LMFDB - L-function 4-1548800-1.1-c1e2-0-0

Dirichlet series

L(s) = 1

− 6·9^-s − 2·11^-s − 12·17^-s + 8·19^-s + 25^-s + 20·41^-s + 2·49^-s − 8·59^-s − 16·67^-s − 28·73^-s + 27·81^-s − 16·83^-s − 12·89^-s + 4·97^-s + 12·99^-s + 16·107^-s + 4·113^-s + 3·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 72·153^-s + 157^-s + 163^-s + ⋯

L(s) = 1

− 2·9^-s − 0.603·11^-s − 2.91·17^-s + 1.83·19^-s + 1/5·25^-s + 3.12·41^-s + 2/7·49^-s − 1.04·59^-s − 1.95·67^-s − 3.27·73^-s + 3·81^-s − 1.75·83^-s − 1.27·89^-s + 0.406·97^-s + 1.20·99^-s + 1.54·107^-s + 0.376·113^-s + 3/11·121^-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 + 5.82·153^-s + 0.0798·157^-s + 0.0783·163^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1548800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1548800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$1548800$ = $2^{9} \cdot 5^{2} \cdot 11^{2}$
Sign:	$1$
Analytic conductor:	$98.7528$
Root analytic conductor:	$3.15237$
Motivic weight:	$1$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 1548800,\ (\ :1/2, 1/2),\ 1)$

Particular Values

$L(1)$	$\approx$	$0.6717433918$
$L(\frac12)$	$\approx$	$0.6717433918$
$L(\frac{3}{2})$		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 $
	5	$C_1$$\times$$C_1$	$ ( 1 - T )( 1 + T ) $
	11	$C_1$	$ ( 1 + T )^{2} $
good	3	$C_2$	$ ( 1 + p T^{2} )^{2} $	2.3.a_g
	7	$C_2$	$ ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) $	2.7.a_ac
	13	$C_2$	$ ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) $	2.13.a_ak
	17	$C_2$	$ ( 1 + 6 T + p T^{2} )^{2} $	2.17.m_cs
	19	$C_2$	$ ( 1 - 4 T + p T^{2} )^{2} $	2.19.ai_cc
	23	$C_2$	$ ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) $	2.23.a_be
	29	$C_2$	$ ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) $	2.29.a_cc
	31	$C_2$	$ ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) $	2.31.a_ac
	37	$C_2$	$ ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) $	2.37.a_aba
	41	$C_2$	$ ( 1 - 10 T + p T^{2} )^{2} $	2.41.au_ha
	43	$C_2$	$ ( 1 + p T^{2} )^{2} $	2.43.a_di
	47	$C_2$	$ ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) $	2.47.a_da
	53	$C_2$	$ ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) $	2.53.a_g
	59	$C_2$	$ ( 1 + 4 T + p T^{2} )^{2} $	2.59.i_fe
	61	$C_2$	$ ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) $	2.61.a_eo
	67	$C_2$	$ ( 1 + 8 T + p T^{2} )^{2} $	2.67.q_hq
	71	$C_2$	$ ( 1 + p T^{2} )^{2} $	2.71.a_fm
	73	$C_2$	$ ( 1 + 14 T + p T^{2} )^{2} $	2.73.bc_ne
	79	$C_2$	$ ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) $	2.79.a_adu
	83	$C_2$	$ ( 1 + 8 T + p T^{2} )^{2} $	2.83.q_iw
	89	$C_2$	$ ( 1 + 6 T + p T^{2} )^{2} $	2.89.m_ig
	97	$C_2$	$ ( 1 - 2 T + p T^{2} )^{2} $	2.97.ae_hq
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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

−7.65072157489903981742781242784, −7.58115128738550044456594424531, −7.20980006114155286473947341742, −6.29670229400537650729444321883, −6.28118370578645402196158692333, −5.68465863116436055914602607048, −5.40941065075534095194244033749, −4.72738046016517462376134223046, −4.42524465686514871694159803405, −3.85885873293086675338957904608, −2.88180193076351113983639438164, −2.85668379629979186422132050434, −2.41167181005319106364125353977, −1.46303061533755639665183373410, −0.34023611415718700769027809145, 0.34023611415718700769027809145, 1.46303061533755639665183373410, 2.41167181005319106364125353977, 2.85668379629979186422132050434, 2.88180193076351113983639438164, 3.85885873293086675338957904608, 4.42524465686514871694159803405, 4.72738046016517462376134223046, 5.40941065075534095194244033749, 5.68465863116436055914602607048, 6.28118370578645402196158692333, 6.29670229400537650729444321883, 7.20980006114155286473947341742, 7.58115128738550044456594424531, 7.65072157489903981742781242784

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.6717433918\)
\(L(\frac12)\)	\(\approx\)	\(0.6717433918\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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