LMFDB - L-function 4-2e2-1.1-c31e2-0-0

Dirichlet series

L(s) = 1

− 6.55e4·2^-s + 1.67e7·3^-s + 3.22e9·4^-s − 2.67e10·5^-s − 1.09e12·6^-s − 2.35e13·7^-s − 1.40e14·8^-s − 1.15e14·9^-s + 1.75e15·10^-s + 2.51e16·11^-s + 5.38e16·12^-s + 1.89e17·13^-s + 1.54e18·14^-s − 4.47e17·15^-s + 5.76e18·16^-s + 9.20e17·17^-s + 7.58e18·18^-s − 1.00e20·19^-s − 8.62e19·20^-s − 3.92e20·21^-s − 1.64e21·22^-s + 5.10e20·23^-s − 2.35e21·24^-s + 4.22e21·25^-s − 1.24e22·26^-s + 1.78e21·27^-s − 7.57e22·28^-s + ⋯

L(s) = 1

− 1.41·2^-s + 0.672·3^-s + 3/2·4^-s − 0.392·5^-s − 0.951·6^-s − 1.87·7^-s − 1.41·8^-s − 0.187·9^-s + 0.554·10^-s + 1.81·11^-s + 1.00·12^-s + 1.02·13^-s + 2.64·14^-s − 0.263·15^-s + 5/4·16^-s + 0.0780·17^-s + 0.264·18^-s − 1.51·19^-s − 0.588·20^-s − 1.25·21^-s − 2.56·22^-s + 0.399·23^-s − 0.951·24^-s + 0.907·25^-s − 1.45·26^-s + 0.116·27^-s − 2.80·28^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+31/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$4$ = $2^{2}$
Sign:	$1$
Analytic conductor:	$148.241$
Root analytic conductor:	$3.48933$
Motivic weight:	$31$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 4,\ (\ :31/2, 31/2),\ 1)$

