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  + 2·4-s − 4·7-s + 16-s − 24·19-s − 8·28-s − 12·31-s + 16·37-s − 32·43-s + 18·49-s + 24·61-s − 2·64-s − 40·67-s − 24·73-s − 48·76-s + 28·79-s + 72·103-s − 40·109-s − 4·112-s − 26·121-s − 24·124-s + 127-s + 131-s + 96·133-s + 137-s + 139-s + 32·148-s + 149-s + ⋯
L(s)  = 1  + 4-s − 1.51·7-s + 1/4·16-s − 5.50·19-s − 1.51·28-s − 2.15·31-s + 2.63·37-s − 4.87·43-s + 18/7·49-s + 3.07·61-s − 1/4·64-s − 4.88·67-s − 2.80·73-s − 5.50·76-s + 3.15·79-s + 7.09·103-s − 3.83·109-s − 0.377·112-s − 2.36·121-s − 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 8.32·133-s + 0.0854·137-s + 0.0848·139-s + 2.63·148-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  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.1563337005$
$L(\frac12)$  $\approx$  $0.1563337005$
$L(\frac{3}{2})$   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 \( ( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
good11 \( ( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \)
17 \( 1 - 32 T^{2} + 478 T^{4} + 1024 T^{6} - 81341 T^{8} + 1024 p^{2} T^{10} + 478 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 12 T + 92 T^{2} + 528 T^{3} + 2487 T^{4} + 528 p T^{5} + 92 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 28 T^{2} + 255 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 62 T^{2} + 1995 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 6 T + 23 T^{2} + 66 T^{3} - 468 T^{4} + 66 p T^{5} + 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 8 T - 8 T^{2} + 16 T^{3} + 1447 T^{4} + 16 p T^{5} - 8 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + 20 T^{2} - 1146 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 + 8 T + 84 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
47 \( 1 - 152 T^{2} + 13198 T^{4} - 834176 T^{6} + 42212419 T^{8} - 834176 p^{2} T^{10} + 13198 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 158 T^{2} + 13753 T^{4} + 883694 T^{6} + 47672164 T^{8} + 883694 p^{2} T^{10} + 13753 p^{4} T^{12} + 158 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 - 38 T^{2} - 4727 T^{4} + 30058 T^{6} + 20937316 T^{8} + 30058 p^{2} T^{10} - 4727 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 - 12 T + 176 T^{2} - 1536 T^{3} + 15591 T^{4} - 1536 p T^{5} + 176 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \)
71 \( ( 1 - 176 T^{2} + 15234 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 12 T + 182 T^{2} + 1608 T^{3} + 16131 T^{4} + 1608 p T^{5} + 182 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 14 T + 7 T^{2} - 434 T^{3} + 13996 T^{4} - 434 p T^{5} + 7 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 278 T^{2} + 32811 T^{4} + 278 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 190 T^{2} + 20643 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−3.57809082300480419805992260702, −3.52730612071327029803041952518, −3.19470665456598952431332159511, −3.18058444351203536781635439010, −3.16898121966913030585047533929, −3.08422113787074866686839713766, −2.97194830648229878207283551245, −2.64053639490025127341516391349, −2.58954535698109018311234781500, −2.54948464887696042426060592854, −2.31668643047815727870522719184, −2.29022266079799523413965184335, −2.11719480203679649183899570274, −2.02829338937267211144962754552, −1.92325686253028341240956056872, −1.72737407971386825805044017987, −1.63844133290403571841484864048, −1.60402582523411515242624414103, −1.43011688794251632095660502580, −1.14657511021093436113004249526, −0.792971381362917272931947788805, −0.70558779521651691287186195459, −0.46521301317213635668169933435, −0.29574091581702251528318757179, −0.05048182065734429726723121525, 0.05048182065734429726723121525, 0.29574091581702251528318757179, 0.46521301317213635668169933435, 0.70558779521651691287186195459, 0.792971381362917272931947788805, 1.14657511021093436113004249526, 1.43011688794251632095660502580, 1.60402582523411515242624414103, 1.63844133290403571841484864048, 1.72737407971386825805044017987, 1.92325686253028341240956056872, 2.02829338937267211144962754552, 2.11719480203679649183899570274, 2.29022266079799523413965184335, 2.31668643047815727870522719184, 2.54948464887696042426060592854, 2.58954535698109018311234781500, 2.64053639490025127341516391349, 2.97194830648229878207283551245, 3.08422113787074866686839713766, 3.16898121966913030585047533929, 3.18058444351203536781635439010, 3.19470665456598952431332159511, 3.52730612071327029803041952518, 3.57809082300480419805992260702

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

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