Properties

Label 16-21e16-1.1-c5e8-0-0
Degree $16$
Conductor $1.431\times 10^{21}$
Sign $1$
Analytic cond. $6.26315\times 10^{14}$
Root an. cond. $8.41006$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $8$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 118·4-s + 6.59e3·16-s − 1.36e4·25-s − 1.91e4·37-s − 4.11e4·43-s − 2.61e5·64-s − 4.23e4·67-s − 2.51e5·79-s + 1.61e6·100-s − 3.44e5·109-s − 1.18e6·121-s + 127-s + 131-s + 137-s + 139-s + 2.25e6·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.84e6·169-s + 4.85e6·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 3.68·4-s + 6.44·16-s − 4.38·25-s − 2.29·37-s − 3.39·43-s − 7.97·64-s − 1.15·67-s − 4.52·79-s + 16.1·100-s − 2.77·109-s − 7.37·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 8.47·148-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 7.64·169-s + 12.5·172-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(6.26315\times 10^{14}\)
Root analytic conductor: \(8.41006\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(8\)
Selberg data: \((16,\ 3^{16} \cdot 7^{16} ,\ ( \ : [5/2]^{8} ),\ 1 )\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( ( 1 + 59 T^{2} + 481 p^{2} T^{4} + 59 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
5 \( ( 1 + 6848 T^{2} + 24891826 T^{4} + 6848 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
11 \( ( 1 + 4908 p^{2} T^{2} + 139898041430 T^{4} + 4908 p^{12} T^{6} + p^{20} T^{8} )^{2} \)
13 \( ( 1 + 1420096 T^{2} + 779793261394 T^{4} + 1420096 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
17 \( ( 1 + 2007936 T^{2} + 3446593247810 T^{4} + 2007936 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
19 \( ( 1 + 7607244 T^{2} + 26715012021686 T^{4} + 7607244 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
23 \( ( 1 + 829636 p T^{2} + 167668754709094 T^{4} + 829636 p^{11} T^{6} + p^{20} T^{8} )^{2} \)
29 \( ( 1 + 21105452 T^{2} + 96069831504310 T^{4} + 21105452 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
31 \( ( 1 - 20130308 T^{2} + 1740348466506118 T^{4} - 20130308 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
37 \( ( 1 + 4784 T + 75114330 T^{2} + 4784 p^{5} T^{3} + p^{10} T^{4} )^{4} \)
41 \( ( 1 + 86798208 T^{2} + 1978782963872930 T^{4} + 86798208 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
43 \( ( 1 + 10296 T + 320264262 T^{2} + 10296 p^{5} T^{3} + p^{10} T^{4} )^{4} \)
47 \( ( 1 + 699949180 T^{2} + 220210640936842566 T^{4} + 699949180 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
53 \( ( 1 - 214678284 T^{2} + 354090238473026774 T^{4} - 214678284 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
59 \( ( 1 + 1120804268 T^{2} + 1333434963591116758 T^{4} + 1120804268 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
61 \( ( 1 + 2189045888 T^{2} + 2563883188164636690 T^{4} + 2189045888 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
67 \( ( 1 + 10584 T + 2091935478 T^{2} + 10584 p^{5} T^{3} + p^{10} T^{4} )^{4} \)
71 \( ( 1 + 686542172 T^{2} + 6627815389407138598 T^{4} + 686542172 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
73 \( ( 1 + 4574308864 T^{2} + 10713295960747278370 T^{4} + 4574308864 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
79 \( ( 1 + 62768 T + 2636722462 T^{2} + 62768 p^{5} T^{3} + p^{10} T^{4} )^{4} \)
83 \( ( 1 + 68498308 p T^{2} + 33005603990673824950 T^{4} + 68498308 p^{11} T^{6} + p^{20} T^{8} )^{2} \)
89 \( ( 1 + 16933432896 T^{2} + \)\(13\!\cdots\!06\)\( T^{4} + 16933432896 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
97 \( ( 1 + 15912910912 T^{2} + \)\(13\!\cdots\!26\)\( T^{4} + 15912910912 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
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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

−4.76595998783021677919650717204, −4.15776924691665043330608042822, −4.14421316306443043148758387150, −4.09165414951758407511774919519, −4.08208364268789818599596325033, −3.87029792190206528835159351968, −3.82601017729652787981938778258, −3.79877785017424152885525445251, −3.79672849118677358618601427160, −3.33911309168299900069244889817, −3.24429887335336888039556620316, −2.95507197048260548174376292431, −2.94014478315800471154889557590, −2.72903553659885590859757714766, −2.54083322087813541063759708801, −2.32879932055782628513131329922, −2.23976325241379451568608829320, −1.95712227525724835287858217206, −1.82810606721939338471105041918, −1.52357218383758525566708223434, −1.39182401357811451403857198434, −1.23180965507702294376864588330, −1.15100219726518430637131665898, −1.11663841645720387559669160235, −1.00216270888322709782583949528, 0, 0, 0, 0, 0, 0, 0, 0, 1.00216270888322709782583949528, 1.11663841645720387559669160235, 1.15100219726518430637131665898, 1.23180965507702294376864588330, 1.39182401357811451403857198434, 1.52357218383758525566708223434, 1.82810606721939338471105041918, 1.95712227525724835287858217206, 2.23976325241379451568608829320, 2.32879932055782628513131329922, 2.54083322087813541063759708801, 2.72903553659885590859757714766, 2.94014478315800471154889557590, 2.95507197048260548174376292431, 3.24429887335336888039556620316, 3.33911309168299900069244889817, 3.79672849118677358618601427160, 3.79877785017424152885525445251, 3.82601017729652787981938778258, 3.87029792190206528835159351968, 4.08208364268789818599596325033, 4.09165414951758407511774919519, 4.14421316306443043148758387150, 4.15776924691665043330608042822, 4.76595998783021677919650717204

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

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