Particular Values

$L(16)$	$\approx$	$0.8093087754$
$L(\frac12)$	$\approx$	$0.8093087754$
$L(\frac{33}{2})$		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 + p^{15} T )^{2} $
good	3	$D_{4}$	$ 1 - 5572168 p T + 180667888754 p^{7} T^{2} - 5572168 p^{32} T^{3} + p^{62} T^{4} $
	5	$D_{4}$	$ 1 + 1070522628 p^{2} T - 44945775705560938 p^{7} T^{2} + 1070522628 p^{33} T^{3} + p^{62} T^{4} $
	7	$D_{4}$	$ 1 + 23503702110608 T + $$13\!\cdots\!14$$ p^{3} T^{2} + 23503702110608 p^{31} T^{3} + p^{62} T^{4} $
	11	$D_{4}$	$ 1 - 2286467043501144 p T + $$37\!\cdots\!06$$ p^{3} T^{2} - 2286467043501144 p^{32} T^{3} + p^{62} T^{4} $
	13	$D_{4}$	$ 1 - 189324344559709324 T + $$33\!\cdots\!22$$ p^{2} T^{2} - 189324344559709324 p^{31} T^{3} + p^{62} T^{4} $
	17	$D_{4}$	$ 1 - 920816829890675172 T + $$16\!\cdots\!86$$ p T^{2} - 920816829890675172 p^{31} T^{3} + p^{62} T^{4} $
	19	$D_{4}$	$ 1 + $$10\!\cdots\!00$$ T + $$55\!\cdots\!02$$ p T^{2} + $$10\!\cdots\!00$$ p^{31} T^{3} + p^{62} T^{4} $
	23	$D_{4}$	$ 1 - 22195146899293731888 p T + $$64\!\cdots\!62$$ p^{2} T^{2} - 22195146899293731888 p^{32} T^{3} + p^{62} T^{4} $
	29	$D_{4}$	$ 1 + 46811908508319393780 p^{2} T + $$46\!\cdots\!38$$ p^{2} T^{2} + 46811908508319393780 p^{33} T^{3} + p^{62} T^{4} $
	31	$D_{4}$	$ 1 - $$49\!\cdots\!84$$ T + $$25\!\cdots\!26$$ T^{2} - $$49\!\cdots\!84$$ p^{31} T^{3} + p^{62} T^{4} $
	37	$D_{4}$	$ 1 - $$18\!\cdots\!52$$ T + $$67\!\cdots\!02$$ T^{2} - $$18\!\cdots\!52$$ p^{31} T^{3} + p^{62} T^{4} $
	41	$D_{4}$	$ 1 + $$29\!\cdots\!36$$ T + $$40\!\cdots\!06$$ T^{2} + $$29\!\cdots\!36$$ p^{31} T^{3} + p^{62} T^{4} $
	43	$D_{4}$	$ 1 + $$29\!\cdots\!36$$ T + $$92\!\cdots\!38$$ T^{2} + $$29\!\cdots\!36$$ p^{31} T^{3} + p^{62} T^{4} $
	47	$D_{4}$	$ 1 - $$13\!\cdots\!52$$ T + $$17\!\cdots\!82$$ T^{2} - $$13\!\cdots\!52$$ p^{31} T^{3} + p^{62} T^{4} $
	53	$D_{4}$	$ 1 + $$45\!\cdots\!36$$ T + $$36\!\cdots\!18$$ T^{2} + $$45\!\cdots\!36$$ p^{31} T^{3} + p^{62} T^{4} $
	59	$D_{4}$	$ 1 + $$25\!\cdots\!60$$ T + $$17\!\cdots\!18$$ T^{2} + $$25\!\cdots\!60$$ p^{31} T^{3} + p^{62} T^{4} $
	61	$D_{4}$	$ 1 - $$73\!\cdots\!64$$ T + $$43\!\cdots\!46$$ T^{2} - $$73\!\cdots\!64$$ p^{31} T^{3} + p^{62} T^{4} $
	67	$D_{4}$	$ 1 - $$21\!\cdots\!92$$ T + $$83\!\cdots\!82$$ T^{2} - $$21\!\cdots\!92$$ p^{31} T^{3} + p^{62} T^{4} $
	71	$D_{4}$	$ 1 - $$36\!\cdots\!84$$ T + $$32\!\cdots\!06$$ T^{2} - $$36\!\cdots\!84$$ p^{31} T^{3} + p^{62} T^{4} $
	73	$D_{4}$	$ 1 - $$16\!\cdots\!84$$ T + $$81\!\cdots\!18$$ T^{2} - $$16\!\cdots\!84$$ p^{31} T^{3} + p^{62} T^{4} $
	79	$D_{4}$	$ 1 + $$43\!\cdots\!20$$ T + $$89\!\cdots\!58$$ T^{2} + $$43\!\cdots\!20$$ p^{31} T^{3} + p^{62} T^{4} $
	83	$D_{4}$	$ 1 - $$41\!\cdots\!24$$ T + $$61\!\cdots\!78$$ T^{2} - $$41\!\cdots\!24$$ p^{31} T^{3} + p^{62} T^{4} $
	89	$D_{4}$	$ 1 - $$52\!\cdots\!20$$ T + $$54\!\cdots\!78$$ T^{2} - $$52\!\cdots\!20$$ p^{31} T^{3} + p^{62} T^{4} $
	97	$D_{4}$	$ 1 - $$93\!\cdots\!72$$ T + $$92\!\cdots\!02$$ T^{2} - $$93\!\cdots\!72$$ p^{31} T^{3} + p^{62} 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

−20.20488582073873889568284906176, −20.01105660898587724230094696828, −18.97780315211315065396241743829, −18.77814365731435503504062121491, −16.98194262506662225952897443350, −16.70058092187662715182165716535, −15.58346985013191643414423181190, −14.70894182662024891153975298837, −13.24032858658847570835527694803, −12.11788598820107532506736370378, −10.94959143486583464217412494644, −9.768600610567284473766511484641, −8.981253192739081945156366818151, −8.389213250405636643321970553291, −6.64626188904998780763796389567, −6.48029438036995491002628402012, −3.72883487562511292311851567497, −3.06058363866767942588973868552, −1.62918593769605815388407559532, −0.48347077370412797938556807682, 0.48347077370412797938556807682, 1.62918593769605815388407559532, 3.06058363866767942588973868552, 3.72883487562511292311851567497, 6.48029438036995491002628402012, 6.64626188904998780763796389567, 8.389213250405636643321970553291, 8.981253192739081945156366818151, 9.768600610567284473766511484641, 10.94959143486583464217412494644, 12.11788598820107532506736370378, 13.24032858658847570835527694803, 14.70894182662024891153975298837, 15.58346985013191643414423181190, 16.70058092187662715182165716535, 16.98194262506662225952897443350, 18.77814365731435503504062121491, 18.97780315211315065396241743829, 20.01105660898587724230094696828, 20.20488582073873889568284906176

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 $\F_{q}$

\(L(16)\)	\(\approx\)	\(0.8093087754\)
\(L(\frac12)\)	\(\approx\)	\(0.8093087754\)
\(L(\frac{33}{2})\)		not available
\(L(1)\)		not available

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