LMFDB - L-function 4-2e8-1.1-c37e2-0-1

Dirichlet series

L(s) = 1

− 1.39e7·3^-s + 5.52e12·5^-s + 3.44e15·7^-s − 8.99e17·9^-s + 2.67e19·11^-s + 5.30e20·13^-s − 7.73e19·15^-s − 8.94e22·17^-s − 3.73e23·19^-s − 4.82e22·21^-s + 2.62e25·23^-s − 2.20e23·25^-s + 1.88e25·27^-s − 1.27e27·29^-s − 2.61e26·31^-s − 3.74e26·33^-s + 1.90e28·35^-s − 6.80e28·37^-s − 7.42e27·39^-s − 1.26e30·41^-s + 2.57e30·43^-s − 4.97e30·45^-s − 4.25e30·47^-s − 1.45e31·49^-s + 1.25e30·51^-s + 1.59e32·53^-s + 1.47e32·55^-s + ⋯

L(s) = 1

− 0.0208·3^-s + 0.648·5^-s + 0.800·7^-s − 1.99·9^-s + 1.44·11^-s + 1.30·13^-s − 0.0135·15^-s − 1.54·17^-s − 0.822·19^-s − 0.0166·21^-s + 1.68·23^-s − 0.00302·25^-s + 0.0624·27^-s − 1.12·29^-s − 0.0671·31^-s − 0.0302·33^-s + 0.518·35^-s − 0.662·37^-s − 0.0272·39^-s − 1.83·41^-s + 1.55·43^-s − 1.29·45^-s − 0.495·47^-s − 0.782·49^-s + 0.0321·51^-s + 2.01·53^-s + 0.939·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+37/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$256$ = $2^{8}$
Sign:	$1$
Analytic conductor:	$19249.4$
Root analytic conductor:	$11.7788$
Motivic weight:	$37$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 256,\ (\ :37/2, 37/2),\ 1)$

