Properties

Degree 16
Conductor $ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s − 24·11-s + 16-s + 12·23-s − 24·37-s − 32·41-s + 16·43-s − 48·44-s − 8·47-s + 24·53-s + 24·59-s − 2·64-s + 24·67-s − 24·79-s − 16·83-s + 16·89-s + 24·92-s + 8·101-s − 24·107-s + 24·109-s + 272·121-s + 127-s + 131-s + 137-s + 139-s − 48·148-s + 149-s + ⋯
L(s)  = 1  + 4-s − 7.23·11-s + 1/4·16-s + 2.50·23-s − 3.94·37-s − 4.99·41-s + 2.43·43-s − 7.23·44-s − 1.16·47-s + 3.29·53-s + 3.12·59-s − 1/4·64-s + 2.93·67-s − 2.70·79-s − 1.75·83-s + 1.69·89-s + 2.50·92-s + 0.796·101-s − 2.32·107-s + 2.29·109-s + 24.7·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.94·148-s + 0.0819·149-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $5.162967072$
$L(\frac12)$  $\approx$  $5.162967072$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{2} \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - 94 T^{4} + p^{4} T^{8} \)
good11 \( 1 + 24 T + 304 T^{2} + 2688 T^{3} + 18481 T^{4} + 104232 T^{5} + 496816 T^{6} + 2036016 T^{7} + 7239280 T^{8} + 2036016 p T^{9} + 496816 p^{2} T^{10} + 104232 p^{3} T^{11} + 18481 p^{4} T^{12} + 2688 p^{5} T^{13} + 304 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 - 80 T^{2} + 3006 T^{4} - 69568 T^{6} + 1087139 T^{8} - 69568 p^{2} T^{10} + 3006 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 28 T^{2} + 192 T^{3} + 26 p T^{4} - 4320 T^{5} + 15824 T^{6} + 58560 T^{7} - 349805 T^{8} + 58560 p T^{9} + 15824 p^{2} T^{10} - 4320 p^{3} T^{11} + 26 p^{5} T^{12} + 192 p^{5} T^{13} - 28 p^{6} T^{14} + p^{8} T^{16} \)
19 \( 1 + 58 T^{2} + 99 p T^{4} - 1872 T^{5} + 44906 T^{6} - 71184 T^{7} + 869156 T^{8} - 71184 p T^{9} + 44906 p^{2} T^{10} - 1872 p^{3} T^{11} + 99 p^{5} T^{12} + 58 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 - 6 T + 57 T^{2} - 270 T^{3} + 1772 T^{4} - 270 p T^{5} + 57 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 36 T^{2} + 470 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( 1 + 100 T^{2} + 5706 T^{4} + 237200 T^{6} + 7904915 T^{8} + 237200 p^{2} T^{10} + 5706 p^{4} T^{12} + 100 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 + 24 T + 216 T^{2} + 1680 T^{3} + 21025 T^{4} + 178920 T^{5} + 986040 T^{6} + 7528032 T^{7} + 58723920 T^{8} + 7528032 p T^{9} + 986040 p^{2} T^{10} + 178920 p^{3} T^{11} + 21025 p^{4} T^{12} + 1680 p^{5} T^{13} + 216 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
41 \( ( 1 + 16 T + 232 T^{2} + 2000 T^{3} + 15591 T^{4} + 2000 p T^{5} + 232 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 8 T + 180 T^{2} - 1000 T^{3} + 11750 T^{4} - 1000 p T^{5} + 180 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 + 8 T - 78 T^{2} - 240 T^{3} + 6305 T^{4} - 3648 T^{5} - 344398 T^{6} - 95176 T^{7} + 11426244 T^{8} - 95176 p T^{9} - 344398 p^{2} T^{10} - 3648 p^{3} T^{11} + 6305 p^{4} T^{12} - 240 p^{5} T^{13} - 78 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 24 T + 362 T^{2} - 4080 T^{3} + 37881 T^{4} - 336960 T^{5} + 2978650 T^{6} - 25456152 T^{7} + 197950532 T^{8} - 25456152 p T^{9} + 2978650 p^{2} T^{10} - 336960 p^{3} T^{11} + 37881 p^{4} T^{12} - 4080 p^{5} T^{13} + 362 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 24 T + 224 T^{2} - 1392 T^{3} + 9358 T^{4} - 16056 T^{5} - 613888 T^{6} + 6806952 T^{7} - 48734957 T^{8} + 6806952 p T^{9} - 613888 p^{2} T^{10} - 16056 p^{3} T^{11} + 9358 p^{4} T^{12} - 1392 p^{5} T^{13} + 