LMFDB - L-function 16-5e8-1.1-c19e8-0-0

Dirichlet series

L(s) = 1

+ 1.28e6·4^-s + 1.47e5·5^-s + 4.82e9·9^-s − 3.37e9·11^-s + 5.37e11·16^-s + 4.54e12·19^-s + 1.89e11·20^-s + 8.86e12·25^-s − 1.88e14·29^-s + 7.21e13·31^-s + 6.20e15·36^-s − 3.31e15·41^-s − 4.34e15·44^-s + 7.08e14·45^-s + 4.00e16·49^-s − 4.96e14·55^-s + 2.59e17·59^-s − 5.82e17·61^-s + 4.13e16·64^-s − 2.61e18·71^-s + 5.85e18·76^-s + 3.93e18·79^-s + 7.89e16·80^-s + 9.64e18·81^-s + 2.12e18·89^-s + 6.68e17·95^-s − 1.62e19·99^-s + ⋯

L(s) = 1

+ 2.45·4^-s + 0.0336·5^-s + 4.14·9^-s − 0.432·11^-s + 1.95·16^-s + 3.23·19^-s + 0.0826·20^-s + 0.464·25^-s − 2.40·29^-s + 0.489·31^-s + 10.1·36^-s − 1.58·41^-s − 1.06·44^-s + 0.139·45^-s + 3.51·49^-s − 0.0145·55^-s + 3.90·59^-s − 6.38·61^-s + 0.287·64^-s − 6.77·71^-s + 7.93·76^-s + 3.69·79^-s + 0.0657·80^-s + 7.13·81^-s + 0.641·89^-s + 0.108·95^-s − 1.79·99^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+19/2)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$390625$ = $5^{8}$
Sign:	$1$
Analytic conductor:	$2.93536\times 10^{8}$
Root analytic conductor:	$3.38243$
Motivic weight:	$19$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 390625,\ (\ :[19/2]^{8}),\ 1)$

