LMFDB - L-function 16-147e8-1.1-c7e8-0-1

Dirichlet series

L(s) = 1

+ 15·2^-s − 108·3^-s + 150·4^-s − 504·5^-s − 1.62e3·6^-s + 975·8^-s + 4.37e3·9^-s − 7.56e3·10^-s + 1.92e3·11^-s − 1.62e4·12^-s + 3.62e4·13^-s + 5.44e4·15^-s + 5.63e3·16^-s + 1.95e4·17^-s + 6.56e4·18^-s − 3.13e4·19^-s − 7.56e4·20^-s + 2.88e4·22^-s − 1.01e5·23^-s − 1.05e5·24^-s + 2.45e5·25^-s + 5.44e5·26^-s + 3.89e5·29^-s + 8.16e5·30^-s − 7.88e4·31^-s + 1.19e5·32^-s − 2.07e5·33^-s + ⋯

L(s) = 1

+ 1.32·2^-s − 2.30·3^-s + 1.17·4^-s − 1.80·5^-s − 3.06·6^-s + 0.673·8^-s + 2·9^-s − 2.39·10^-s + 0.434·11^-s − 2.70·12^-s + 4.58·13^-s + 4.16·15^-s + 0.343·16^-s + 0.966·17^-s + 2.65·18^-s − 1.04·19^-s − 2.11·20^-s + 0.576·22^-s − 1.73·23^-s − 1.55·24^-s + 3.14·25^-s + 6.07·26^-s + 2.96·29^-s + 5.52·30^-s − 0.475·31^-s + 0.646·32^-s − 1.00·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$3^{8} \cdot 7^{16}$
Sign:	$1$
Analytic conductor:	$1.97724\times 10^{13}$
Root analytic conductor:	$6.77647$
Motivic weight:	$7$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 3^{8} \cdot 7^{16} ,\ ( \ : [7/2]^{8} ),\ 1 )$

