Properties

Label 16-39e16-1.1-c3e8-0-1
Degree $16$
Conductor $2.864\times 10^{25}$
Sign $1$
Analytic cond. $4.20690\times 10^{15}$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·4-s − 22·7-s + 43·16-s + 244·19-s − 323·25-s + 264·28-s + 242·31-s + 1.01e3·37-s + 74·43-s − 981·49-s + 1.14e3·61-s + 394·64-s − 2.19e3·67-s − 2.17e3·73-s − 2.92e3·76-s + 1.86e3·79-s − 4.37e3·97-s + 3.87e3·100-s + 3.45e3·103-s + 4.85e3·109-s − 946·112-s − 928·121-s − 2.90e3·124-s + 127-s + 131-s − 5.36e3·133-s + 137-s + ⋯
L(s)  = 1  − 3/2·4-s − 1.18·7-s + 0.671·16-s + 2.94·19-s − 2.58·25-s + 1.78·28-s + 1.40·31-s + 4.52·37-s + 0.262·43-s − 2.86·49-s + 2.40·61-s + 0.769·64-s − 4.00·67-s − 3.48·73-s − 4.41·76-s + 2.65·79-s − 4.57·97-s + 3.87·100-s + 3.30·103-s + 4.26·109-s − 0.798·112-s − 0.697·121-s − 2.10·124-s + 0.000698·127-s + 0.000666·131-s − 3.49·133-s + 0.000623·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(4.20690\times 10^{15}\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 13^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(15.56157013\)
\(L(\frac12)\) \(\approx\) \(15.56157013\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + 3 p^{2} T^{2} + 101 T^{4} + 151 p T^{6} + 31 p^{5} T^{8} + 151 p^{7} T^{10} + 101 p^{12} T^{12} + 3 p^{20} T^{14} + p^{24} T^{16} \)
5 \( 1 + 323 T^{2} + 14109 p T^{4} + 2362538 p T^{6} + 1522988542 T^{8} + 2362538 p^{7} T^{10} + 14109 p^{13} T^{12} + 323 p^{18} T^{14} + p^{24} T^{16} \)
7 \( ( 1 + 11 T + 96 p T^{2} + 425 T^{3} + 185810 T^{4} + 425 p^{3} T^{5} + 96 p^{7} T^{6} + 11 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
11 \( 1 + 928 T^{2} + 4646848 T^{4} + 4183083552 T^{6} + 10154734587870 T^{8} + 4183083552 p^{6} T^{10} + 4646848 p^{12} T^{12} + 928 p^{18} T^{14} + p^{24} T^{16} \)
17 \( 1 + 9879 T^{2} + 66841397 T^{4} + 413456180966 T^{6} + 2043761199640142 T^{8} + 413456180966 p^{6} T^{10} + 66841397 p^{12} T^{12} + 9879 p^{18} T^{14} + p^{24} T^{16} \)
19 \( ( 1 - 122 T + 19262 T^{2} - 1598054 T^{3} + 184664834 T^{4} - 1598054 p^{3} T^{5} + 19262 p^{6} T^{6} - 122 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
23 \( 1 + 50984 T^{2} + 1184277008 T^{4} + 17460450114648 T^{6} + 214285541615044382 T^{8} + 17460450114648 p^{6} T^{10} + 1184277008 p^{12} T^{12} + 50984 p^{18} T^{14} + p^{24} T^{16} \)
29 \( 1 + 55747 T^{2} + 2689371553 T^{4} + 97997970647298 T^{6} + 2493016456293686670 T^{8} + 97997970647298 p^{6} T^{10} + 2689371553 p^{12} T^{12} + 55747 p^{18} T^{14} + p^{24} T^{16} \)
31 \( ( 1 - 121 T + 64928 T^{2} - 1397453 T^{3} + 1743770334 T^{4} - 1397453 p^{3} T^{5} + 64928 p^{6} T^{6} - 121 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
37 \( ( 1 - 509 T + 232763 T^{2} - 63937384 T^{3} + 17269851438 T^{4} - 63937384 p^{3} T^{5} + 232763 p^{6} T^{6} - 509 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
41 \( 1 + 325831 T^{2} + 55427369189 T^{4} + 6215115063879894 T^{6} + \)\(49\!\cdots\!62\)\( T^{8} + 6215115063879894 p^{6} T^{10} + 55427369189 p^{12} T^{12} + 325831 p^{18} T^{14} + p^{24} T^{16} \)
43 \( ( 1 - 37 T + 171856 T^{2} + 580529 T^{3} + 16008374778 T^{4} + 580529 p^{3} T^{5} + 171856 p^{6} T^{6} - 37 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
47 \( 1 + 673128 T^{2} + 205725969680 T^{4} + 38035116822467736 T^{6} + \)\(47\!\cdots\!38\)\( T^{8} + 38035116822467736 p^{6} T^{10} + 205725969680 p^{12} T^{12} + 673128 p^{18} T^{14} + p^{24} T^{16} \)
53 \( 1 + 457499 T^{2} + 135151726097 T^{4} + 29356704635913714 T^{6} + \)\(48\!\cdots\!10\)\( T^{8} + 29356704635913714 p^{6} T^{10} + 135151726097 p^{12} T^{12} + 457499 p^{18} T^{14} + p^{24} T^{16} \)
59 \( 1 + 1197976 T^{2} + 658848403772 T^{4} + 224645430532147560 T^{6} + \)\(53\!\cdots\!30\)\( T^{8} + 224645430532147560 p^{6} T^{10} + 658848403772 p^{12} T^{12} + 1197976 p^{18} T^{14} + p^{24} T^{16} \)
61 \( ( 1 - 574 T + 837230 T^{2} - 366110348 T^{3} + 277714385799 T^{4} - 366110348 p^{3} T^{5} + 837230 p^{6} T^{6} - 574 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
67 \( ( 1 + 1099 T + 1300616 T^{2} + 940063721 T^{3} + 604138073322 T^{4} + 940063721 p^{3} T^{5} + 1300616 p^{6} T^{6} + 1099 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
71 \( 1 + 1071656 T^{2} + 748386640784 T^{4} + 370617385955897688 T^{6} + \)\(15\!\cdots\!98\)\( T^{8} + 370617385955897688 p^{6} T^{10} + 748386640784 p^{12} T^{12} + 1071656 p^{18} T^{14} + p^{24} T^{16} \)
73 \( ( 1 + 1088 T + 1503042 T^{2} + 1067919104 T^{3} + 887747129507 T^{4} + 1067919104 p^{3} T^{5} + 1503042 p^{6} T^{6} + 1088 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
79 \( ( 1 - 931 T + 2149008 T^{2} - 1338612247 T^{3} + 1623052949342 T^{4} - 1338612247 p^{3} T^{5} + 2149008 p^{6} T^{6} - 931 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
83 \( 1 + 3704064 T^{2} + 6275226483968 T^{4} + 6440475301192582208 T^{6} + \)\(44\!\cdots\!02\)\( T^{8} + 6440475301192582208 p^{6} T^{10} + 6275226483968 p^{12} T^{12} + 3704064 p^{18} T^{14} + p^{24} T^{16} \)
89 \( 1 + 2583560 T^{2} + 3976347469404 T^{4} + 4123207445713521592 T^{6} + \)\(33\!\cdots\!14\)\( T^{8} + 4123207445713521592 p^{6} T^{10} + 3976347469404 p^{12} T^{12} + 2583560 p^{18} T^{14} + p^{24} T^{16} \)
97 \( ( 1 + 2185 T + 3491318 T^{2} + 4399067099 T^{3} + 4998430507914 T^{4} + 4399067099 p^{3} T^{5} + 3491318 p^{6} T^{6} + 2185 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
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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

