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  − 4·2-s + 6·4-s − 15·16-s + 24·17-s + 24·19-s − 12·31-s + 24·32-s − 96·34-s − 96·38-s − 24·47-s − 10·49-s + 12·53-s + 24·61-s + 48·62-s − 6·64-s + 144·68-s + 144·76-s − 28·79-s + 96·94-s + 40·98-s − 48·106-s + 12·107-s + 40·109-s − 26·121-s − 96·122-s − 72·124-s + 127-s + ⋯
L(s)  = 1  − 2.82·2-s + 3·4-s − 3.75·16-s + 5.82·17-s + 5.50·19-s − 2.15·31-s + 4.24·32-s − 16.4·34-s − 15.5·38-s − 3.50·47-s − 1.42·49-s + 1.64·53-s + 3.07·61-s + 6.09·62-s − 3/4·64-s + 17.4·68-s + 16.5·76-s − 3.15·79-s + 9.90·94-s + 4.04·98-s − 4.66·106-s + 1.16·107-s + 3.83·109-s − 2.36·121-s − 8.69·122-s − 6.46·124-s + 0.0887·127-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  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $3.165757436$
$L(\frac12)$  $\approx$  $3.165757436$
$L(\frac{3}{2})$   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 \( ( 1 + T + T^{2} )^{4} \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
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 \( ( 1 - 12 T + 88 T^{2} - 480 T^{3} + 2127 T^{4} - 480 p T^{5} + 88 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( 1 + 80 T^{2} + 3214 T^{4} + 35840 T^{6} - 485165 T^{8} + 35840 p^{2} T^{10} + 3214 p^{4} T^{12} + 80 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 20 T^{2} - 1146 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 104 T^{2} + 5250 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 12 T + 148 T^{2} + 1200 T^{3} + 10047 T^{4} + 1200 p T^{5} + 148 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 6 T - 61 T^{2} + 54 T^{3} + 4692 T^{4} + 54 p T^{5} - 61 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( ( 1 + 34 T^{2} - 3333 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 176 T^{2} + 15234 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( 1 - 220 T^{2} + 26794 T^{4} - 2408560 T^{6} + 180497395 T^{8} - 2408560 p^{2} T^{10} + 26794 p^{4} T^{12} - 220 p^{6} T^{14} + p^{8} T^{16} \)
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

−3.45914942624014748654530272389, −3.33569688737505150169847251417, −3.33179311236744859505928227049, −3.27105060440942964533861536903, −3.26786958719261464420846891917, −3.26641393561378319125480958864, −2.85621654618056608942195261847, −2.67182802459757513713878044661, −2.66986614817986451806889984468, −2.51364843617247239667898348552, −2.39618119285263659760705717789, −2.10038395458285428453921527112, −1.99040342845526136539756612032, −1.76695826659496434204368133450, −1.65896241700584935050024241503, −1.51264886013267997542706791999, −1.42381403937196745603907718074, −1.28141558610562689170863959346, −1.19987309093797477791013480713, −1.03274135121225810912700855272, −1.02623631116594382939961784368, −0.71519075067656998607478234768, −0.66437495239029434762558683254, −0.37980302078076454295286044726, −0.27649738723943874449910614904, 0.27649738723943874449910614904, 0.37980302078076454295286044726, 0.66437495239029434762558683254, 0.71519075067656998607478234768, 1.02623631116594382939961784368, 1.03274135121225810912700855272, 1.19987309093797477791013480713, 1.28141558610562689170863959346, 1.42381403937196745603907718074, 1.51264886013267997542706791999, 1.65896241700584935050024241503, 1.76695826659496434204368133450, 1.99040342845526136539756612032, 2.10038395458285428453921527112, 2.39618119285263659760705717789, 2.51364843617247239667898348552, 2.66986614817986451806889984468, 2.67182802459757513713878044661, 2.85621654618056608942195261847, 3.26641393561378319125480958864, 3.26786958719261464420846891917, 3.27105060440942964533861536903, 3.33179311236744859505928227049, 3.33569688737505150169847251417, 3.45914942624014748654530272389

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

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