LMFDB - L-function 4-2e8-1.1-c25e2-0-1

Dirichlet series

L(s) = 1

+ 8.99e5·3^-s − 3.99e8·5^-s + 4.05e10·7^-s − 1.82e10·9^-s + 1.30e12·11^-s + 3.27e13·13^-s − 3.59e14·15^-s + 4.62e15·17^-s − 2.46e16·19^-s + 3.64e16·21^-s + 3.10e16·23^-s − 2.65e17·25^-s + 1.34e15·27^-s − 3.32e18·29^-s + 6.42e18·31^-s + 1.17e18·33^-s − 1.61e19·35^-s + 3.16e19·37^-s + 2.94e19·39^-s − 1.63e19·41^-s + 3.41e20·43^-s + 7.27e18·45^-s + 1.11e21·47^-s + 7.37e20·49^-s + 4.16e21·51^-s + 7.61e21·53^-s − 5.21e20·55^-s + ⋯

L(s) = 1

+ 0.977·3^-s − 0.731·5^-s + 1.10·7^-s − 0.0215·9^-s + 0.125·11^-s + 0.389·13^-s − 0.714·15^-s + 1.92·17^-s − 2.55·19^-s + 1.08·21^-s + 0.295·23^-s − 0.891·25^-s + 0.00172·27^-s − 1.74·29^-s + 1.46·31^-s + 0.122·33^-s − 0.809·35^-s + 0.791·37^-s + 0.380·39^-s − 0.112·41^-s + 1.30·43^-s + 0.0157·45^-s + 1.40·47^-s + 0.549·49^-s + 1.88·51^-s + 2.12·53^-s − 0.0918·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$256$ = $2^{8}$
Sign:	$1$
Analytic conductor:	$4014.42$
Root analytic conductor:	$7.95986$
Motivic weight:	$25$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 256,\ (\ :25/2, 25/2),\ 1)$