Particular Values

$L(19)$	$\approx$	$3.744803325$
$L(\frac12)$	$\approx$	$3.744803325$
$L(\frac{39}{2})$		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 + 518200 p^{3} T + 15239131479430 p^{10} T^{2} + 518200 p^{40} T^{3} + p^{74} T^{4} $
	5	$D_{4}$	$ 1 - 221183375436 p^{2} T + $$39\!\cdots\!58$$ p^{7} T^{2} - 221183375436 p^{39} T^{3} + p^{74} T^{4} $
	7	$D_{4}$	$ 1 - 70376407214000 p^{2} T + $$15\!\cdots\!50$$ p^{5} T^{2} - 70376407214000 p^{39} T^{3} + p^{74} T^{4} $
	11	$D_{4}$	$ 1 - 2430366941349867096 p T + $$53\!\cdots\!46$$ p^{3} T^{2} - 2430366941349867096 p^{38} T^{3} + p^{74} T^{4} $
	13	$D_{4}$	$ 1 - 40813980127225629100 p T + $$14\!\cdots\!70$$ p^{3} T^{2} - 40813980127225629100 p^{38} T^{3} + p^{74} T^{4} $
	17	$D_{4}$	$ 1 + $$52\!\cdots\!00$$ p T + $$17\!\cdots\!10$$ p^{3} T^{2} + $$52\!\cdots\!00$$ p^{38} T^{3} + p^{74} T^{4} $
	19	$D_{4}$	$ 1 + $$37\!\cdots\!20$$ T + $$10\!\cdots\!62$$ p T^{2} + $$37\!\cdots\!20$$ p^{37} T^{3} + p^{74} T^{4} $
	23	$D_{4}$	$ 1 - $$26\!\cdots\!00$$ T + $$21\!\cdots\!70$$ p T^{2} - $$26\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	29	$D_{4}$	$ 1 + $$43\!\cdots\!80$$ p T + $$26\!\cdots\!98$$ p^{2} T^{2} + $$43\!\cdots\!80$$ p^{38} T^{3} + p^{74} T^{4} $
	31	$D_{4}$	$ 1 + $$84\!\cdots\!04$$ p T + $$25\!\cdots\!06$$ p^{2} T^{2} + $$84\!\cdots\!04$$ p^{38} T^{3} + p^{74} T^{4} $
	37	$D_{4}$	$ 1 + $$49\!\cdots\!00$$ p^{2} T + $$13\!\cdots\!90$$ T^{2} + $$49\!\cdots\!00$$ p^{39} T^{3} + p^{74} T^{4} $
	41	$D_{4}$	$ 1 + $$12\!\cdots\!36$$ T + $$12\!\cdots\!86$$ T^{2} + $$12\!\cdots\!36$$ p^{37} T^{3} + p^{74} T^{4} $
	43	$D_{4}$	$ 1 - $$25\!\cdots\!00$$ T + $$44\!\cdots\!50$$ T^{2} - $$25\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	47	$D_{4}$	$ 1 + $$42\!\cdots\!00$$ T + $$14\!\cdots\!70$$ T^{2} + $$42\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	53	$D_{4}$	$ 1 - $$15\!\cdots\!00$$ T + $$16\!\cdots\!70$$ T^{2} - $$15\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	59	$D_{4}$	$ 1 - $$23\!\cdots\!40$$ T + $$64\!\cdots\!38$$ T^{2} - $$23\!\cdots\!40$$ p^{37} T^{3} + p^{74} T^{4} $
	61	$D_{4}$	$ 1 - $$10\!\cdots\!44$$ T + $$10\!\cdots\!26$$ T^{2} - $$10\!\cdots\!44$$ p^{37} T^{3} + p^{74} T^{4} $
	67	$D_{4}$	$ 1 - $$10\!\cdots\!00$$ T + $$94\!\cdots\!30$$ T^{2} - $$10\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	71	$D_{4}$	$ 1 - $$74\!\cdots\!16$$ T + $$58\!\cdots\!46$$ T^{2} - $$74\!\cdots\!16$$ p^{37} T^{3} + p^{74} T^{4} $
	73	$D_{4}$	$ 1 - $$19\!\cdots\!00$$ T + $$18\!\cdots\!10$$ T^{2} - $$19\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	79	$D_{4}$	$ 1 + $$27\!\cdots\!80$$ T + $$64\!\cdots\!42$$ p T^{2} + $$27\!\cdots\!80$$ p^{37} T^{3} + p^{74} T^{4} $
	83	$D_{4}$	$ 1 - $$47\!\cdots\!00$$ T + $$24\!\cdots\!30$$ T^{2} - $$47\!\cdots\!00$$ p^{37} T^{3} + p^{74} T^{4} $
	89	$D_{4}$	$ 1 + $$13\!\cdots\!60$$ T + $$23\!\cdots\!58$$ T^{2} + $$13\!\cdots\!60$$ p^{37} T^{3} + p^{74} T^{4} $
	97	$D_{4}$	$ 1 - $$60\!\cdots\!00$$ T + $$57\!\cdots\!70$$ T^{2} - $$60\!\cdots\!00$$ p^{37} T^{3} + p^{74} 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

−11.63570038206931187760117896643, −11.61817963187377350838378567945, −10.93548851042354001621649290320, −10.66444627583628319518712175507, −9.320569638082412155012610814067, −9.082530952924149311279851224062, −8.392111776729606655268157506374, −8.381354059317065715614190486156, −6.84304770847050538803454861751, −6.75655037235575088644195409181, −5.87874172726784002714066863420, −5.55352843763540579957358294669, −4.83259897192983661503065920965, −4.07263606092347850393129449613, −3.49906419536323136957132158656, −2.85712715797902631722937602909, −1.92473319049223611386570096839, −1.89017386356784304373828656014, −0.914544740680475232016605877677, −0.44125494373399518763772837622, 0.44125494373399518763772837622, 0.914544740680475232016605877677, 1.89017386356784304373828656014, 1.92473319049223611386570096839, 2.85712715797902631722937602909, 3.49906419536323136957132158656, 4.07263606092347850393129449613, 4.83259897192983661503065920965, 5.55352843763540579957358294669, 5.87874172726784002714066863420, 6.75655037235575088644195409181, 6.84304770847050538803454861751, 8.381354059317065715614190486156, 8.392111776729606655268157506374, 9.082530952924149311279851224062, 9.320569638082412155012610814067, 10.66444627583628319518712175507, 10.93548851042354001621649290320, 11.61817963187377350838378567945, 11.63570038206931187760117896643

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 $\F_{q}$

\(L(19)\)	\(\approx\)	\(3.744803325\)
\(L(\frac12)\)	\(\approx\)	\(3.744803325\)
\(L(\frac{39}{2})\)		not available
\(L(1)\)		not available

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