LMFDB - L-function 4-2e8-1.1-c33e2-0-0

Dirichlet series

L(s) = 1

− 3.79e7·3^-s − 1.81e11·5^-s + 6.71e13·7^-s − 8.85e15·9^-s − 1.33e17·11^-s − 2.98e18·13^-s + 6.86e18·15^-s − 7.93e19·17^-s + 1.36e21·19^-s − 2.54e21·21^-s − 2.61e22·23^-s − 1.95e23·25^-s + 5.14e23·27^-s − 1.67e24·29^-s + 6.23e24·31^-s + 5.07e24·33^-s − 1.21e25·35^-s − 1.04e26·37^-s + 1.13e26·39^-s + 2.77e26·41^-s − 1.56e27·43^-s + 1.60e27·45^-s − 5.42e27·47^-s − 4.23e27·49^-s + 3.00e27·51^-s − 2.68e28·53^-s + 2.42e28·55^-s + ⋯

L(s) = 1

− 0.508·3^-s − 0.530·5^-s + 0.763·7^-s − 1.59·9^-s − 0.878·11^-s − 1.24·13^-s + 0.269·15^-s − 0.395·17^-s + 1.08·19^-s − 0.388·21^-s − 0.889·23^-s − 1.68·25^-s + 1.24·27^-s − 1.24·29^-s + 1.54·31^-s + 0.446·33^-s − 0.405·35^-s − 1.39·37^-s + 0.632·39^-s + 0.679·41^-s − 1.74·43^-s + 0.844·45^-s − 1.39·47^-s − 0.547·49^-s + 0.201·51^-s − 0.952·53^-s + 0.466·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+33/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$256$ = $2^{8}$
Sign:	$1$
Analytic conductor:	$12182.0$
Root analytic conductor:	$10.5058$
Motivic weight:	$33$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 256,\ (\ :33/2, 33/2),\ 1)$

