LMFDB - L-function 8-75e4-1.1-c2e4-0-0

Dirichlet series

L(s) = 1

− 72·11^-s + 7·16^-s + 88·31^-s − 72·41^-s + 8·61^-s + 288·71^-s − 9·81^-s − 432·101^-s + 2.75e3·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 173^-s − 504·176^-s + 179^-s + 181^-s + 191^-s + 193^-s + 197^-s + 199^-s + 211^-s + ⋯

L(s) = 1

− 6.54·11^-s + 7/16·16^-s + 2.83·31^-s − 1.75·41^-s + 8/61·61^-s + 4.05·71^-s − 1/9·81^-s − 4.27·101^-s + 22.7·121^-s + 0.00787·127^-s + 0.00763·131^-s + 0.00729·137^-s + 0.00719·139^-s + 0.00671·149^-s + 0.00662·151^-s + 0.00636·157^-s + 0.00613·163^-s + 0.00598·167^-s + 0.00578·173^-s − 2.86·176^-s + 0.00558·179^-s + 0.00552·181^-s + 0.00523·191^-s + 0.00518·193^-s + 0.00507·197^-s + 0.00502·199^-s + 0.00473·211^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]

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

Invariants

Degree:	$8$
Conductor:	$31640625$ = $3^{4} \cdot 5^{8}$
Sign:	$1$
Analytic conductor:	$17.4415$
Root analytic conductor:	$1.42954$
Motivic weight:	$2$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 31640625,\ (\ :1, 1, 1, 1),\ 1)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.5464015786$
$L(\frac12)$	$\approx$	$0.5464015786$
$L(2)$		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	3	$C_2^2$	$ 1 + p^{2} T^{4} $
	5		$ 1 $
good	2	$C_2^3$	$ 1 - 7 T^{4} + p^{8} T^{8} $
	7	$C_2^3$	$ 1 - 4702 T^{4} + p^{8} T^{8} $
	11	$C_2$	$ ( 1 + 18 T + p^{2} T^{2} )^{4} $
	13	$C_2^3$	$ 1 - 4222 T^{4} + p^{8} T^{8} $
	17	$C_2^3$	$ 1 + 113858 T^{4} + p^{8} T^{8} $
	19	$C_2^2$	$ ( 1 - 622 T^{2} + p^{4} T^{4} )^{2} $
	23	$C_2^3$	$ 1 - 475582 T^{4} + p^{8} T^{8} $
	29	$C_1$$\times$$C_1$	$ ( 1 - p T )^{4}( 1 + p T )^{4} $
	31	$C_2$	$ ( 1 - 22 T + p^{2} T^{2} )^{4} $
	37	$C_2^3$	$ 1 + 3168578 T^{4} + p^{8} T^{8} $
	41	$C_2$	$ ( 1 + 18 T + p^{2} T^{2} )^{4} $
	43	$C_2^3$	$ 1 - 2956702 T^{4} + p^{8} T^{8} $
	47	$C_2^3$	$ 1 - 9478462 T^{4} + p^{8} T^{8} $
	53	$C_2^3$	$ 1 + 15243938 T^{4} + p^{8} T^{8} $
	59	$C_2^2$	$ ( 1 + 1138 T^{2} + p^{4} T^{4} )^{2} $
	61	$C_2$	$ ( 1 - 2 T + p^{2} T^{2} )^{4} $
	67	$C_2^3$	$ 1 - 14394142 T^{4} + p^{8} T^{8} $
	71	$C_2$	$ ( 1 - 72 T + p^{2} T^{2} )^{4} $
	73	$C_2^3$	$ 1 - 10963582 T^{4} + p^{8} T^{8} $
	79	$C_2^2$	$ ( 1 - 7582 T^{2} + p^{4} T^{4} )^{2} $
	83	$C_2^3$	$ 1 - 31395742 T^{4} + p^{8} T^{8} $
	89	$C_2^2$	$ ( 1 - 7742 T^{2} + p^{4} T^{4} )^{2} $
	97	$C_2^3$	$ 1 - 171536062 T^{4} + p^{8} T^{8} $
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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

−10.43272677271919347671140114634, −10.32854973849335610960309268242, −10.08845407323857217880959204922, −9.739890412006283037553083325122, −9.737129714682633257585266768364, −8.941822631590589330935127162669, −8.297992431837071455606485224541, −8.242542159109238893010863315701, −8.196381258044041905517853924566, −7.84828196928130944765311894468, −7.66229350958728451736624291298, −7.27028704252744003276067801359, −6.64449596392523289051848591996, −6.54992475316093479185306346998, −5.71175058250511466770320981869, −5.53765216867931320478999536059, −5.12847651918852031711671823561, −5.11961660639705437874096027671, −4.76273541405197502558615016847, −4.10780310830882473457413560765, −3.10813538946266115431392591894, −3.00536279826085397227320469675, −2.47970882258389291612008452390, −2.26277832427149618763724769302, −0.40958377309226192183101428220, 0.40958377309226192183101428220, 2.26277832427149618763724769302, 2.47970882258389291612008452390, 3.00536279826085397227320469675, 3.10813538946266115431392591894, 4.10780310830882473457413560765, 4.76273541405197502558615016847, 5.11961660639705437874096027671, 5.12847651918852031711671823561, 5.53765216867931320478999536059, 5.71175058250511466770320981869, 6.54992475316093479185306346998, 6.64449596392523289051848591996, 7.27028704252744003276067801359, 7.66229350958728451736624291298, 7.84828196928130944765311894468, 8.196381258044041905517853924566, 8.242542159109238893010863315701, 8.297992431837071455606485224541, 8.941822631590589330935127162669, 9.737129714682633257585266768364, 9.739890412006283037553083325122, 10.08845407323857217880959204922, 10.32854973849335610960309268242, 10.43272677271919347671140114634

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.5464015786\)
\(L(\frac12)\)	\(\approx\)	\(0.5464015786\)
\(L(2)\)		not available
\(L(1)\)		not available

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