LMFDB - L-function 8-5e8-1.1-c33e4-0-0

Dirichlet series

L(s) = 1

+ 2.52e9·4^-s + 1.91e16·9^-s + 2.67e17·11^-s + 3.80e19·16^-s + 2.72e21·19^-s + 3.34e24·29^-s − 1.24e25·31^-s + 4.83e25·36^-s + 5.55e26·41^-s + 6.76e26·44^-s + 1.29e28·49^-s + 6.11e29·59^-s − 1.14e28·61^-s + 3.62e29·64^-s − 5.33e30·71^-s + 6.87e30·76^-s + 1.70e31·79^-s + 2.14e32·81^-s − 2.65e32·89^-s + 5.12e33·99^-s + 1.36e33·101^-s − 9.95e33·109^-s + 8.46e33·116^-s − 3.79e33·121^-s − 3.15e34·124^-s + 127^-s + 131^-s + ⋯

L(s) = 1

+ 0.294·4^-s + 3.44·9^-s + 1.75·11^-s + 0.515·16^-s + 2.16·19^-s + 2.48·29^-s − 3.08·31^-s + 1.01·36^-s + 1.35·41^-s + 0.516·44^-s + 1.67·49^-s + 3.69·59^-s − 0.0400·61^-s + 0.572·64^-s − 1.51·71^-s + 0.636·76^-s + 0.834·79^-s + 6.94·81^-s − 1.81·89^-s + 6.04·99^-s + 1.15·101^-s − 2.40·109^-s + 0.731·116^-s − 0.163·121^-s − 0.906·124^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+33/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$390625$ = $5^{8}$
Sign:	$1$
Analytic conductor:	$8.84553\times 10^{8}$
Root analytic conductor:	$13.1322$
Motivic weight:	$33$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 390625,\ (\ :33/2, 33/2, 33/2, 33/2),\ 1)$

Particular Values

$L(17)$	$\approx$	$15.96775858$
$L(\frac12)$	$\approx$	$15.96775858$
$L(\frac{35}{2})$		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	5		$ 1 $
good	2	$D_4\times C_2$	$ 1 - 9873575 p^{8} T^{2} - 1888135309767 p^{24} T^{4} - 9873575 p^{74} T^{6} + p^{132} T^{8} $
	3	$D_4\times C_2$	$ 1 - 26254126321100 p^{6} T^{2} + $$39\!\cdots\!22$$ p^{18} T^{4} - 26254126321100 p^{72} T^{6} + p^{132} T^{8} $
	7	$D_4\times C_2$	$ 1 - $$26\!\cdots\!00$$ p^{2} T^{2} + $$21\!\cdots\!98$$ p^{8} T^{4} - $$26\!\cdots\!00$$ p^{68} T^{6} + p^{132} T^{8} $
	11	$D_{4}$	$ ( 1 - 12170165040174024 p T + $$23\!\cdots\!66$$ p^{2} T^{2} - 12170165040174024 p^{34} T^{3} + p^{66} T^{4} )^{2} $
	13	$D_4\times C_2$	$ 1 - $$10\!\cdots\!00$$ p^{2} T^{2} + $$30\!\cdots\!02$$ p^{6} T^{4} - $$10\!\cdots\!00$$ p^{68} T^{6} + p^{132} T^{8} $
	17	$D_4\times C_2$	$ 1 + $$15\!\cdots\!00$$ T^{2} + $$95\!\cdots\!42$$ p^{2} T^{4} + $$15\!\cdots\!00$$ p^{66} T^{6} + p^{132} T^{8} $
	19	$D_{4}$	$ ( 1 - $$13\!\cdots\!00$$ T + $$17\!\cdots\!22$$ p T^{2} - $$13\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	23	$D_4\times C_2$	$ 1 - $$24\!\cdots\!00$$ T^{2} + $$52\!\cdots\!82$$ p^{2} T^{4} - $$24\!\cdots\!00$$ p^{66} T^{6} + p^{132} T^{8} $
	29	$D_{4}$	$ ( 1 - $$57\!\cdots\!00$$ p T + $$46\!\cdots\!58$$ p^{2} T^{2} - $$57\!\cdots\!00$$ p^{34} T^{3} + p^{66} T^{4} )^{2} $
	31	$D_{4}$	$ ( 1 + $$20\!\cdots\!36$$ p T + $$30\!\cdots\!86$$ p^{2} T^{2} + $$20\!\cdots\!36$$ p^{34} T^{3} + p^{66} T^{4} )^{2} $
	37	$D_4\times C_2$	$ 1 - $$16\!\cdots\!00$$ T^{2} + $$12\!\cdots\!18$$ T^{4} - $$16\!\cdots\!00$$ p^{66} T^{6} + p^{132} T^{8} $
	41	$D_{4}$	$ ( 1 - $$27\!\cdots\!44$$ T + $$22\!\cdots\!26$$ T^{2} - $$27\!\cdots\!44$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	43	$D_4\times C_2$	$ 1 - $$48\!\cdots\!00$$ T^{2} - $$10\!\cdots\!02$$ T^{4} - $$48\!\cdots\!00$$ p^{66} T^{6} + p^{132} T^{8} $
	47	$D_4\times C_2$	$ 1 - $$45\!\cdots\!00$$ T^{2} + $$95\!\cdots\!58$$ T^{4} - $$45\!\cdots\!00$$ p^{66} T^{6} + p^{132} T^{8} $
	53	$D_4\times C_2$	$ 1 - $$18\!\cdots\!00$$ T^{2} + $$17\!\cdots\!58$$ T^{4} - $$18\!\cdots\!00$$ p^{66} T^{6} + p^{132} T^{8} $
	59	$D_{4}$	$ ( 1 - $$30\!\cdots\!00$$ T + $$77\!\cdots\!58$$ T^{2} - $$30\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	61	$D_{4}$	$ ( 1 + $$57\!\cdots\!36$$ T + $$16\!\cdots\!86$$ T^{2} + $$57\!\cdots\!36$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	67	$D_4\times C_2$	$ 1 - $$58\!\cdots\!00$$ T^{2} + $$14\!\cdots\!38$$ T^{4} - $$58\!\cdots\!00$$ p^{66} T^{6} + p^{132} T^{8} $
	71	$D_{4}$	$ ( 1 + $$26\!\cdots\!76$$ T + $$22\!\cdots\!66$$ T^{2} + $$26\!\cdots\!76$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	73	$D_4\times C_2$	$ 1 - $$37\!\cdots\!00$$ T^{2} + $$22\!\cdots\!78$$ T^{4} - $$37\!\cdots\!00$$ p^{66} T^{6} + p^{132} T^{8} $
	79	$D_{4}$	$ ( 1 - $$85\!\cdots\!00$$ T + $$85\!\cdots\!78$$ T^{2} - $$85\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	83	$D_4\times C_2$	$ 1 - $$66\!\cdots\!00$$ T^{2} + $$19\!\cdots\!38$$ T^{4} - $$66\!\cdots\!00$$ p^{66} T^{6} + p^{132} T^{8} $
	89	$D_{4}$	$ ( 1 + $$13\!\cdots\!00$$ T + $$47\!\cdots\!38$$ T^{2} + $$13\!\cdots\!00$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	97	$D_4\times C_2$	$ 1 - $$70\!\cdots\!00$$ T^{2} + $$34\!\cdots\!58$$ T^{4} - $$70\!\cdots\!00$$ p^{66} T^{6} + p^{132} 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