Particular Values

$L(13)$	$\approx$	$4.768220647$
$L(\frac12)$	$\approx$	$4.768220647$
$L(\frac{27}{2})$		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 - 33320 p^{3} T + 1135214918 p^{6} T^{2} - 33320 p^{28} T^{3} + p^{50} T^{4} $
	5	$D_{4}$	$ 1 + 399350196 T + 17004419092170526 p^{2} T^{2} + 399350196 p^{25} T^{3} + p^{50} T^{4} $
	7	$D_{4}$	$ 1 - 5788351760 p T + 376748861434354830 p^{4} T^{2} - 5788351760 p^{26} T^{3} + p^{50} T^{4} $
	11	$D_{4}$	$ 1 - 1306289379240 T + $$16\!\cdots\!62$$ p^{2} T^{2} - 1306289379240 p^{25} T^{3} + p^{50} T^{4} $
	13	$D_{4}$	$ 1 - 32729023492060 T + $$10\!\cdots\!14$$ p T^{2} - 32729023492060 p^{25} T^{3} + p^{50} T^{4} $
	17	$D_{4}$	$ 1 - 4624697010380580 T + $$88\!\cdots\!94$$ p T^{2} - 4624697010380580 p^{25} T^{3} + p^{50} T^{4} $
	19	$D_{4}$	$ 1 + 1298632725303928 p T + $$47\!\cdots\!06$$ p^{3} T^{2} + 1298632725303928 p^{26} T^{3} + p^{50} T^{4} $
	23	$D_{4}$	$ 1 - 1349692516511280 p T + $$37\!\cdots\!70$$ p^{2} T^{2} - 1349692516511280 p^{26} T^{3} + p^{50} T^{4} $
	29	$D_{4}$	$ 1 + 3328137384414404868 T + $$98\!\cdots\!54$$ T^{2} + 3328137384414404868 p^{25} T^{3} + p^{50} T^{4} $
	31	$D_{4}$	$ 1 - 6421803981270308288 T + $$37\!\cdots\!38$$ T^{2} - 6421803981270308288 p^{25} T^{3} + p^{50} T^{4} $
	37	$D_{4}$	$ 1 - 31687565809212923020 T + $$14\!\cdots\!50$$ T^{2} - 31687565809212923020 p^{25} T^{3} + p^{50} T^{4} $
	41	$D_{4}$	$ 1 + 16316808963958618668 T + $$35\!\cdots\!58$$ T^{2} + 16316808963958618668 p^{25} T^{3} + p^{50} T^{4} $
	43	$D_{4}$	$ 1 - $$34\!\cdots\!20$$ T + $$15\!\cdots\!86$$ T^{2} - $$34\!\cdots\!20$$ p^{25} T^{3} + p^{50} T^{4} $
	47	$D_{4}$	$ 1 - $$11\!\cdots\!00$$ T + $$14\!\cdots\!98$$ T^{2} - $$11\!\cdots\!00$$ p^{25} T^{3} + p^{50} T^{4} $
	53	$D_{4}$	$ 1 - $$76\!\cdots\!20$$ T + $$39\!\cdots\!30$$ T^{2} - $$76\!\cdots\!20$$ p^{25} T^{3} + p^{50} T^{4} $
	59	$D_{4}$	$ 1 - $$44\!\cdots\!16$$ T - $$94\!\cdots\!38$$ T^{2} - $$44\!\cdots\!16$$ p^{25} T^{3} + p^{50} T^{4} $
	61	$D_{4}$	$ 1 + $$10\!\cdots\!36$$ T + $$86\!\cdots\!26$$ T^{2} + $$10\!\cdots\!36$$ p^{25} T^{3} + p^{50} T^{4} $
	67	$D_{4}$	$ 1 - $$27\!\cdots\!80$$ T + $$85\!\cdots\!10$$ T^{2} - $$27\!\cdots\!80$$ p^{25} T^{3} + p^{50} T^{4} $
	71	$D_{4}$	$ 1 - $$31\!\cdots\!04$$ T + $$35\!\cdots\!06$$ T^{2} - $$31\!\cdots\!04$$ p^{25} T^{3} + p^{50} T^{4} $
	73	$D_{4}$	$ 1 - $$39\!\cdots\!60$$ T + $$10\!\cdots\!62$$ T^{2} - $$39\!\cdots\!60$$ p^{25} T^{3} + p^{50} T^{4} $
	79	$D_{4}$	$ 1 - $$11\!\cdots\!04$$ T + $$80\!\cdots\!02$$ T^{2} - $$11\!\cdots\!04$$ p^{25} T^{3} + p^{50} T^{4} $
	83	$D_{4}$	$ 1 - $$25\!\cdots\!00$$ T + $$33\!\cdots\!30$$ T^{2} - $$25\!\cdots\!00$$ p^{25} T^{3} + p^{50} T^{4} $
	89	$D_{4}$	$ 1 - $$39\!\cdots\!68$$ T + $$14\!\cdots\!54$$ T^{2} - $$39\!\cdots\!68$$ p^{25} T^{3} + p^{50} T^{4} $
	97	$D_{4}$	$ 1 + $$82\!\cdots\!40$$ T + $$79\!\cdots\!30$$ T^{2} + $$82\!\cdots\!40$$ p^{25} T^{3} + p^{50} 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

−13.80152269093836801695896176843, −13.31124145399028256112686772845, −12.14048763053446263415875297538, −12.11577773459279919610333981566, −10.88771204240397283423236304718, −10.78117864941364878933052231029, −9.644919492513740894364065042545, −8.938559351388101803554025744300, −8.135187747209364789561911869829, −8.081174718138896845782987673218, −7.36395724376649930299443280559, −6.28169174889196463877527173696, −5.58736078024020582626922649598, −4.68856044500820112157426913614, −3.76136085318798941217036727950, −3.75141192429493631614077681815, −2.35526273028551314834433292814, −2.24150642115672016463226189830, −1.10795310023170851169367664492, −0.56776397141431924432087652530, 0.56776397141431924432087652530, 1.10795310023170851169367664492, 2.24150642115672016463226189830, 2.35526273028551314834433292814, 3.75141192429493631614077681815, 3.76136085318798941217036727950, 4.68856044500820112157426913614, 5.58736078024020582626922649598, 6.28169174889196463877527173696, 7.36395724376649930299443280559, 8.081174718138896845782987673218, 8.135187747209364789561911869829, 8.938559351388101803554025744300, 9.644919492513740894364065042545, 10.78117864941364878933052231029, 10.88771204240397283423236304718, 12.11577773459279919610333981566, 12.14048763053446263415875297538, 13.31124145399028256112686772845, 13.80152269093836801695896176843

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 $\F_{q}$

\(L(13)\)	\(\approx\)	\(4.768220647\)
\(L(\frac12)\)	\(\approx\)	\(4.768220647\)
\(L(\frac{27}{2})\)		not available
\(L(1)\)		not available

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