Particular Values

$L(4)$	$\approx$	$0.7612539810$
$L(\frac12)$	$\approx$	$0.7612539810$
$L(\frac{9}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ ( 1 + p^{3} T + p^{6} T^{2} )^{4} $
	7	$ 1 $
good	2	$ 1 - 15 T + 75 T^{2} + 75 p T^{3} - 1127 p^{2} T^{4} - 495 p^{7} T^{5} + 84375 p^{5} T^{6} - 129105 p^{8} T^{7} + 794793 p^{8} T^{8} - 129105 p^{15} T^{9} + 84375 p^{19} T^{10} - 495 p^{28} T^{11} - 1127 p^{30} T^{12} + 75 p^{36} T^{13} + 75 p^{42} T^{14} - 15 p^{49} T^{15} + p^{56} T^{16} $
	5	$ 1 + 504 T + 8152 T^{2} - 40445424 T^{3} - 4617639614 T^{4} + 557714358072 p T^{5} + 28195826245408 p^{2} T^{6} - 1736252096189688 p^{3} T^{7} - 211783543053184349 p^{4} T^{8} - 1736252096189688 p^{10} T^{9} + 28195826245408 p^{16} T^{10} + 557714358072 p^{22} T^{11} - 4617639614 p^{28} T^{12} - 40445424 p^{35} T^{13} + 8152 p^{42} T^{14} + 504 p^{49} T^{15} + p^{56} T^{16} $
	11	$ 1 - 1920 T - 43617504 T^{2} + 250306451040 T^{3} + 851974453842130 T^{4} - 7328729425220879760 T^{5} + $$85\!\cdots\!84$$ T^{6} + $$85\!\cdots\!20$$ T^{7} - $$38\!\cdots\!41$$ T^{8} + $$85\!\cdots\!20$$ p^{7} T^{9} + $$85\!\cdots\!84$$ p^{14} T^{10} - 7328729425220879760 p^{21} T^{11} + 851974453842130 p^{28} T^{12} + 250306451040 p^{35} T^{13} - 43617504 p^{42} T^{14} - 1920 p^{49} T^{15} + p^{56} T^{16} $
	13	$ ( 1 - 18144 T + 274398964 T^{2} - 180541542816 p T^{3} + 21711173630808534 T^{4} - 180541542816 p^{8} T^{5} + 274398964 p^{14} T^{6} - 18144 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	17	$ 1 - 1152 p T - 606984152 T^{2} + 21277920917088 T^{3} + 23895638307807058 T^{4} - $$65\!\cdots\!32$$ T^{5} + $$30\!\cdots\!16$$ T^{6} + $$54\!\cdots\!44$$ T^{7} + $$21\!\cdots\!03$$ T^{8} + $$54\!\cdots\!44$$ p^{7} T^{9} + $$30\!\cdots\!16$$ p^{14} T^{10} - $$65\!\cdots\!32$$ p^{21} T^{11} + 23895638307807058 p^{28} T^{12} + 21277920917088 p^{35} T^{13} - 606984152 p^{42} T^{14} - 1152 p^{50} T^{15} + p^{56} T^{16} $
	19	$ 1 + 31320 T - 1195603708 T^{2} - 75817594337040 T^{3} + 44627959461466 T^{4} + $$65\!\cdots\!80$$ T^{5} + $$12\!\cdots\!52$$ T^{6} - $$12\!\cdots\!40$$ p T^{7} - $$14\!\cdots\!81$$ T^{8} - $$12\!\cdots\!40$$ p^{8} T^{9} + $$12\!\cdots\!52$$ p^{14} T^{10} + $$65\!\cdots\!80$$ p^{21} T^{11} + 44627959461466 p^{28} T^{12} - 75817594337040 p^{35} T^{13} - 1195603708 p^{42} T^{14} + 31320 p^{49} T^{15} + p^{56} T^{16} $
	23	$ 1 + 101160 T - 4708078416 T^{2} - 386150051343600 T^{3} + 49684725721742722114 T^{4} + $$74\!\cdots\!00$$ p T^{5} - $$23\!\cdots\!84$$ T^{6} - $$14\!\cdots\!40$$ T^{7} + $$10\!\cdots\!79$$ T^{8} - $$14\!\cdots\!40$$ p^{7} T^{9} - $$23\!\cdots\!84$$ p^{14} T^{10} + $$74\!\cdots\!00$$ p^{22} T^{11} + 49684725721742722114 p^{28} T^{12} - 386150051343600 p^{35} T^{13} - 4708078416 p^{42} T^{14} + 101160 p^{49} T^{15} + p^{56} T^{16} $
	29	$ ( 1 - 194832 T + 68169175020 T^{2} - 9181092556616112 T^{3} + $$17\!\cdots\!94$$ T^{4} - 9181092556616112 p^{7} T^{5} + 68169175020 p^{14} T^{6} - 194832 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	31	$ 1 + 78840 T - 16183724812 T^{2} - 10658302272654480 T^{3} - $$10\!\cdots\!34$$ T^{4} + $$14\!\cdots\!00$$ T^{5} + $$34\!\cdots\!68$$ T^{6} + $$20\!\cdots\!40$$ T^{7} - $$59\!\cdots\!81$$ T^{8} + $$20\!\cdots\!40$$ p^{7} T^{9} + $$34\!\cdots\!68$$ p^{14} T^{10} + $$14\!\cdots\!00$$ p^{21} T^{11} - $$10\!\cdots\!34$$ p^{28} T^{12} - 10658302272654480 p^{35} T^{13} - 16183724812 p^{42} T^{14} + 78840 p^{49} T^{15} + p^{56} T^{16} $
	37	$ 1 + 128640 T - 259896387244 T^{2} - 10652201423243520 T^{3} + $$39\!\cdots\!14$$ T^{4} - $$20\!\cdots\!80$$ T^{5} - $$42\!\cdots\!36$$ T^{6} + $$72\!\cdots\!40$$ T^{7} + $$37\!\cdots\!59$$ T^{8} + $$72\!\cdots\!40$$ p^{7} T^{9} - $$42\!\cdots\!36$$ p^{14} T^{10} - $$20\!\cdots\!80$$ p^{21} T^{11} + $$39\!\cdots\!14$$ p^{28} T^{12} - 10652201423243520 p^{35} T^{13} - 259896387244 p^{42} T^{14} + 128640 p^{49} T^{15} + p^{56} T^{16} $
	41	$ ( 1 - 365040 T + 391725770744 T^{2} - 146719702189581120 T^{3} + $$84\!\cdots\!06$$ T^{4} - 146719702189581120 p^{7} T^{5} + 391725770744 p^{14} T^{6} - 365040 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	43	$ ( 1 - 449520 T + 911801001196 T^{2} - 338923246405434480 T^{3} + $$35\!\cdots\!42$$ T^{4} - 338923246405434480 p^{7} T^{5} + 911801001196 p^{14} T^{6} - 449520 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	47	$ 1 - 1575792 T + 306134096068 T^{2} + 691098568700217120 T^{3} - $$22\!\cdots\!30$$ T^{4} - $$12\!\cdots\!48$$ T^{5} - $$17\!\cdots\!56$$ T^{6} - $$77\!\cdots\!40$$ T^{7} + $$15\!\cdots\!79$$ T^{8} - $$77\!\cdots\!40$$ p^{7} T^{9} - $$17\!\cdots\!56$$ p^{14} T^{10} - $$12\!\cdots\!48$$ p^{21} T^{11} - $$22\!\cdots\!30$$ p^{28} T^{12} + 691098568700217120 p^{35} T^{13} + 306134096068 p^{42} T^{14} - 1575792 p^{49} T^{15} + p^{56} T^{16} $
