LMFDB - L-function 4-1045e2-1.1-c0e2-0-0

Dirichlet series

L(s) = 1

− 2·2^-s + 2·4^-s − 2·8^-s + 2·11^-s + 2·13^-s + 3·16^-s − 4·22^-s − 2·23^-s − 25^-s − 4·26^-s − 4·32^-s + 4·44^-s + 4·46^-s − 2·47^-s + 2·50^-s + 4·52^-s + 4·64^-s − 81^-s − 4·88^-s − 4·92^-s + 4·94^-s − 2·100^-s − 4·104^-s + 2·107^-s + 3·121^-s + 127^-s − 4·128^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1092025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$1092025$ = $5^{2} \cdot 11^{2} \cdot 19^{2}$
Sign:	$1$
Analytic conductor:	$0.271986$
Root analytic conductor:	$0.722165$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 1092025,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3777158548$
$L(\frac12)$	$\approx$	$0.3777158548$
$L(1)$		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	5	$C_2$	$ 1 + T^{2} $
	11	$C_1$	$ ( 1 - T )^{2} $
	19	$C_2$	$ 1 + T^{2} $
good	2	$C_1$$\times$$C_2$	$ ( 1 + T )^{2}( 1 + T^{2} ) $
	3	$C_2^2$	$ 1 + T^{4} $
	7	$C_2^2$	$ 1 + T^{4} $
	13	$C_1$$\times$$C_2$	$ ( 1 - T )^{2}( 1 + T^{2} ) $
	17	$C_2^2$	$ 1 + T^{4} $
	23	$C_1$$\times$$C_2$	$ ( 1 + T )^{2}( 1 + T^{2} ) $
	29	$C_2$	$ ( 1 + T^{2} )^{2} $
	31	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	37	$C_2^2$	$ 1 + T^{4} $
	41	$C_2$	$ ( 1 + T^{2} )^{2} $
	43	$C_2^2$	$ 1 + T^{4} $
	47	$C_1$$\times$$C_2$	$ ( 1 + T )^{2}( 1 + T^{2} ) $
	53	$C_2^2$	$ 1 + T^{4} $
	59	$C_2$	$ ( 1 + T^{2} )^{2} $
	61	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	67	$C_2^2$	$ 1 + T^{4} $
	71	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	73	$C_2^2$	$ 1 + T^{4} $
	79	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	83	$C_2^2$	$ 1 + T^{4} $
	89	$C_2$	$ ( 1 + T^{2} )^{2} $
	97	$C_2^2$	$ 1 + 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

−10.33475460401745601792993923074, −9.693456368266116561304895484405, −9.397071014794155632236721369133, −9.190234876684105935306729063950, −8.707675254976155888493077512234, −8.333479885990558255116293997436, −8.013869389305407121251535938426, −7.85204055585456626447240482814, −6.88366514467458759264822792460, −6.75263552876485218216078419294, −6.19626198096216368649407430673, −5.77642914467062816249419478091, −5.68826381795151903456043795518, −4.40768063736441103055922973967, −4.05035672074264593152838961880, −3.42725120732277916088107447536, −3.24554637044310628553852979076, −1.79926170213146696960084929396, −1.75206183071507496831304329491, −0.868387995510878913450839233923, 0.868387995510878913450839233923, 1.75206183071507496831304329491, 1.79926170213146696960084929396, 3.24554637044310628553852979076, 3.42725120732277916088107447536, 4.05035672074264593152838961880, 4.40768063736441103055922973967, 5.68826381795151903456043795518, 5.77642914467062816249419478091, 6.19626198096216368649407430673, 6.75263552876485218216078419294, 6.88366514467458759264822792460, 7.85204055585456626447240482814, 8.013869389305407121251535938426, 8.333479885990558255116293997436, 8.707675254976155888493077512234, 9.190234876684105935306729063950, 9.397071014794155632236721369133, 9.693456368266116561304895484405, 10.33475460401745601792993923074

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.3777158548\)
\(L(\frac12)\)	\(\approx\)	\(0.3777158548\)
\(L(1)\)		not available
\(L(1)\)		not available

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