LMFDB - L-function 4-480e2-1.1-c1e2-0-27

Dirichlet series

L(s) = 1

+ 4·7^-s + 9^-s + 4·23^-s − 25^-s + 4·41^-s + 12·47^-s − 2·49^-s + 4·63^-s + 8·71^-s − 4·73^-s + 81^-s + 4·89^-s − 4·97^-s + 12·103^-s − 8·113^-s − 10·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 16·161^-s + 163^-s + 167^-s + 6·169^-s + ⋯

L(s) = 1

+ 1.51·7^-s + 1/3·9^-s + 0.834·23^-s − 1/5·25^-s + 0.624·41^-s + 1.75·47^-s − 2/7·49^-s + 0.503·63^-s + 0.949·71^-s − 0.468·73^-s + 1/9·81^-s + 0.423·89^-s − 0.406·97^-s + 1.18·103^-s − 0.752·113^-s − 0.909·121^-s + 0.0887·127^-s + 0.0873·131^-s + 0.0854·137^-s + 0.0848·139^-s + 0.0819·149^-s + 0.0813·151^-s + 0.0798·157^-s + 1.26·161^-s + 0.0783·163^-s + 0.0773·167^-s + 6/13·169^-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}

\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree:	$4$
Conductor:	$230400$ = $2^{10} \cdot 3^{2} \cdot 5^{2}$
Sign:	$1$
Analytic conductor:	$14.6905$
Root analytic conductor:	$1.95775$
Motivic weight:	$1$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	yes
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 230400,\ (\ :1/2, 1/2),\ 1)$

Particular Values

$L(1)$	$\approx$	$2.221587558$
$L(\frac12)$	$\approx$	$2.221587558$
$L(\frac{3}{2})$		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		$1$
	3	$C_1$ $\times$ $C_1$	$( 1 - T )( 1 + T )$
	5	$C_2$	$1 + T^{2}$
good	7	$C_2$	$( 1 - 2 T + p T^{2} )^{2}$	2.7.ae_s
	11	$C_2^2$	$1 + 10 T^{2} + p^{2} T^{4}$	2.11.a_k
	13	$C_2^2$	$1 - 6 T^{2} + p^{2} T^{4}$	2.13.a_ag
	17	$C_2$	$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$	2.17.a_s
	19	$C_2$	$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$	2.19.a_aba
	23	$C_2$ $\times$ $C_2$	$( 1 - 4 T + p T^{2} )( 1 + p T^{2} )$	2.23.ae_bu
	29	$C_2$	$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$	2.29.a_ag
	31	$C_2$	$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$	2.31.a_bu
	37	$C_2$	$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$	2.37.a_k
	41	$C_2$ $\times$ $C_2$	$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$	2.41.ae_cs
	43	$C_2^2$	$1 + 6 T^{2} + p^{2} T^{4}$	2.43.a_g
	47	$C_2$ $\times$ $C_2$	$( 1 - 12 T + p T^{2} )( 1 + p T^{2} )$	2.47.am_dq
	53	$C_2^2$	$1 - 22 T^{2} + p^{2} T^{4}$	2.53.a_aw
	59	$C_2^2$	$1 + 90 T^{2} + p^{2} T^{4}$	2.59.a_dm
	61	$C_2^2$	$1 - 10 T^{2} + p^{2} T^{4}$	2.61.a_ak
	67	$C_2$	$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$	2.67.a_ak
	71	$C_2$ $\times$ $C_2$	$( 1 - 8 T + p T^{2} )( 1 + p T^{2} )$	2.71.ai_fm
	73	$C_2$ $\times$ $C_2$	$( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} )$	2.73.e_g
	79	$C_2$	$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$	2.79.a_fm
	83	$C_2^2$	$1 - 58 T^{2} + p^{2} T^{4}$	2.83.a_acg
	89	$C_2$ $\times$ $C_2$	$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$	2.89.ae_gk
	97	$C_2$ $\times$ $C_2$	$( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )$	2.97.e_ha
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L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.010491145617605083476351870773, −8.460687659062073508331132456542, −8.061548317513178034733944314162, −7.59291462867186493700473932851, −7.24489141892793086305399814747, −6.65388597794988806319202028461, −6.07248787764964529117609675479, −5.45297028291658394314709952790, −5.03796141753130864842855351444, −4.51416044297849093294124555918, −4.06028195628447761837367673348, −3.30054061241319217567250675863, −2.48949295169962412612476552574, −1.80359649052988157033946617775, −1.00760419767862181709979731337, 1.00760419767862181709979731337, 1.80359649052988157033946617775, 2.48949295169962412612476552574, 3.30054061241319217567250675863, 4.06028195628447761837367673348, 4.51416044297849093294124555918, 5.03796141753130864842855351444, 5.45297028291658394314709952790, 6.07248787764964529117609675479, 6.65388597794988806319202028461, 7.24489141892793086305399814747, 7.59291462867186493700473932851, 8.061548317513178034733944314162, 8.460687659062073508331132456542, 9.010491145617605083476351870773

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Dirichlet series

Functional equation

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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	4-480e2-1.1-c1e2-0-27
Degree	$4$
Conductor	$230400$
Sign	$1$
Analytic cond.	$14.6905$
Root an. cond.	$1.95775$
Motivic weight	$1$
Arithmetic	yes
Rational	yes
Primitive	yes
Self-dual	yes
Analytic rank	$0$

Properties

Origins

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