LMFDB - L-function 4-226e2-1.1-c1e2-0-0

Dirichlet series

L(s) = 1

− 2·2^-s + 3·4^-s − 8·7^-s − 4·8^-s + 4·9^-s − 8·13^-s + 16·14^-s + 5·16^-s − 8·18^-s + 2·25^-s + 16·26^-s − 24·28^-s − 8·31^-s − 6·32^-s + 12·36^-s + 24·41^-s + 34·49^-s − 4·50^-s − 24·52^-s + 24·53^-s + 32·56^-s + 4·61^-s + 16·62^-s − 32·63^-s + 7·64^-s − 16·72^-s + 7·81^-s + ⋯

L(s) = 1

− 1.41·2^-s + 3/2·4^-s − 3.02·7^-s − 1.41·8^-s + 4/3·9^-s − 2.21·13^-s + 4.27·14^-s + 5/4·16^-s − 1.88·18^-s + 2/5·25^-s + 3.13·26^-s − 4.53·28^-s − 1.43·31^-s − 1.06·32^-s + 2·36^-s + 3.74·41^-s + 34/7·49^-s − 0.565·50^-s − 3.32·52^-s + 3.29·53^-s + 4.27·56^-s + 0.512·61^-s + 2.03·62^-s − 4.03·63^-s + 7/8·64^-s − 1.88·72^-s + 7/9·81^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51076 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 51076 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$51076$ = $2^{2} \cdot 113^{2}$
Sign:	$1$
Analytic conductor:	$3.25665$
Root analytic conductor:	$1.34336$
Motivic weight:	$1$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 51076,\ (\ :1/2, 1/2),\ 1)$

Particular Values

$L(1)$	$\approx$	$0.3612874878$
$L(\frac12)$	$\approx$	$0.3612874878$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$\Gal(F_p)$	$F_p(T)$
bad	2	$C_1$	$ ( 1 + T )^{2} $
	113	$C_2$	$ 1 - 18 T + p T^{2} $
good	3	$C_2^2$	$ 1 - 4 T^{2} + p^{2} T^{4} $
	5	$C_2^2$	$ 1 - 2 T^{2} + p^{2} T^{4} $
	7	$C_2$	$ ( 1 + 4 T + p T^{2} )^{2} $
	11	$C_2$	$ ( 1 + p T^{2} )^{2} $
	13	$C_2$	$ ( 1 + 4 T + p T^{2} )^{2} $
	17	$C_2$	$ ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) $
	19	$C_2^2$	$ 1 - 20 T^{2} + p^{2} T^{4} $
	23	$C_2^2$	$ 1 + 4 T^{2} + p^{2} T^{4} $
	29	$C_2^2$	$ 1 - 50 T^{2} + p^{2} T^{4} $
	31	$C_2$	$ ( 1 + 4 T + p T^{2} )^{2} $
	37	$C_2^2$	$ 1 - 2 T^{2} + p^{2} T^{4} $
	41	$C_2$	$ ( 1 - 12 T + p T^{2} )^{2} $
	43	$C_2^2$	$ 1 - 68 T^{2} + p^{2} T^{4} $
	47	$C_2^2$	$ 1 + 4 T^{2} + p^{2} T^{4} $
	53	$C_2$	$ ( 1 - 12 T + p T^{2} )^{2} $
	59	$C_2^2$	$ 1 - 68 T^{2} + p^{2} T^{4} $
	61	$C_2$	$ ( 1 - 2 T + p T^{2} )^{2} $
	67	$C_2^2$	$ 1 - 116 T^{2} + p^{2} T^{4} $
	71	$C_2^2$	$ 1 - 140 T^{2} + p^{2} T^{4} $
	73	$C_2$	$ ( 1 - p T^{2} )^{2} $
	79	$C_2^2$	$ 1 + 4 T^{2} + p^{2} T^{4} $
	83	$C_2$	$ ( 1 + p T^{2} )^{2} $
	89	$C_2$	$ ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) $
	97	$C_2$	$ ( 1 + 10 T + p T^{2} )^{2} $
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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

−12.42044212405947585736859845485, −12.31141079418763191491288505633, −11.40221014391323909181777223038, −10.70096008605305733963445737444, −10.24724742728993739816746233272, −9.910540253202443326284011209513, −9.638314203437256002275773930041, −9.194661205689948471468017094572, −9.001857723805051246630429721182, −7.87234642956242869761994818953, −7.32351798829068601037656538487, −6.95339628049080709813870601373, −6.89257024341965846620786518238, −5.96594262407378662303220721319, −5.54515067934607623186954799462, −4.31394680844603180552755485570, −3.69535666331519582135264799592, −2.74714091808275711222503104768, −2.37442649096191884035668307047, −0.60205853275041224072904327123, 0.60205853275041224072904327123, 2.37442649096191884035668307047, 2.74714091808275711222503104768, 3.69535666331519582135264799592, 4.31394680844603180552755485570, 5.54515067934607623186954799462, 5.96594262407378662303220721319, 6.89257024341965846620786518238, 6.95339628049080709813870601373, 7.32351798829068601037656538487, 7.87234642956242869761994818953, 9.001857723805051246630429721182, 9.194661205689948471468017094572, 9.638314203437256002275773930041, 9.910540253202443326284011209513, 10.24724742728993739816746233272, 10.70096008605305733963445737444, 11.40221014391323909181777223038, 12.31141079418763191491288505633, 12.42044212405947585736859845485

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3612874878\)
\(L(\frac12)\)	\(\approx\)	\(0.3612874878\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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