Particular Values

$L(17)$	$\approx$	$0.03649727191$
$L(\frac12)$	$\approx$	$0.03649727191$
$L(\frac{35}{2})$		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		$ 1 $
good	3	$D_{4}$	$ 1 + 1404440 p^{3} T + 522714408050 p^{9} T^{2} + 1404440 p^{36} T^{3} + p^{66} T^{4} $
	5	$D_{4}$	$ 1 + 1448492292 p^{3} T + 585507855025144558 p^{8} T^{2} + 1448492292 p^{36} T^{3} + p^{66} T^{4} $
	7	$D_{4}$	$ 1 - 9593297152400 p T + $$36\!\cdots\!50$$ p^{4} T^{2} - 9593297152400 p^{34} T^{3} + p^{66} T^{4} $
	11	$D_{4}$	$ 1 + 12170165040174024 p T + $$23\!\cdots\!66$$ p^{2} T^{2} + 12170165040174024 p^{34} T^{3} + p^{66} T^{4} $
	13	$D_{4}$	$ 1 + 229354652173803380 p T + $$61\!\cdots\!50$$ p^{3} T^{2} + 229354652173803380 p^{34} T^{3} + p^{66} T^{4} $
	17	$D_{4}$	$ 1 + 79361149261175525340 T - $$26\!\cdots\!50$$ p T^{2} + 79361149261175525340 p^{33} T^{3} + p^{66} T^{4} $
	19	$D_{4}$	$ 1 - $$13\!\cdots\!00$$ T + $$17\!\cdots\!22$$ p T^{2} - $$13\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} $
	23	$D_{4}$	$ 1 + $$26\!\cdots\!40$$ T + $$68\!\cdots\!50$$ p T^{2} + $$26\!\cdots\!40$$ p^{33} T^{3} + p^{66} T^{4} $
	29	$D_{4}$	$ 1 + $$57\!\cdots\!00$$ p T + $$46\!\cdots\!58$$ p^{2} T^{2} + $$57\!\cdots\!00$$ p^{34} T^{3} + p^{66} T^{4} $
	31	$D_{4}$	$ 1 - $$20\!\cdots\!36$$ p T + $$30\!\cdots\!86$$ p^{2} T^{2} - $$20\!\cdots\!36$$ p^{34} T^{3} + p^{66} T^{4} $
	37	$D_{4}$	$ 1 + $$10\!\cdots\!20$$ T + $$13\!\cdots\!50$$ T^{2} + $$10\!\cdots\!20$$ p^{33} T^{3} + p^{66} T^{4} $
	41	$D_{4}$	$ 1 - $$27\!\cdots\!44$$ T + $$22\!\cdots\!26$$ T^{2} - $$27\!\cdots\!44$$ p^{33} T^{3} + p^{66} T^{4} $
	43	$D_{4}$	$ 1 + $$15\!\cdots\!00$$ T + $$12\!\cdots\!50$$ T^{2} + $$15\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} $
	47	$D_{4}$	$ 1 + $$11\!\cdots\!20$$ p T + $$37\!\cdots\!50$$ T^{2} + $$11\!\cdots\!20$$ p^{34} T^{3} + p^{66} T^{4} $
	53	$D_{4}$	$ 1 + $$26\!\cdots\!20$$ T + $$12\!\cdots\!50$$ T^{2} + $$26\!\cdots\!20$$ p^{33} T^{3} + p^{66} T^{4} $
	59	$D_{4}$	$ 1 - $$30\!\cdots\!00$$ T + $$77\!\cdots\!58$$ T^{2} - $$30\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} $
	61	$D_{4}$	$ 1 + $$57\!\cdots\!36$$ T + $$16\!\cdots\!86$$ T^{2} + $$57\!\cdots\!36$$ p^{33} T^{3} + p^{66} T^{4} $
	67	$D_{4}$	$ 1 + $$15\!\cdots\!60$$ T + $$41\!\cdots\!50$$ T^{2} + $$15\!\cdots\!60$$ p^{33} T^{3} + p^{66} T^{4} $
	71	$D_{4}$	$ 1 - $$26\!\cdots\!76$$ T + $$22\!\cdots\!66$$ T^{2} - $$26\!\cdots\!76$$ p^{33} T^{3} + p^{66} T^{4} $
	73	$D_{4}$	$ 1 - $$94\!\cdots\!40$$ T + $$19\!\cdots\!50$$ T^{2} - $$94\!\cdots\!40$$ p^{33} T^{3} + p^{66} T^{4} $
	79	$D_{4}$	$ 1 - $$85\!\cdots\!00$$ T + $$85\!\cdots\!78$$ T^{2} - $$85\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} $
	83	$D_{4}$	$ 1 + $$29\!\cdots\!20$$ T + $$37\!\cdots\!50$$ T^{2} + $$29\!\cdots\!20$$ p^{33} T^{3} + p^{66} T^{4} $
	89	$D_{4}$	$ 1 - $$13\!\cdots\!00$$ T + $$47\!\cdots\!38$$ T^{2} - $$13\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} $
	97	$D_{4}$	$ 1 + $$36\!\cdots\!60$$ T + $$42\!\cdots\!50$$ T^{2} + $$36\!\cdots\!60$$ p^{33} T^{3} + p^{66} T^{4} $
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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

−12.11741156293658344462034692248, −12.00513888651495600751595498413, −11.29999584220204753661098657890, −11.10773914479327891515264527690, −9.870865696969853055179130645699, −9.835709323323277353440559334502, −8.542107698709462892330623056654, −8.209860984286909586520096005948, −7.67916853481324454796199046259, −7.04161386478204239761336523495, −6.02936079225254323042517202720, −5.63762040844751398840477284084, −4.89415390986715155249161414570, −4.68285482124470133772131360789, −3.45239371718155382683723068754, −3.15343772930008076716608946691, −2.09668008381621235150357278359, −1.97335887214323248851084993514, −0.72264428829493771535027805478, −0.05698198268677698107713723635, 0.05698198268677698107713723635, 0.72264428829493771535027805478, 1.97335887214323248851084993514, 2.09668008381621235150357278359, 3.15343772930008076716608946691, 3.45239371718155382683723068754, 4.68285482124470133772131360789, 4.89415390986715155249161414570, 5.63762040844751398840477284084, 6.02936079225254323042517202720, 7.04161386478204239761336523495, 7.67916853481324454796199046259, 8.209860984286909586520096005948, 8.542107698709462892330623056654, 9.835709323323277353440559334502, 9.870865696969853055179130645699, 11.10773914479327891515264527690, 11.29999584220204753661098657890, 12.00513888651495600751595498413, 12.11741156293658344462034692248

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 $\F_{q}$

\(L(17)\)	\(\approx\)	\(0.03649727191\)
\(L(\frac12)\)	\(\approx\)	\(0.03649727191\)
\(L(\frac{35}{2})\)		not available
\(L(1)\)		not available

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