LMFDB - L-function 4-3120e2-1.1-c0e2-0-0

Dirichlet series

L(s) = 1

− 2·2^-s − 2·3^-s + 3·4^-s + 4·6^-s − 4·8^-s + 3·9^-s − 2·11^-s − 6·12^-s + 5·16^-s − 6·18^-s + 4·22^-s + 8·24^-s − 25^-s − 4·27^-s − 6·32^-s + 4·33^-s + 9·36^-s − 6·44^-s − 2·47^-s − 10·48^-s + 2·50^-s + 8·54^-s − 2·59^-s − 2·61^-s + 7·64^-s − 8·66^-s + 4·71^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$9734400$ = $2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}$
Sign:	$1$
Analytic conductor:	$2.42450$
Root analytic conductor:	$1.24783$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 9734400,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.03361031529$
$L(\frac12)$	$\approx$	$0.03361031529$
$L(1)$		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	2	$C_1$	$ ( 1 + T )^{2} $
	3	$C_1$	$ ( 1 + T )^{2} $
	5	$C_2$	$ 1 + T^{2} $
	13	$C_2$	$ 1 + T^{2} $
good	7	$C_2^2$	$ 1 + T^{4} $
	11	$C_1$$\times$$C_2$	$ ( 1 + T )^{2}( 1 + T^{2} ) $
	17	$C_2^2$	$ 1 + T^{4} $
	19	$C_2^2$	$ 1 + T^{4} $
	23	$C_2^2$	$ 1 + T^{4} $
	29	$C_2^2$	$ 1 + T^{4} $
	31	$C_2$	$ ( 1 + T^{2} )^{2} $
	37	$C_2$	$ ( 1 + T^{2} )^{2} $
	41	$C_2$	$ ( 1 + T^{2} )^{2} $
	43	$C_2$	$ ( 1 + T^{2} )^{2} $
	47	$C_1$$\times$$C_2$	$ ( 1 + T )^{2}( 1 + T^{2} ) $
	53	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	59	$C_1$$\times$$C_2$	$ ( 1 + T )^{2}( 1 + T^{2} ) $
	61	$C_1$$\times$$C_2$	$ ( 1 + T )^{2}( 1 + T^{2} ) $
	67	$C_2$	$ ( 1 + T^{2} )^{2} $
	71	$C_1$	$ ( 1 - T )^{4} $
	73	$C_2^2$	$ 1 + T^{4} $
	79	$C_2$	$ ( 1 + T^{2} )^{2} $
	83	$C_2$	$ ( 1 + T^{2} )^{2} $
	89	$C_1$	$ ( 1 + T )^{4} $
	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

−9.507170464792910245138997848198, −8.504103696041919693249171546958, −8.301069495687874322331030872027, −7.78094914515781151292707484117, −7.70063365520891063885660621895, −7.29596449489977323877486213399, −6.88384552490354203445996416339, −6.39466669168248110730928321720, −6.25996206647103381674040089934, −5.78253419618698056919096680866, −5.43629065668004695806064202251, −5.00424299342966177043749033129, −4.76125000540659101128154805114, −3.85165948722256170796859737615, −3.49295383250308019507920194441, −2.62026593299130443529441701431, −2.43282253719499938358967323645, −1.46875902122967954705443555760, −1.42027577008181672466879146213, −0.18230458639774769804195917253, 0.18230458639774769804195917253, 1.42027577008181672466879146213, 1.46875902122967954705443555760, 2.43282253719499938358967323645, 2.62026593299130443529441701431, 3.49295383250308019507920194441, 3.85165948722256170796859737615, 4.76125000540659101128154805114, 5.00424299342966177043749033129, 5.43629065668004695806064202251, 5.78253419618698056919096680866, 6.25996206647103381674040089934, 6.39466669168248110730928321720, 6.88384552490354203445996416339, 7.29596449489977323877486213399, 7.70063365520891063885660621895, 7.78094914515781151292707484117, 8.301069495687874322331030872027, 8.504103696041919693249171546958, 9.507170464792910245138997848198

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.03361031529\)
\(L(\frac12)\)	\(\approx\)	\(0.03361031529\)
\(L(1)\)		not available
\(L(1)\)		not available

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