LMFDB - L-function 4-28e2-1.1-c1e2-0-1

Dirichlet series

L(s) = 1

− 2^-s − 4^-s + 3·8^-s − 6·9^-s − 16^-s + 6·18^-s + 10·25^-s − 4·29^-s − 5·32^-s + 6·36^-s + 12·37^-s − 7·49^-s − 10·50^-s − 20·53^-s + 4·58^-s + 7·64^-s − 18·72^-s − 12·74^-s + 27·81^-s + 7·98^-s − 10·100^-s + 20·106^-s − 36·109^-s + 4·113^-s + 4·116^-s − 6·121^-s + 127^-s + ⋯

L(s) = 1

− 0.707·2^-s − 1/2·4^-s + 1.06·8^-s − 2·9^-s − 1/4·16^-s + 1.41·18^-s + 2·25^-s − 0.742·29^-s − 0.883·32^-s + 36^-s + 1.97·37^-s − 49^-s − 1.41·50^-s − 2.74·53^-s + 0.525·58^-s + 7/8·64^-s − 2.12·72^-s − 1.39·74^-s + 3·81^-s + 0.707·98^-s − 100^-s + 1.94·106^-s − 3.44·109^-s + 0.376·113^-s + 0.371·116^-s − 0.545·121^-s + 0.0887·127^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$1$
Analytic conductor:	$0.0499885$
Root analytic conductor:	$0.472843$
Motivic weight:	$1$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 784,\ (\ :1/2, 1/2),\ 1)$

Particular Values

$L(1)$	$\approx$	$0.3090315375$
$L(\frac12)$	$\approx$	$0.3090315375$
$L(\frac{3}{2})$		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	$C_2$	$ 1 + T + p T^{2} $
	7	$C_2$	$ 1 + p T^{2} $
good	3	$C_2$	$ ( 1 + p T^{2} )^{2} $
	5	$C_2$	$ ( 1 - p T^{2} )^{2} $
	11	$C_2$	$ ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) $
	13	$C_2$	$ ( 1 - p T^{2} )^{2} $
	17	$C_2$	$ ( 1 - p T^{2} )^{2} $
	19	$C_2$	$ ( 1 + p T^{2} )^{2} $
	23	$C_2$	$ ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) $
	29	$C_2$	$ ( 1 + 2 T + p T^{2} )^{2} $
	31	$C_2$	$ ( 1 + p T^{2} )^{2} $
	37	$C_2$	$ ( 1 - 6 T + p T^{2} )^{2} $
	41	$C_2$	$ ( 1 - p T^{2} )^{2} $
	43	$C_2$	$ ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) $
	47	$C_2$	$ ( 1 + p T^{2} )^{2} $
	53	$C_2$	$ ( 1 + 10 T + p T^{2} )^{2} $
	59	$C_2$	$ ( 1 + p T^{2} )^{2} $
	61	$C_2$	$ ( 1 - p T^{2} )^{2} $
	67	$C_2$	$ ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) $
	71	$C_2$	$ ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) $
	73	$C_2$	$ ( 1 - p T^{2} )^{2} $
	79	$C_2$	$ ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) $
	83	$C_2$	$ ( 1 + p T^{2} )^{2} $
	89	$C_2$	$ ( 1 - p T^{2} )^{2} $
	97	$C_2$	$ ( 1 - 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

−17.36103568531052980967719794657, −17.28002321670383671839280006782, −16.51354540129250993514876248337, −16.20713218939645642377157653893, −14.98415704814163214532725277147, −14.46184169942106912708858944273, −14.18991637382708731070571093951, −13.24557983896369712639090803793, −12.76869743899601511049677332802, −11.80582065632319781891202715000, −11.00726626863206248848760142427, −10.77438049141535870105484223016, −9.487120310487436012715062545611, −9.195135941073282046423675747516, −8.288881670273938005137198112079, −7.916820437477627041447330894700, −6.62673790339788372172641120178, −5.61837662151843458306893694869, −4.67922456886986568351537135220, −3.05576516303848460288434731668, 3.05576516303848460288434731668, 4.67922456886986568351537135220, 5.61837662151843458306893694869, 6.62673790339788372172641120178, 7.916820437477627041447330894700, 8.288881670273938005137198112079, 9.195135941073282046423675747516, 9.487120310487436012715062545611, 10.77438049141535870105484223016, 11.00726626863206248848760142427, 11.80582065632319781891202715000, 12.76869743899601511049677332802, 13.24557983896369712639090803793, 14.18991637382708731070571093951, 14.46184169942106912708858944273, 14.98415704814163214532725277147, 16.20713218939645642377157653893, 16.51354540129250993514876248337, 17.28002321670383671839280006782, 17.36103568531052980967719794657

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3090315375\)
\(L(\frac12)\)	\(\approx\)	\(0.3090315375\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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