	53	$ 1 - 1448160 T - 217257586860 T^{2} + 372567057484827840 T^{3} + $$52\!\cdots\!62$$ T^{4} - $$30\!\cdots\!80$$ T^{5} + $$17\!\cdots\!00$$ T^{6} - $$15\!\cdots\!80$$ p T^{7} - $$70\!\cdots\!13$$ p^{2} T^{8} - $$15\!\cdots\!80$$ p^{8} T^{9} + $$17\!\cdots\!00$$ p^{14} T^{10} - $$30\!\cdots\!80$$ p^{21} T^{11} + $$52\!\cdots\!62$$ p^{28} T^{12} + 372567057484827840 p^{35} T^{13} - 217257586860 p^{42} T^{14} - 1448160 p^{49} T^{15} + p^{56} T^{16} $
	59	$ 1 + 3280320 T + 4401496666132 T^{2} + 6335039987350911360 T^{3} + $$62\!\cdots\!06$$ T^{4} - $$36\!\cdots\!20$$ T^{5} + $$63\!\cdots\!72$$ T^{6} + $$25\!\cdots\!40$$ T^{7} + $$18\!\cdots\!39$$ T^{8} + $$25\!\cdots\!40$$ p^{7} T^{9} + $$63\!\cdots\!72$$ p^{14} T^{10} - $$36\!\cdots\!20$$ p^{21} T^{11} + $$62\!\cdots\!06$$ p^{28} T^{12} + 6335039987350911360 p^{35} T^{13} + 4401496666132 p^{42} T^{14} + 3280320 p^{49} T^{15} + p^{56} T^{16} $
	61	$ 1 + 606960 T - 5638071288196 T^{2} + 432857490606283680 T^{3} + $$14\!\cdots\!30$$ T^{4} - $$10\!\cdots\!40$$ T^{5} - $$48\!\cdots\!84$$ T^{6} + $$27\!\cdots\!80$$ T^{7} - $$19\!\cdots\!41$$ T^{8} + $$27\!\cdots\!80$$ p^{7} T^{9} - $$48\!\cdots\!84$$ p^{14} T^{10} - $$10\!\cdots\!40$$ p^{21} T^{11} + $$14\!\cdots\!30$$ p^{28} T^{12} + 432857490606283680 p^{35} T^{13} - 5638071288196 p^{42} T^{14} + 606960 p^{49} T^{15} + p^{56} T^{16} $
	67	$ 1 + 3492880 T - 4053064665564 T^{2} - 32954975534844335200 T^{3} - $$45\!\cdots\!46$$ T^{4} - $$64\!\cdots\!00$$ T^{5} - $$22\!\cdots\!76$$ T^{6} + $$86\!\cdots\!20$$ T^{7} + $$54\!\cdots\!39$$ T^{8} + $$86\!\cdots\!20$$ p^{7} T^{9} - $$22\!\cdots\!76$$ p^{14} T^{10} - $$64\!\cdots\!00$$ p^{21} T^{11} - $$45\!\cdots\!46$$ p^{28} T^{12} - 32954975534844335200 p^{35} T^{13} - 4053064665564 p^{42} T^{14} + 3492880 p^{49} T^{15} + p^{56} T^{16} $
	71	$ ( 1 - 984 T + 23313454396080 T^{2} - 10877264513013635016 T^{3} + $$26\!\cdots\!74$$ T^{4} - 10877264513013635016 p^{7} T^{5} + 23313454396080 p^{14} T^{6} - 984 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	73	$ 1 + 10981440 T + 52184200249292 T^{2} + 91864376880944423040 T^{3} - $$18\!\cdots\!70$$ T^{4} - $$11\!\cdots\!20$$ T^{5} + $$64\!\cdots\!72$$ T^{6} + $$26\!\cdots\!60$$ p T^{7} + $$94\!\cdots\!99$$ T^{8} + $$26\!\cdots\!60$$ p^{8} T^{9} + $$64\!\cdots\!72$$ p^{14} T^{10} - $$11\!\cdots\!20$$ p^{21} T^{11} - $$18\!\cdots\!70$$ p^{28} T^{12} + 91864376880944423040 p^{35} T^{13} + 52184200249292 p^{42} T^{14} + 10981440 p^{49} T^{15} + p^{56} T^{16} $
	79	$ 1 + 4654544 T - 28361900310156 T^{2} - $$16\!\cdots\!96$$ T^{3} + $$42\!\cdots\!06$$ T^{4} + $$28\!\cdots\!52$$ T^{5} + $$33\!\cdots\!96$$ T^{6} - $$32\!\cdots\!88$$ T^{7} - $$16\!\cdots\!13$$ T^{8} - $$32\!\cdots\!88$$ p^{7} T^{9} + $$33\!\cdots\!96$$ p^{14} T^{10} + $$28\!\cdots\!52$$ p^{21} T^{11} + $$42\!\cdots\!06$$ p^{28} T^{12} - $$16\!\cdots\!96$$ p^{35} T^{13} - 28361900310156 p^{42} T^{14} + 4654544 p^{49} T^{15} + p^{56} T^{16} $
	83	$ ( 1 - 8126496 T + 100452515205644 T^{2} - $$57\!\cdots\!12$$ T^{3} + $$40\!\cdots\!74$$ T^{4} - $$57\!\cdots\!12$$ p^{7} T^{5} + 100452515205644 p^{14} T^{6} - 8126496 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	89	$ 1 - 11272320 T + 60652848406216 T^{2} - $$95\!\cdots\!80$$ T^{3} + $$76\!\cdots\!50$$ T^{4} - $$36\!\cdots\!60$$ T^{5} + $$46\!\cdots\!84$$ T^{6} - $$32\!\cdots\!60$$ T^{7} + $$13\!\cdots\!59$$ T^{8} - $$32\!\cdots\!60$$ p^{7} T^{9} + $$46\!\cdots\!84$$ p^{14} T^{10} - $$36\!\cdots\!60$$ p^{21} T^{11} + $$76\!\cdots\!50$$ p^{28} T^{12} - $$95\!\cdots\!80$$ p^{35} T^{13} + 60652848406216 p^{42} T^{14} - 11272320 p^{49} T^{15} + p^{56} T^{16} $
	97	$ ( 1 - 6572448 T + 235163205768916 T^{2} - $$14\!\cdots\!84$$ T^{3} + $$26\!\cdots\!14$$ T^{4} - $$14\!\cdots\!84$$ p^{7} T^{5} + 235163205768916 p^{14} T^{6} - 6572448 p^{21} T^{7} + p^{28} T^{8} )^{2} $
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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