−3.47275882137356083105717394382, −3.38786393750793213237699149251, −3.35270152792110552979633087080, −3.28720847252529450340796642212, −3.07009648985944735954270919420, −3.00258304103630215619634211508, −2.90718879027924682294491916107, −2.70321509999014958749612960624, −2.59924426086718583736948297908, −2.58855974955339248906487103834, −2.35992612053334781140831337171, −2.16307086424769569624798648740, −1.95618809470098879740109268363, −1.76765274860768698619404287873, −1.62215405411838707900674976178, −1.53098363887141625123271184495, −1.41712725266854264927426848139, −1.34649130385158759088539605799, −0.962209282377226122256399074178, −0.826147854525943561136768423378, −0.58531447736850017812077306988, −0.54058540867024542797197543457, −0.46950192917112332773387893470, −0.40216176888026715276851624571, −0.26650693767055300953773789971, 0.26650693767055300953773789971, 0.40216176888026715276851624571, 0.46950192917112332773387893470, 0.54058540867024542797197543457, 0.58531447736850017812077306988, 0.826147854525943561136768423378, 0.962209282377226122256399074178, 1.34649130385158759088539605799, 1.41712725266854264927426848139, 1.53098363887141625123271184495, 1.62215405411838707900674976178, 1.76765274860768698619404287873, 1.95618809470098879740109268363, 2.16307086424769569624798648740, 2.35992612053334781140831337171, 2.58855974955339248906487103834, 2.59924426086718583736948297908, 2.70321509999014958749612960624, 2.90718879027924682294491916107, 3.00258304103630215619634211508, 3.07009648985944735954270919420, 3.28720847252529450340796642212, 3.35270152792110552979633087080, 3.38786393750793213237699149251, 3.47275882137356083105717394382

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

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