LMFDB - L-function 16-432e8-1.1-c7e8-0-3

Dirichlet series

L(s) = 1

− 54·5^-s + 44·7^-s + 2.17e3·11^-s − 6.39e3·13^-s + 5.19e4·17^-s − 9.07e4·19^-s − 2.02e3·23^-s + 7.09e4·25^-s − 2.83e5·29^-s + 812·31^-s − 2.37e3·35^-s + 4.41e5·37^-s − 6.10e5·41^-s + 8.43e4·43^-s + 8.55e5·47^-s + 1.01e6·49^-s + 1.84e6·53^-s − 1.17e5·55^-s + 3.80e5·59^-s − 3.15e6·61^-s + 3.45e5·65^-s + 9.96e6·67^-s + 8.97e6·71^-s + 2.92e7·73^-s + 9.55e4·77^-s + 1.73e7·79^-s − 1.15e7·83^-s + ⋯

L(s) = 1

− 0.193·5^-s + 0.0484·7^-s + 0.492·11^-s − 0.807·13^-s + 2.56·17^-s − 3.03·19^-s − 0.0347·23^-s + 0.908·25^-s − 2.15·29^-s + 0.00489·31^-s − 0.00936·35^-s + 1.43·37^-s − 1.38·41^-s + 0.161·43^-s + 1.20·47^-s + 1.22·49^-s + 1.70·53^-s − 0.0950·55^-s + 0.241·59^-s − 1.77·61^-s + 0.156·65^-s + 4.04·67^-s + 2.97·71^-s + 8.79·73^-s + 0.0238·77^-s + 3.94·79^-s − 2.21·83^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$2^{32} \cdot 3^{24}$
Sign:	$1$
Analytic conductor:	$1.09999\times 10^{17}$
Root analytic conductor:	$11.6168$
Motivic weight:	$7$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{32} \cdot 3^{24} ,\ ( \ : [7/2]^{8} ),\ 1 )$