−7.42643418739594355350047933479, −7.01283143231191852436439322892, −6.95320191734799576983374617497, −6.52073576851840968158247973157, −6.44954667074274223118465155057, −5.94057826848992868277443194776, −5.36236872423600023931550743743, −5.31248398465302009492686864020, −5.18605420308471064939932757654, −4.43730711573230299268352256426, −4.30592944987151372285920308095, −4.18562675086109187871626062473, −3.74067515513565405365706615506, −3.60001395140453920284843399498, −3.50746022152091158879019733381, −2.72423542890499863245897275430, −2.66075507469925669054284849425, −2.19966231801485240598957879163, −1.74554174370435379989627101769, −1.70607888546296212809527741994, −1.13266830489321155152831898329, −1.07153670297011045903868857601, −1.06578092453934911451802601196, −0.804767480488882017166037510230, −0.23478862722530027689647920418, 0.23478862722530027689647920418, 0.804767480488882017166037510230, 1.06578092453934911451802601196, 1.07153670297011045903868857601, 1.13266830489321155152831898329, 1.70607888546296212809527741994, 1.74554174370435379989627101769, 2.19966231801485240598957879163, 2.66075507469925669054284849425, 2.72423542890499863245897275430, 3.50746022152091158879019733381, 3.60001395140453920284843399498, 3.74067515513565405365706615506, 4.18562675086109187871626062473, 4.30592944987151372285920308095, 4.43730711573230299268352256426, 5.18605420308471064939932757654, 5.31248398465302009492686864020, 5.36236872423600023931550743743, 5.94057826848992868277443194776, 6.44954667074274223118465155057, 6.52073576851840968158247973157, 6.95320191734799576983374617497, 7.01283143231191852436439322892, 7.42643418739594355350047933479

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 $\F_{q}$

\(L(17)\)	\(\approx\)	\(15.96775858\)
\(L(\frac12)\)	\(\approx\)	\(15.96775858\)
\(L(\frac{35}{2})\)		not available
\(L(1)\)		not available

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