LMFDB - L-function 8-75e4-1.1-c4e4-0-4

Dirichlet series

L(s) = 1

+ 44·4^-s + 72·9^-s + 940·16^-s − 1.23e3·19^-s + 128·31^-s + 3.16e3·36^-s − 9.13e3·49^-s − 3.71e3·61^-s + 8.80e3·64^-s − 5.42e4·76^-s − 32·79^-s − 1.37e3·81^-s + 2.54e4·109^-s + 3.98e4·121^-s + 5.63e3·124^-s + 127^-s + 131^-s + 137^-s + 139^-s + 6.76e4·144^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 5.56e3·169^-s − 8.87e4·171^-s + ⋯

L(s) = 1

+ 11/4·4^-s + 8/9·9^-s + 3.67·16^-s − 3.41·19^-s + 0.133·31^-s + 22/9·36^-s − 3.80·49^-s − 0.997·61^-s + 2.14·64^-s − 9.38·76^-s − 0.00512·79^-s − 0.209·81^-s + 2.13·109^-s + 2.72·121^-s + 0.366·124^-s + 6.20e−5·127^-s + 5.82e−5·131^-s + 5.32e−5·137^-s + 5.17e−5·139^-s + 3.26·144^-s + 4.50e−5·149^-s + 4.38e−5·151^-s + 4.05e−5·157^-s + 3.76e−5·163^-s + 3.58e−5·167^-s + 0.194·169^-s − 3.03·171^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$31640625$ = $3^{4} \cdot 5^{8}$
Sign:	$1$
Analytic conductor:	$3612.62$
Root analytic conductor:	$2.78437$
Motivic weight:	$4$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 31640625,\ (\ :2, 2, 2, 2),\ 1)$

Particular Values

$L(\frac{5}{2})$	$\approx$	$5.016314517$
$L(\frac12)$	$\approx$	$5.016314517$
$L(3)$		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	3	$C_2^2$	$ 1 - 8 p^{2} T^{2} + p^{8} T^{4} $
	5		$ 1 $
good	2	$C_2^2$	$ ( 1 - 11 p T^{2} + p^{8} T^{4} )^{2} $
	7	$C_2^2$	$ ( 1 + 4568 T^{2} + p^{8} T^{4} )^{2} $
	11	$C_2^2$	$ ( 1 - 19922 T^{2} + p^{8} T^{4} )^{2} $
	13	$C_2^2$	$ ( 1 - 214 p T^{2} + p^{8} T^{4} )^{2} $
	17	$C_2^2$	$ ( 1 - 89602 T^{2} + p^{8} T^{4} )^{2} $
	19	$C_2$	$ ( 1 + 308 T + p^{4} T^{2} )^{4} $
	23	$C_2^2$	$ ( 1 - 388072 T^{2} + p^{8} T^{4} )^{2} $
	29	$C_2^2$	$ ( 1 - 1377122 T^{2} + p^{8} T^{4} )^{2} $
	31	$C_2$	$ ( 1 - 32 T + p^{4} T^{2} )^{4} $
	37	$C_2^2$	$ ( 1 + 2097218 T^{2} + p^{8} T^{4} )^{2} $
	41	$C_2^2$	$ ( 1 - 1324862 T^{2} + p^{8} T^{4} )^{2} $
	43	$C_2^2$	$ ( 1 + 154328 T^{2} + p^{8} T^{4} )^{2} $
	47	$C_2^2$	$ ( 1 - 3784072 T^{2} + p^{8} T^{4} )^{2} $
	53	$C_2^2$	$ ( 1 - 13773922 T^{2} + p^{8} T^{4} )^{2} $
	59	$C_2^2$	$ ( 1 - 8500562 T^{2} + p^{8} T^{4} )^{2} $
	61	$C_2$	$ ( 1 + 928 T + p^{4} T^{2} )^{4} $
	67	$C_2^2$	$ ( 1 + 33618968 T^{2} + p^{8} T^{4} )^{2} $
	71	$C_2^2$	$ ( 1 - 29257922 T^{2} + p^{8} T^{4} )^{2} $
	73	$C_2^2$	$ ( 1 + 38971298 T^{2} + p^{8} T^{4} )^{2} $
	79	$C_2$	$ ( 1 + 8 T + p^{4} T^{2} )^{4} $
	83	$C_2^2$	$ ( 1 - 75232552 T^{2} + p^{8} T^{4} )^{2} $
	89	$C_2^2$	$ ( 1 - 39222722 T^{2} + p^{8} T^{4} )^{2} $
	97	$C_2^2$	$ ( 1 + 170764898 T^{2} + p^{8} T^{4} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−10.25949964943031990456708898712, −9.863971345531952065343425164016, −9.369103140582373527845874025808, −9.290404020332520169555084294406, −8.674978203427884348229391751629, −8.194938513935151130105712395360, −8.131128173456404989091139422749, −7.968936007023284562318892157503, −7.21248292970861316165984481824, −7.06132179658095188282352393732, −6.82944989024557793775279229477, −6.68107233291403665662742846401, −6.10577109875096889924542365741, −6.00656225657649521388237613816, −5.83908078730190760662338582089, −4.83301708454504493064409526956, −4.53526844587481686923438187718, −4.39555455396191878173550644546, −3.57009961599535403304284743719, −3.17000569511315101721430677944, −2.74068181530285147226818461064, −2.09357639942449417823164059186, −1.74114622374841319534927830500, −1.73793635366936271801659769751, −0.46398618199337959240624110096, 0.46398618199337959240624110096, 1.73793635366936271801659769751, 1.74114622374841319534927830500, 2.09357639942449417823164059186, 2.74068181530285147226818461064, 3.17000569511315101721430677944, 3.57009961599535403304284743719, 4.39555455396191878173550644546, 4.53526844587481686923438187718, 4.83301708454504493064409526956, 5.83908078730190760662338582089, 6.00656225657649521388237613816, 6.10577109875096889924542365741, 6.68107233291403665662742846401, 6.82944989024557793775279229477, 7.06132179658095188282352393732, 7.21248292970861316165984481824, 7.968936007023284562318892157503, 8.131128173456404989091139422749, 8.194938513935151130105712395360, 8.674978203427884348229391751629, 9.290404020332520169555084294406, 9.369103140582373527845874025808, 9.863971345531952065343425164016, 10.25949964943031990456708898712

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 $\F_{q}$

\(L(\frac{5}{2})\)	\(\approx\)	\(5.016314517\)
\(L(\frac12)\)	\(\approx\)	\(5.016314517\)
\(L(3)\)		not available
\(L(1)\)		not available

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