224 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 124 T^{2} + 6186 T^{4} + 30240 T^{5} + 293552 T^{6} + 4411200 T^{7} + 18080963 T^{8} + 4411200 p T^{9} + 293552 p^{2} T^{10} + 30240 p^{3} T^{11} + 6186 p^{4} T^{12} + 124 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 24 T + 332 T^{2} - 3696 T^{3} + 30922 T^{4} - 206568 T^{5} + 963056 T^{6} - 1103736 T^{7} - 11820413 T^{8} - 1103736 p T^{9} + 963056 p^{2} T^{10} - 206568 p^{3} T^{11} + 30922 p^{4} T^{12} - 3696 p^{5} T^{13} + 332 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 4371368 p^{2} T^{10} + 41532 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 + 156 T^{2} + 12202 T^{4} + 230256 T^{6} - 10850829 T^{8} + 230256 p^{2} T^{10} + 12202 p^{4} T^{12} + 156 p^{6} T^{14} + p^{8} T^{16} \)
79 \( 1 + 24 T + 252 T^{2} + 1584 T^{3} + 5050 T^{4} + 17832 T^{5} - 542160 T^{6} - 18801768 T^{7} - 226869549 T^{8} - 18801768 p T^{9} - 542160 p^{2} T^{10} + 17832 p^{3} T^{11} + 5050 p^{4} T^{12} + 1584 p^{5} T^{13} + 252 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
83 \( ( 1 + 8 T + 148 T^{2} + 1192 T^{3} + 18438 T^{4} + 1192 p T^{5} + 148 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 - 16 T - 48 T^{2} + 1824 T^{3} + 2078 T^{4} - 135504 T^{5} + 38912 T^{6} + 7970384 T^{7} - 72309597 T^{8} + 7970384 p T^{9} + 38912 p^{2} T^{10} - 135504 p^{3} T^{11} + 2078 p^{4} T^{12} + 1824 p^{5} T^{13} - 48 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 24 T^{2} + 19036 T^{4} - 653400 T^{6} + 157960902 T^{8} - 653400 p^{2} T^{10} + 19036 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.53345289779169516689751430260, −3.52605672019096578168114958964, −3.38553663913116955537356845714, −3.07970134418292686914415470137, −2.97926145368555365668444617717, −2.92616631411907260065898799591, −2.85392276138053913478938824054, −2.80820857365368323887647466723, −2.62606557595000936560621918606, −2.50865997447267313464070974330, −2.50373110256644709819093863215, −2.35992896738636617296814834189, −2.08869691299227641257062703458, −2.07947607705290411439155733421, −1.91416895571848540848711294336, −1.80075146859170239376575391977, −1.77156187178959820831441426959, −1.53602425015116208156085436635, −1.30803123899090426892365242905, −1.13953809679200660032134813389, −0.74327576339253188545193901259, −0.51726468778802505605677853855, −0.50037622870713479249850232395, −0.36426027656358802423075337262, −0.31615537077247545733565601265, 0.31615537077247545733565601265, 0.36426027656358802423075337262, 0.50037622870713479249850232395, 0.51726468778802505605677853855, 0.74327576339253188545193901259, 1.13953809679200660032134813389, 1.30803123899090426892365242905, 1.53602425015116208156085436635, 1.77156187178959820831441426959, 1.80075146859170239376575391977, 1.91416895571848540848711294336, 2.07947607705290411439155733421, 2.08869691299227641257062703458, 2.35992896738636617296814834189, 2.50373110256644709819093863215, 2.50865997447267313464070974330, 2.62606557595000936560621918606, 2.80820857365368323887647466723, 2.85392276138053913478938824054, 2.92616631411907260065898799591, 2.97926145368555365668444617717, 3.07970134418292686914415470137, 3.38553663913116955537356845714, 3.52605672019096578168114958964, 3.53345289779169516689751430260

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

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