Particular Values

$L(4)$	$\approx$	$16.87035584$
$L(\frac12)$	$\approx$	$16.87035584$
$L(\frac{9}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
good	5	$ 1 + 54 T - 68069 T^{2} + 4918698 p T^{3} + 100639333 p^{2} T^{4} - 15046343892 p^{3} T^{5} + 1278685874146 p^{4} T^{6} + 8465234686416 p^{6} T^{7} - 164191074046454 p^{8} T^{8} + 8465234686416 p^{13} T^{9} + 1278685874146 p^{18} T^{10} - 15046343892 p^{24} T^{11} + 100639333 p^{30} T^{12} + 4918698 p^{36} T^{13} - 68069 p^{42} T^{14} + 54 p^{49} T^{15} + p^{56} T^{16} $
	7	$ 1 - 44 T - 1009599 T^{2} + 1270970228 T^{3} + 480821631533 T^{4} - 1161819355358904 T^{5} + 1261515981612730498 T^{6} + $$70\!\cdots\!08$$ T^{7} - $$12\!\cdots\!14$$ T^{8} + $$70\!\cdots\!08$$ p^{7} T^{9} + 1261515981612730498 p^{14} T^{10} - 1161819355358904 p^{21} T^{11} + 480821631533 p^{28} T^{12} + 1270970228 p^{35} T^{13} - 1009599 p^{42} T^{14} - 44 p^{49} T^{15} + p^{56} T^{16} $
	11	$ 1 - 2172 T - 2620966 p T^{2} + 146940958056 T^{3} - 11758344640799 T^{4} - 1925579145610582056 T^{5} + $$25\!\cdots\!98$$ p T^{6} + $$12\!\cdots\!04$$ T^{7} + $$70\!\cdots\!12$$ T^{8} + $$12\!\cdots\!04$$ p^{7} T^{9} + $$25\!\cdots\!98$$ p^{15} T^{10} - 1925579145610582056 p^{21} T^{11} - 11758344640799 p^{28} T^{12} + 146940958056 p^{35} T^{13} - 2620966 p^{43} T^{14} - 2172 p^{49} T^{15} + p^{56} T^{16} $
	13	$ 1 + 6398 T - 197873697 T^{2} - 744359220134 T^{3} + 27660128824949825 T^{4} + 59811842745933963420 T^{5} - $$19\!\cdots\!10$$ p T^{6} - $$13\!\cdots\!20$$ T^{7} + $$18\!\cdots\!06$$ T^{8} - $$13\!\cdots\!20$$ p^{7} T^{9} - $$19\!\cdots\!10$$ p^{15} T^{10} + 59811842745933963420 p^{21} T^{11} + 27660128824949825 p^{28} T^{12} - 744359220134 p^{35} T^{13} - 197873697 p^{42} T^{14} + 6398 p^{49} T^{15} + p^{56} T^{16} $
	17	$ ( 1 - 25986 T + 41527057 p T^{2} + 411810814782 T^{3} + 32553470452427988 T^{4} + 411810814782 p^{7} T^{5} + 41527057 p^{15} T^{6} - 25986 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	19	$ ( 1 + 45356 T + 2846091223 T^{2} + 3581875083404 p T^{3} + 2914017634216626904 T^{4} + 3581875083404 p^{8} T^{5} + 2846091223 p^{14} T^{6} + 45356 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	23	$ 1 + 2028 T - 7206476375 T^{2} + 415064256498060 T^{3} + 28416628801554830125 T^{4} - $$22\!\cdots\!32$$ T^{5} + $$35\!\cdots\!30$$ T^{6} + $$52\!\cdots\!24$$ T^{7} - $$29\!\cdots\!74$$ T^{8} + $$52\!\cdots\!24$$ p^{7} T^{9} + $$35\!\cdots\!30$$ p^{14} T^{10} - $$22\!\cdots\!32$$ p^{21} T^{11} + 28416628801554830125 p^{28} T^{12} + 415064256498060 p^{35} T^{13} - 7206476375 p^{42} T^{14} + 2028 p^{49} T^{15} + p^{56} T^{16} $
	29	$ 1 + 9762 p T + 30586949983 T^{2} + 418893317840382 T^{3} - $$44\!\cdots\!23$$ T^{4} - $$99\!\cdots\!00$$ T^{5} - $$59\!\cdots\!10$$ T^{6} + $$65\!\cdots\!96$$ T^{7} + $$16\!\cdots\!54$$ T^{8} + $$65\!\cdots\!96$$ p^{7} T^{9} - $$59\!\cdots\!10$$ p^{14} T^{10} - $$99\!\cdots\!00$$ p^{21} T^{11} - $$44\!\cdots\!23$$ p^{28} T^{12} + 418893317840382 p^{35} T^{13} + 30586949983 p^{42} T^{14} + 9762 p^{50} T^{15} + p^{56} T^{16} $
	31	$ 1 - 812 T - 30449005155 T^{2} + 12671245330443572 T^{3} + $$86\!\cdots\!25$$ T^{4} - $$36\!\cdots\!16$$ T^{5} + $$84\!\cdots\!14$$ T^{6} + $$10\!\cdots\!40$$ T^{7} - $$23\!\cdots\!78$$ T^{8} + $$10\!\cdots\!40$$ p^{7} T^{9} + $$84\!\cdots\!14$$ p^{14} T^{10} - $$36\!\cdots\!16$$ p^{21} T^{11} + $$86\!\cdots\!25$$ p^{28} T^{12} + 12671245330443572 p^{35} T^{13} - 30449005155 p^{42} T^{14} - 812 p^{49} T^{15} + p^{56} T^{16} $
	37	$ ( 1 - 220844 T + 310586900824 T^{2} - 54661996045973396 T^{3} + $$42\!\cdots\!54$$ T^{4} - 54661996045973396 p^{7} T^{5} + 310586900824 p^{14} T^{6} - 220844 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	41	$ 1 + 610704 T - 273078654194 T^{2} - 126790945167366432 T^{3} + $$90\!\cdots\!77$$ T^{4} + $$14\!\cdots\!44$$ T^{5} - $$22\!\cdots\!38$$ T^{6} - $$23\!\cdots\!72$$ T^{7} + $$32\!\cdots\!56$$ T^{8} - $$23\!\cdots\!72$$ p^{7} T^{9} - $$22\!\cdots\!38$$ p^{14} T^{10} + $$14\!\cdots\!44$$ p^{21} T^{11} + $$90\!\cdots\!77$$ p^{28} T^{12} - 126790945167366432 p^{35} T^{13} - 273078654194 p^{42} T^{14} + 610704 p^{49} T^{15} + p^{56} T^{16} $
