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  + 8·13-s − 2·16-s − 16·23-s − 16·37-s − 8·43-s + 8·47-s − 32·53-s − 8·59-s − 32·61-s + 16·67-s + 8·83-s + 16·89-s + 16·97-s − 8·103-s + 32·107-s − 32·113-s + 64·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + ⋯
L(s)  = 1  + 2.21·13-s − 1/2·16-s − 3.33·23-s − 2.63·37-s − 1.21·43-s + 1.16·47-s − 4.39·53-s − 1.04·59-s − 4.09·61-s + 1.95·67-s + 0.878·83-s + 1.69·89-s + 1.62·97-s − 0.788·103-s + 3.09·107-s − 3.01·113-s + 5.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.46·169-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  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $4.003943495$
$L(\frac12)$  $\approx$  $4.003943495$
$L(\frac{3}{2})$   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 \( ( 1 + T^{4} )^{2} \)
3 \( 1 \)
5 \( 1 \)
7 \( ( 1 + T^{4} )^{2} \)
good11 \( ( 1 - 32 T^{2} + 466 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( 1 - 8 T + 32 T^{2} - 128 T^{3} + 574 T^{4} - 2360 T^{5} + 8704 T^{6} - 35112 T^{7} + 139491 T^{8} - 35112 p T^{9} + 8704 p^{2} T^{10} - 2360 p^{3} T^{11} + 574 p^{4} T^{12} - 128 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 96 T^{3} + 158 T^{4} + 2400 T^{5} + 4608 T^{6} + 8544 T^{7} - 207741 T^{8} + 8544 p T^{9} + 4608 p^{2} T^{10} + 2400 p^{3} T^{11} + 158 p^{4} T^{12} - 96 p^{5} T^{13} + p^{8} T^{16} \)
19 \( ( 1 - 48 T^{2} + 1202 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( 1 + 16 T + 128 T^{2} + 800 T^{3} + 4670 T^{4} + 26576 T^{5} + 147456 T^{6} + 35120 p T^{7} + 7827 p^{2} T^{8} + 35120 p^{2} T^{9} + 147456 p^{2} T^{10} + 26576 p^{3} T^{11} + 4670 p^{4} T^{12} + 800 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
29 \( ( 1 + 2 p T^{2} + 96 T^{3} + 2019 T^{4} + 96 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 42 T^{2} - 192 T^{3} + 1139 T^{4} - 192 p T^{5} + 42 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( 1 + 16 T + 128 T^{2} + 784 T^{3} + 4324 T^{4} + 27760 T^{5} + 198016 T^{6} + 1322736 T^{7} + 8422566 T^{8} + 1322736 p T^{9} + 198016 p^{2} T^{10} + 27760 p^{3} T^{11} + 4324 p^{4} T^{12} + 784 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 172 T^{2} + 15786 T^{4} - 1010672 T^{6} + 47929043 T^{8} - 1010672 p^{2} T^{10} + 15786 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 + 8 T + 32 T^{2} + 368 T^{3} + 1438 T^{4} - 5752 T^{5} - 24320 T^{6} - 270552 T^{7} - 3004701 T^{8} - 270552 p T^{9} - 24320 p^{2} T^{10} - 5752 p^{3} T^{11} + 1438 p^{4} T^{12} + 368 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 - 8 T + 32 T^{2} - 520 T^{3} + 2012 T^{4} + 17432 T^{5} - 68640 T^{6} + 1046296 T^{7} - 15494202 T^{8} + 1046296 p T^{9} - 68640 p^{2} T^{10} + 17432 p^{3} T^{11} + 2012 p^{4} T^{12} - 520 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 32 T + 512 T^{2} + 6592 T^{3} + 79454 T^{4} + 821920 T^{5} + 7348224 T^{6} + 61872608 T^{7} + 480643491 T^{8} + 61872608 p T^{9} + 7348224 p^{2} T^{10} + 821920 p^{3} T^{11} + 79454 p^{4} T^{12} + 6592 p^{5} T^{13} + 512 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \)
59 \( ( 1 + 4 T + 170 T^{2} + 856 T^{3} + 13027 T^{4} + 856 p T^{5} + 170 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 16 T + 222 T^{2} + 2480 T^{3} + 20579 T^{4} + 2480 p T^{5} + 222 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 16 T + 128 T^{2} - 1648 T^{3} + 12196 T^{4} + 9104 T^{5} - 348800 T^{6} + 6630000 T^{7} - 95097114 T^{8} + 6630000 p T^{9} - 348800 p^{2} T^{10} + 9104 p^{3} T^{11} + 12196 p^{4} T^{12} - 1648 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
71 \( ( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( 1 + 28 p T^{4} + 20460870 T^{8} + 28 p^{5} T^{12} + p^{8} T^{16} \)
79 \( 1 - 320 T^{2} + 58500 T^{4} - 7347136 T^{6} + 669209222 T^{8} - 7347136 p^{2} T^{10} + 58500 p^{4} T^{12} - 320 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 - 8 T + 32 T^{2} - 160 T^{3} + 10142 T^{4} - 122968 T^{5} + 672000 T^{6} - 7632056 T^{7} + 75142083 T^{8} - 7632056 p T^{9} + 672000 p^{2} T^{10} - 122968 p^{3} T^{11} + 10142 p^{4} T^{12} - 160 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 - 8 T + 316 T^{2} - 1912 T^{3} + 40422 T^{4} - 1912 p T^{5} + 316 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 16 T + 128 T^{2} - 1136 T^{3} + 10556 T^{4} - 116816 T^{5} + 1163136 T^{6} - 11493168 T^{7} + 113239174 T^{8} - 11493168 p T^{9} + 1163136 p^{2} T^{10} - 116816 p^{3} T^{11} + 10556 p^{4} T^{12} - 1136 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
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

−3.67227151550542258534234427383, −3.56374265251577514429304408744, −3.36210693942609757049395135595, −3.15684952951806546684801507907, −3.13947930292168024544584912797, −3.12282890035675066197319588295, −2.92823324811952693780306745833, −2.84600117698028953473279023127, −2.77771424095374270735509726658, −2.52691134046953549203251126106, −2.29638417366889197514547395971, −2.12207885649123457522371909137, −1.97656019902731778130060265939, −1.92167938491223065497947516093, −1.77938070022417431632864397314, −1.75354013429345891697626495462, −1.74711884951506192461190227099, −1.54405428099446043313228903992, −1.40767654736903383798807150691, −1.09667169838984893080014557780, −0.965016983142027748135988071982, −0.63018672653389835222683019207, −0.57858624825547086064757433847, −0.27258980048435326874109320390, −0.23975981702927033148530612956, 0.23975981702927033148530612956, 0.27258980048435326874109320390, 0.57858624825547086064757433847, 0.63018672653389835222683019207, 0.965016983142027748135988071982, 1.09667169838984893080014557780, 1.40767654736903383798807150691, 1.54405428099446043313228903992, 1.74711884951506192461190227099, 1.75354013429345891697626495462, 1.77938070022417431632864397314, 1.92167938491223065497947516093, 1.97656019902731778130060265939, 2.12207885649123457522371909137, 2.29638417366889197514547395971, 2.52691134046953549203251126106, 2.77771424095374270735509726658, 2.84600117698028953473279023127, 2.92823324811952693780306745833, 3.12282890035675066197319588295, 3.13947930292168024544584912797, 3.15684952951806546684801507907, 3.36210693942609757049395135595, 3.56374265251577514429304408744, 3.67227151550542258534234427383

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

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