Particular Values

$L(10)$	$\approx$	$31.13554850$
$L(\frac12)$	$\approx$	$31.13554850$
$L(\frac{21}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	5	$ 1 - 1176 p^{3} T - 113242244 p^{7} T^{2} + 15081850008 p^{12} T^{3} + 389129703582 p^{21} T^{4} + 15081850008 p^{31} T^{5} - 113242244 p^{45} T^{6} - 1176 p^{60} T^{7} + p^{76} T^{8} $
good	2	$ 1 - 321695 p^{2} T^{2} + 4369211371 p^{8} T^{4} - 12043934255735 p^{16} T^{6} + 6568673708573269 p^{26} T^{8} - 12043934255735 p^{54} T^{10} + 4369211371 p^{84} T^{12} - 321695 p^{116} T^{14} + p^{152} T^{16} $
	3	$ 1 - 535747240 p^{2} T^{2} + 18664213771033564 p^{6} T^{4} - $$15\!\cdots\!40$$ p^{13} T^{6} + $$89\!\cdots\!34$$ p^{18} T^{8} - $$15\!\cdots\!40$$ p^{51} T^{10} + 18664213771033564 p^{82} T^{12} - 535747240 p^{116} T^{14} + p^{152} T^{16} $
	7	$ 1 - 40030951735584200 T^{2} + $$23\!\cdots\!72$$ p^{3} T^{4} - $$47\!\cdots\!00$$ p^{4} T^{6} + $$24\!\cdots\!06$$ p^{8} T^{8} - $$47\!\cdots\!00$$ p^{42} T^{10} + $$23\!\cdots\!72$$ p^{79} T^{12} - 40030951735584200 p^{114} T^{14} + p^{152} T^{16} $
	11	$ ( 1 + 1689787632 T + 44995983407219867948 T^{2} + $$51\!\cdots\!44$$ p T^{3} + $$12\!\cdots\!70$$ p^{2} T^{4} + $$51\!\cdots\!44$$ p^{20} T^{5} + 44995983407219867948 p^{38} T^{6} + 1689787632 p^{57} T^{7} + p^{76} T^{8} )^{2} $
	13	$ 1 - $$73\!\cdots\!20$$ T^{2} + $$16\!\cdots\!64$$ p^{2} T^{4} - $$23\!\cdots\!40$$ p^{4} T^{6} + $$24\!\cdots\!94$$ p^{6} T^{8} - $$23\!\cdots\!40$$ p^{42} T^{10} + $$16\!\cdots\!64$$ p^{78} T^{12} - $$73\!\cdots\!20$$ p^{114} T^{14} + p^{152} T^{16} $
	17	$ 1 - $$52\!\cdots\!60$$ p^{2} T^{2} + $$72\!\cdots\!48$$ p^{5} T^{4} - $$10\!\cdots\!60$$ p^{7} T^{6} + $$17\!\cdots\!46$$ p^{8} T^{8} - $$10\!\cdots\!60$$ p^{45} T^{10} + $$72\!\cdots\!48$$ p^{81} T^{12} - $$52\!\cdots\!60$$ p^{116} T^{14} + p^{152} T^{16} $
	19	$ ( 1 - 2273594190320 T + $$24\!\cdots\!64$$ p T^{2} - $$77\!\cdots\!40$$ T^{3} + $$14\!\cdots\!46$$ T^{4} - $$77\!\cdots\!40$$ p^{19} T^{5} + $$24\!\cdots\!64$$ p^{39} T^{6} - 2273594190320 p^{57} T^{7} + p^{76} T^{8} )^{2} $
	23	$ 1 - $$26\!\cdots\!80$$ T^{2} + $$48\!\cdots\!76$$ T^{4} - $$55\!\cdots\!60$$ T^{6} + $$48\!\cdots\!66$$ T^{8} - $$55\!\cdots\!60$$ p^{38} T^{10} + $$48\!\cdots\!76$$ p^{76} T^{12} - $$26\!\cdots\!80$$ p^{114} T^{14} + p^{152} T^{16} $
	29	$ ( 1 + 94111150172520 T + $$18\!\cdots\!76$$ T^{2} + $$85\!\cdots\!40$$ T^{3} + $$12\!\cdots\!66$$ T^{4} + $$85\!\cdots\!40$$ p^{19} T^{5} + $$18\!\cdots\!76$$ p^{38} T^{6} + 94111150172520 p^{57} T^{7} + p^{76} T^{8} )^{2} $
	31	$ ( 1 - 36057503343488 T + $$32\!\cdots\!88$$ T^{2} - $$22\!\cdots\!36$$ T^{3} + $$45\!\cdots\!70$$ T^{4} - $$22\!\cdots\!36$$ p^{19} T^{5} + $$32\!\cdots\!88$$ p^{38} T^{6} - 36057503343488 p^{57} T^{7} + p^{76} T^{8} )^{2} $
	37	$ 1 - $$22\!\cdots\!20$$ T^{2} + $$21\!\cdots\!16$$ T^{4} - $$13\!\cdots\!40$$ T^{6} + $$72\!\cdots\!46$$ T^{8} - $$13\!\cdots\!40$$ p^{38} T^{10} + $$21\!\cdots\!16$$ p^{76} T^{12} - $$22\!\cdots\!20$$ p^{114} T^{14} + p^{152} T^{16} $
	41	$ ( 1 + 1657134714428952 T + $$15\!\cdots\!08$$ T^{2} + $$20\!\cdots\!04$$ T^{3} + $$95\!\cdots\!70$$ T^{4} + $$20\!\cdots\!04$$ p^{19} T^{5} + $$15\!\cdots\!08$$ p^{38} T^{6} + 1657134714428952 p^{57} T^{7} + p^{76} T^{8} )^{2} $
	43	$ 1 - $$66\!\cdots\!00$$ T^{2} + $$20\!\cdots\!96$$ T^{4} - $$39\!\cdots\!00$$ T^{6} + $$52\!\cdots\!06$$ T^{8} - $$39\!\cdots\!00$$ p^{38} T^{10} + $$20\!\cdots\!96$$ p^{76} T^{12} - $$66\!\cdots\!00$$ p^{114} T^{14} + p^{152} T^{16} $
	47	$ 1 - $$24\!\cdots\!60$$ T^{2} + $$31\!\cdots\!56$$ T^{4} - $$28\!\cdots\!20$$ T^{6} + $$19\!\cdots\!26$$ T^{8} - $$28\!\cdots\!20$$ p^{38} T^{10} + $$31\!\cdots\!56$$ p^{76} T^{12} - $$24\!\cdots\!60$$ p^{114} T^{14} + p^{152} T^{16} $
	53	$ 1 - $$19\!\cdots\!60$$ T^{2} + $$24\!\cdots\!56$$ T^{4} - $$20\!\cdots\!20$$ T^{6} + $$13\!\cdots\!26$$ T^{8} - $$20\!\cdots\!20$$ p^{38} T^{10} + $$24\!\cdots\!56$$ p^{76} T^{12} - $$19\!\cdots\!60$$ p^{114} T^{14} + p^{152} T^{16} $
	59	$ ( 1 - 129824016389006160 T + $$17\!\cdots\!56$$ T^{2} - $$14\!\cdots\!20$$ T^{3} + $$11\!\cdots\!26$$ T^{4} - $$14\!\cdots\!20$$ p^{19} T^{5} + $$17\!\cdots\!56$$ p^{38} T^{6} - 129824016389006160 p^{57} T^{7} + p^{76} T^{8} )^{2} $