	43	$ 1 - 84380 T - 773918972010 T^{2} + 203230048416963080 T^{3} + $$32\!\cdots\!17$$ T^{4} - $$95\!\cdots\!00$$ T^{5} - $$82\!\cdots\!50$$ T^{6} + $$13\!\cdots\!40$$ T^{7} + $$19\!\cdots\!88$$ T^{8} + $$13\!\cdots\!40$$ p^{7} T^{9} - $$82\!\cdots\!50$$ p^{14} T^{10} - $$95\!\cdots\!00$$ p^{21} T^{11} + $$32\!\cdots\!17$$ p^{28} T^{12} + 203230048416963080 p^{35} T^{13} - 773918972010 p^{42} T^{14} - 84380 p^{49} T^{15} + p^{56} T^{16} $
	47	$ 1 - 855708 T - 1354935601655 T^{2} + 659637997148165028 T^{3} + $$17\!\cdots\!89$$ T^{4} - $$45\!\cdots\!48$$ T^{5} - $$12\!\cdots\!26$$ T^{6} + $$73\!\cdots\!16$$ T^{7} + $$77\!\cdots\!50$$ T^{8} + $$73\!\cdots\!16$$ p^{7} T^{9} - $$12\!\cdots\!26$$ p^{14} T^{10} - $$45\!\cdots\!48$$ p^{21} T^{11} + $$17\!\cdots\!89$$ p^{28} T^{12} + 659637997148165028 p^{35} T^{13} - 1354935601655 p^{42} T^{14} - 855708 p^{49} T^{15} + p^{56} T^{16} $
	53	$ ( 1 - 921300 T + 3778080260792 T^{2} - 2920012701559961580 T^{3} + $$61\!\cdots\!50$$ T^{4} - 2920012701559961580 p^{7} T^{5} + 3778080260792 p^{14} T^{6} - 921300 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	59	$ 1 - 380796 T - 6111449961842 T^{2} + 1705054701602628456 T^{3} + $$16\!\cdots\!77$$ T^{4} - $$15\!\cdots\!04$$ T^{5} - $$54\!\cdots\!98$$ T^{6} - $$84\!\cdots\!84$$ T^{7} + $$17\!\cdots\!52$$ T^{8} - $$84\!\cdots\!84$$ p^{7} T^{9} - $$54\!\cdots\!98$$ p^{14} T^{10} - $$15\!\cdots\!04$$ p^{21} T^{11} + $$16\!\cdots\!77$$ p^{28} T^{12} + 1705054701602628456 p^{35} T^{13} - 6111449961842 p^{42} T^{14} - 380796 p^{49} T^{15} + p^{56} T^{16} $
	61	$ 1 + 3151130 T - 961541754885 T^{2} - 10665947337440259938 T^{3} - $$32\!\cdots\!15$$ T^{4} + $$12\!\cdots\!64$$ T^{5} - $$25\!\cdots\!14$$ T^{6} - $$21\!\cdots\!20$$ T^{7} + $$67\!\cdots\!38$$ T^{8} - $$21\!\cdots\!20$$ p^{7} T^{9} - $$25\!\cdots\!14$$ p^{14} T^{10} + $$12\!\cdots\!64$$ p^{21} T^{11} - $$32\!\cdots\!15$$ p^{28} T^{12} - 10665947337440259938 p^{35} T^{13} - 961541754885 p^{42} T^{14} + 3151130 p^{49} T^{15} + p^{56} T^{16} $
	67	$ 1 - 9961580 T + 39289102796430 T^{2} - $$12\!\cdots\!28$$ T^{3} + $$57\!\cdots\!73$$ T^{4} - $$19\!\cdots\!28$$ T^{5} + $$45\!\cdots\!26$$ T^{6} - $$13\!\cdots\!32$$ T^{7} + $$40\!\cdots\!60$$ T^{8} - $$13\!\cdots\!32$$ p^{7} T^{9} + $$45\!\cdots\!26$$ p^{14} T^{10} - $$19\!\cdots\!28$$ p^{21} T^{11} + $$57\!\cdots\!73$$ p^{28} T^{12} - $$12\!\cdots\!28$$ p^{35} T^{13} + 39289102796430 p^{42} T^{14} - 9961580 p^{49} T^{15} + p^{56} T^{16} $
	71	$ ( 1 - 4485192 T + 38499321804620 T^{2} - $$11\!\cdots\!04$$ T^{3} + $$53\!\cdots\!06$$ T^{4} - $$11\!\cdots\!04$$ p^{7} T^{5} + 38499321804620 p^{14} T^{6} - 4485192 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	73	$ ( 1 - 14617466 T + 124150261877713 T^{2} - $$67\!\cdots\!62$$ T^{3} + $$26\!\cdots\!48$$ T^{4} - $$67\!\cdots\!62$$ p^{7} T^{5} + 124150261877713 p^{14} T^{6} - 14617466 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	79	$ 1 - 17309396 T + 115868794688781 T^{2} - $$62\!\cdots\!84$$ T^{3} + $$50\!\cdots\!01$$ T^{4} - $$31\!\cdots\!80$$ T^{5} + $$13\!\cdots\!22$$ T^{6} - $$68\!\cdots\!92$$ T^{7} + $$36\!\cdots\!22$$ T^{8} - $$68\!\cdots\!92$$ p^{7} T^{9} + $$13\!\cdots\!22$$ p^{14} T^{10} - $$31\!\cdots\!80$$ p^{21} T^{11} + $$50\!\cdots\!01$$ p^{28} T^{12} - $$62\!\cdots\!84$$ p^{35} T^{13} + 115868794688781 p^{42} T^{14} - 17309396 p^{49} T^{15} + p^{56} T^{16} $
	83	$ 1 + 11520192 T + 2288609056513 T^{2} - $$33\!\cdots\!72$$ T^{3} + $$82\!\cdots\!53$$ T^{4} + $$15\!\cdots\!60$$ T^{5} + $$17\!\cdots\!90$$ T^{6} - $$13\!\cdots\!96$$ T^{7} + $$79\!\cdots\!22$$ T^{8} - $$13\!\cdots\!96$$ p^{7} T^{9} + $$17\!\cdots\!90$$ p^{14} T^{10} + $$15\!\cdots\!60$$ p^{21} T^{11} + $$82\!\cdots\!53$$ p^{28} T^{12} - $$33\!\cdots\!72$$ p^{35} T^{13} + 2288609056513 p^{42} T^{14} + 11520192 p^{49} T^{15} + p^{56} T^{16} $
	89	$ ( 1 + 13033032 T + 98491258269644 T^{2} + $$38\!\cdots\!28$$ T^{3} + $$22\!\cdots\!90$$ T^{4} + $$38\!\cdots\!28$$ p^{7} T^{5} + 98491258269644 p^{14} T^{6} + 13033032 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	97	$ 1 + 22003112 T + 54505731669414 T^{2} - $$10\!\cdots\!44$$ T^{3} + $$13\!\cdots\!09$$ T^{4} + $$22\!\cdots\!28$$ T^{5} - $$18\!\cdots\!78$$ T^{6} + $$20\!\cdots\!24$$ p T^{7} + $$15\!\cdots\!12$$ T^{8} + $$20\!\cdots\!24$$ p^{8} T^{9} - $$18\!\cdots\!78$$ p^{14} T^{10} + $$22\!\cdots\!28$$ p^{21} T^{11} + $$13\!\cdots\!09$$ p^{28} T^{12} - $$10\!\cdots\!44$$ p^{35} T^{13} + 54505731669414 p^{42} T^{14} + 22003112 p^{49} T^{15} + p^{56} 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