−4.41852420961095087200008853935, −4.36496173635525434787727685211, −4.34951516730648503494869603041, −4.28430354580104729973503262554, −3.74893711809114633918503906496, −3.71477900098064444757032531323, −3.70759187947794039587030055259, −3.56011387611497406119922586134, −3.30697009858515842234474833342, −3.25572884094153473883629733115, −3.01464677831036009769022629561, −2.52247107699978731449253638756, −2.49196067599503610818291753363, −2.48583615380552137908197188421, −2.32997580228663982321653467067, −1.72372403479761316494863089348, −1.52973337534301434627614944702, −1.35619670645607308027649283627, −1.04912685453215505068028790205, −1.02598910390899609476080311948, −0.998571549406775535855489345688, −0.935336161176410135619719004504, −0.46903833264959969224262404179, −0.36075800282848570390308241618, −0.05725028538328262131754424993, 0.05725028538328262131754424993, 0.36075800282848570390308241618, 0.46903833264959969224262404179, 0.935336161176410135619719004504, 0.998571549406775535855489345688, 1.02598910390899609476080311948, 1.04912685453215505068028790205, 1.35619670645607308027649283627, 1.52973337534301434627614944702, 1.72372403479761316494863089348, 2.32997580228663982321653467067, 2.48583615380552137908197188421, 2.49196067599503610818291753363, 2.52247107699978731449253638756, 3.01464677831036009769022629561, 3.25572884094153473883629733115, 3.30697009858515842234474833342, 3.56011387611497406119922586134, 3.70759187947794039587030055259, 3.71477900098064444757032531323, 3.74893711809114633918503906496, 4.28430354580104729973503262554, 4.34951516730648503494869603041, 4.36496173635525434787727685211, 4.41852420961095087200008853935

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(0.7612539810\)
\(L(\frac12)\)	\(\approx\)	\(0.7612539810\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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