	61	$ ( 1 + 291446781376613032 T + $$58\!\cdots\!48$$ T^{2} + $$77\!\cdots\!84$$ T^{3} + $$82\!\cdots\!70$$ T^{4} + $$77\!\cdots\!84$$ p^{19} T^{5} + $$58\!\cdots\!48$$ p^{38} T^{6} + 291446781376613032 p^{57} T^{7} + p^{76} T^{8} )^{2} $
	67	$ 1 - $$24\!\cdots\!40$$ T^{2} + $$29\!\cdots\!36$$ T^{4} - $$23\!\cdots\!80$$ T^{6} + $$13\!\cdots\!86$$ T^{8} - $$23\!\cdots\!80$$ p^{38} T^{10} + $$29\!\cdots\!36$$ p^{76} T^{12} - $$24\!\cdots\!40$$ p^{114} T^{14} + p^{152} T^{16} $
	71	$ ( 1 + 1308503800301463072 T + $$93\!\cdots\!68$$ T^{2} + $$48\!\cdots\!24$$ T^{3} + $$20\!\cdots\!70$$ T^{4} + $$48\!\cdots\!24$$ p^{19} T^{5} + $$93\!\cdots\!68$$ p^{38} T^{6} + 1308503800301463072 p^{57} T^{7} + p^{76} T^{8} )^{2} $
	73	$ 1 - $$77\!\cdots\!80$$ T^{2} + $$33\!\cdots\!76$$ T^{4} - $$95\!\cdots\!60$$ T^{6} + $$24\!\cdots\!66$$ T^{8} - $$95\!\cdots\!60$$ p^{38} T^{10} + $$33\!\cdots\!76$$ p^{76} T^{12} - $$77\!\cdots\!80$$ p^{114} T^{14} + p^{152} T^{16} $
	79	$ ( 1 - 1967546045554930880 T + $$57\!\cdots\!76$$ T^{2} - $$69\!\cdots\!60$$ T^{3} + $$10\!\cdots\!66$$ T^{4} - $$69\!\cdots\!60$$ p^{19} T^{5} + $$57\!\cdots\!76$$ p^{38} T^{6} - 1967546045554930880 p^{57} T^{7} + p^{76} T^{8} )^{2} $
	83	$ 1 - $$12\!\cdots\!40$$ T^{2} + $$74\!\cdots\!36$$ T^{4} - $$30\!\cdots\!80$$ T^{6} + $$99\!\cdots\!86$$ T^{8} - $$30\!\cdots\!80$$ p^{38} T^{10} + $$74\!\cdots\!36$$ p^{76} T^{12} - $$12\!\cdots\!40$$ p^{114} T^{14} + p^{152} T^{16} $
	89	$ ( 1 - 1060070550818460840 T + $$28\!\cdots\!36$$ T^{2} - $$12\!\cdots\!80$$ T^{3} + $$40\!\cdots\!86$$ T^{4} - $$12\!\cdots\!80$$ p^{19} T^{5} + $$28\!\cdots\!36$$ p^{38} T^{6} - 1060070550818460840 p^{57} T^{7} + p^{76} T^{8} )^{2} $
	97	$ 1 - $$32\!\cdots\!60$$ T^{2} + $$51\!\cdots\!56$$ T^{4} - $$50\!\cdots\!20$$ T^{6} + $$33\!\cdots\!26$$ T^{8} - $$50\!\cdots\!20$$ p^{38} T^{10} + $$51\!\cdots\!56$$ p^{76} T^{12} - $$32\!\cdots\!60$$ p^{114} T^{14} + p^{152} T^{16} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−7.39573164421913238834310301911, −7.25883133510291488012789057423, −7.04473472253404703331352963041, −6.61734897906604316339147424740, −6.33679891528075839509365594569, −6.29600916637654116772491843110, −5.57642900716856323219850371192, −5.45240308834153565353557527992, −5.31572913527155533852192511861, −4.75903219157888522097274595731, −4.35891039511971729915563440192, −4.32148374013859165094166361759, −3.98130355863727855174632213388, −3.62906517344969903661762936904, −3.07478043977055208750766695212, −3.01561736054509416054805534549, −2.82873058678629568874298440152, −2.21183286693286771553106214346, −1.80674809275836228471380955443, −1.74882732092898545758802220120, −1.69129196686179634757714001565, −1.31084057141007667986305372199, −0.920789572771209375375736521027, −0.78821774480713855965924585185, −0.31970905131449460091552508910, 0.31970905131449460091552508910, 0.78821774480713855965924585185, 0.920789572771209375375736521027, 1.31084057141007667986305372199, 1.69129196686179634757714001565, 1.74882732092898545758802220120, 1.80674809275836228471380955443, 2.21183286693286771553106214346, 2.82873058678629568874298440152, 3.01561736054509416054805534549, 3.07478043977055208750766695212, 3.62906517344969903661762936904, 3.98130355863727855174632213388, 4.32148374013859165094166361759, 4.35891039511971729915563440192, 4.75903219157888522097274595731, 5.31572913527155533852192511861, 5.45240308834153565353557527992, 5.57642900716856323219850371192, 6.29600916637654116772491843110, 6.33679891528075839509365594569, 6.61734897906604316339147424740, 7.04473472253404703331352963041, 7.25883133510291488012789057423, 7.39573164421913238834310301911

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.

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 $\F_{q}$

\(L(10)\)	\(\approx\)	\(31.13554850\)
\(L(\frac12)\)	\(\approx\)	\(31.13554850\)
\(L(\frac{21}{2})\)		not available
\(L(1)\)		not available

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