489539145373, 5.79282805802190037060522883054, 6.25588444438717162541363592433, 6.67124480121587678737305448806, 6.79139461715931586476520847496, 6.87360197021863132651143148562, 7.11478437130649698482413304070, 7.32456289254254282086171325256

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Elliptic curves over $\Q$
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Properties

Origins

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Dirichlet series

Functional equation

Invariants

Particular Values

Euler product

Imaginary part of the first few zeros on the critical line

Graph of the $Z$ -function along the critical line

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	8-832e4-1.1-c1e4-0-16
Degree	$8$
Conductor	$479174066176$
Sign	$1$
Analytic cond.	$1948.05$
Root an. cond.	$2.57750$
Motivic weight	$1$
Arithmetic	yes
Rational	yes
Primitive	no
Self-dual	yes
Analytic rank	$0$

Properties

Origins

Origins of factors

Downloads

Learn more

Particular Values

Imaginary part of the first few zeros on the critical line

Graph of the ZZZ-function along the critical line

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Graph of the $Z$ -function along the critical line