−3.61736181657935906906627909966, −3.52366160960457751232821106665, −3.46367693328949438159361325649, −3.41917470447167880302541652097, −3.23671974462435365897605545000, −3.20150397345815132744362967567, −2.67508937046666429495443603065, −2.55704627192278150123446585929, −2.53070196437177291287671959007, −2.37961719851535475217832413493, −2.16075656490207212085601160717, −2.13259926248899336148652025725, −2.04932648240606525855168385190, −2.00531845911620717767006913602, −1.77762184339628448674058993277, −1.42572824374460861309092272231, −1.32988087876749103427718507807, −1.06419106067135776764797931481, −0.937051244370309225004016583921, −0.850840071910250033871632131131, −0.71150686374736577038201650353, −0.69145256994406387332545795697, −0.41024283139760190793560874348, −0.30999806153145096448375542658, −0.17368341660347211104701821253, 0.17368341660347211104701821253, 0.30999806153145096448375542658, 0.41024283139760190793560874348, 0.69145256994406387332545795697, 0.71150686374736577038201650353, 0.850840071910250033871632131131, 0.937051244370309225004016583921, 1.06419106067135776764797931481, 1.32988087876749103427718507807, 1.42572824374460861309092272231, 1.77762184339628448674058993277, 2.00531845911620717767006913602, 2.04932648240606525855168385190, 2.13259926248899336148652025725, 2.16075656490207212085601160717, 2.37961719851535475217832413493, 2.53070196437177291287671959007, 2.55704627192278150123446585929, 2.67508937046666429495443603065, 3.20150397345815132744362967567, 3.23671974462435365897605545000, 3.41917470447167880302541652097, 3.46367693328949438159361325649, 3.52366160960457751232821106665, 3.61736181657935906906627909966

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 $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(4)\)	\(\approx\)	\(16.87035584\)
\(L(\frac12)\)	\(\approx\)	\(16.87035584\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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