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

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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  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $3.683761425$
$L(\frac12)$  $\approx$  $3.683761425$
$L(\frac{3}{2})$   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 \( 1 + 16 T + 128 T^{2} + 1136 T^{3} + 10556 T^{4} + 116816 T^{5} + 1163136 T^{6} + 11493168 T^{7} + 113239174 T^{8} + 11493168 p T^{9} + 1163136 p^{2} T^{10} + 116816 p^{3} T^{11} + 10556 p^{4} T^{12} + 1136 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.55396518073639348193880556878, −3.51835709902715486372016589097, −3.18343246105206921119910253007, −3.10827393343650203989420938924, −3.08516076419862647878181931438, −3.02114468244778674067375509310, −2.99400679836436862778162835879, −2.88911616398679639984627737041, −2.57776330589408099818493215056, −2.44862624316212779760033996747, −2.39398410414490926861789224403, −2.29683296931882682510921698734, −2.18108823679994157617920148507, −2.07574004074383983011991093315, −2.05037955079324852595825770398, −1.65035383270839368215730839981, −1.46866451412770780430076149114, −1.43772437434668039705318438539, −1.31406883418170422371505121307, −0.998958831000997447776626346292, −0.945303765170667093007102829602, −0.793576430557116122452167975156, −0.63566087153005768704016647823, −0.27727964515687568696094245146, −0.19918555011668512923708670613, 0.19918555011668512923708670613, 0.27727964515687568696094245146, 0.63566087153005768704016647823, 0.793576430557116122452167975156, 0.945303765170667093007102829602, 0.998958831000997447776626346292, 1.31406883418170422371505121307, 1.43772437434668039705318438539, 1.46866451412770780430076149114, 1.65035383270839368215730839981, 2.05037955079324852595825770398, 2.07574004074383983011991093315, 2.18108823679994157617920148507, 2.29683296931882682510921698734, 2.39398410414490926861789224403, 2.44862624316212779760033996747, 2.57776330589408099818493215056, 2.88911616398679639984627737041, 2.99400679836436862778162835879, 3.02114468244778674067375509310, 3.08516076419862647878181931438, 3.10827393343650203989420938924, 3.18343246105206921119910253007, 3.51835709902715486372016589097, 3.55396518073639348193880556878

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

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