# Properties

 Degree 16 Conductor $2^{40} \cdot 3^{24} \cdot 7^{8}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·5-s − 16·19-s + 16·23-s + 28·25-s + 24·29-s − 8·43-s − 8·47-s − 4·49-s − 8·67-s + 32·71-s + 8·73-s − 128·95-s − 32·97-s + 16·101-s + 128·115-s + 44·121-s + 80·125-s + 127-s + 131-s + 137-s + 139-s + 192·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
 L(s)  = 1 + 3.57·5-s − 3.67·19-s + 3.33·23-s + 28/5·25-s + 4.45·29-s − 1.21·43-s − 1.16·47-s − 4/7·49-s − 0.977·67-s + 3.79·71-s + 0.936·73-s − 13.1·95-s − 3.24·97-s + 1.59·101-s + 11.9·115-s + 4·121-s + 7.15·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 15.9·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24} \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^{40} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$2^{40} \cdot 3^{24} \cdot 7^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{6048} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(16,\ 2^{40} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$ $L(1)$ $\approx$ $63.97419920$ $L(\frac12)$ $\approx$ $63.97419920$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 16. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$( 1 + T^{2} )^{4}$$
good5 $$( 1 - 4 T + 2 p T^{2} - 32 T^{3} + 87 T^{4} - 32 p T^{5} + 2 p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
11 $$1 - 4 p T^{2} + 994 T^{4} - 15968 T^{6} + 198763 T^{8} - 15968 p^{2} T^{10} + 994 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16}$$
13 $$1 - 8 T^{2} + 228 T^{4} - 4504 T^{6} + 20006 T^{8} - 4504 p^{2} T^{10} + 228 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16}$$
17 $$( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4}$$
19 $$( 1 + 8 T + 30 T^{2} + 112 T^{3} + 611 T^{4} + 112 p T^{5} + 30 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
23 $$( 1 - 8 T + 2 p T^{2} - 88 T^{3} + 231 T^{4} - 88 p T^{5} + 2 p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
29 $$( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4}$$
31 $$1 - 156 T^{2} + 12106 T^{4} - 610416 T^{6} + 22035219 T^{8} - 610416 p^{2} T^{10} + 12106 p^{4} T^{12} - 156 p^{6} T^{14} + p^{8} T^{16}$$
37 $$1 - 164 T^{2} + 14154 T^{4} - 829168 T^{6} + 35617331 T^{8} - 829168 p^{2} T^{10} + 14154 p^{4} T^{12} - 164 p^{6} T^{14} + p^{8} T^{16}$$
41 $$1 - 116 T^{2} + 11362 T^{4} - 660608 T^{6} + 32960203 T^{8} - 660608 p^{2} T^{10} + 11362 p^{4} T^{12} - 116 p^{6} T^{14} + p^{8} T^{16}$$
43 $$( 1 + 4 T + 84 T^{2} + 44 T^{3} + 3002 T^{4} + 44 p T^{5} + 84 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
47 $$( 1 + 4 T + 76 T^{2} + 44 T^{3} + 2154 T^{4} + 44 p T^{5} + 76 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
53 $$( 1 + 124 T^{2} + 192 T^{3} + 7734 T^{4} + 192 p T^{5} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
59 $$1 - 232 T^{2} + 28836 T^{4} - 2582456 T^{6} + 175948646 T^{8} - 2582456 p^{2} T^{10} + 28836 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16}$$
61 $$( 1 - 92 T^{2} + 6486 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
67 $$( 1 + 4 T + 228 T^{2} + 620 T^{3} + 21530 T^{4} + 620 p T^{5} + 228 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
71 $$( 1 - 16 T + 310 T^{2} - 3320 T^{3} + 33807 T^{4} - 3320 p T^{5} + 310 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
73 $$( 1 - 4 T + 156 T^{2} - 20 p T^{3} + 11402 T^{4} - 20 p^{2} T^{5} + 156 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
79 $$1 - 248 T^{2} + 37188 T^{4} - 3949096 T^{6} + 344639558 T^{8} - 3949096 p^{2} T^{10} + 37188 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16}$$
83 $$1 - 232 T^{2} + 34596 T^{4} - 3949112 T^{6} + 355979174 T^{8} - 3949112 p^{2} T^{10} + 34596 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16}$$
89 $$1 - 292 T^{2} + 47890 T^{4} - 6052768 T^{6} + 614802619 T^{8} - 6052768 p^{2} T^{10} + 47890 p^{4} T^{12} - 292 p^{6} T^{14} + p^{8} T^{16}$$
97 $$( 1 + 16 T + 300 T^{2} + 4208 T^{3} + 41318 T^{4} + 4208 p T^{5} + 300 p^{2} T^{6} + 16 p^{3} T^{7} + 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

−3.04637826612850192862577202271, −2.97174111818743424554710859370, −2.87099715177283997738541561912, −2.84854939083634840573830341900, −2.84635270026773653283050701686, −2.83316748512958257491657654022, −2.78314527451617909408218542862, −2.61910101344009414843902186909, −2.29620108616098144077442724052, −2.18248130541583922713924155289, −1.96520035562596015841153736045, −1.95921825406442540299731609109, −1.95590941314841440997262091269, −1.88837341919346447752905858732, −1.78649613979293731993243945337, −1.64063992526293053522789000613, −1.54508478891026504594513082786, −1.32579090498005799832329908452, −1.07084058956806449728231798869, −1.03921693088365904270909527807, −0.806271150537698766524519076454, −0.67620212561380455741458993931, −0.57624787825082764553617777649, −0.50314665263146546899600447676, −0.26003176139400920359723001397, 0.26003176139400920359723001397, 0.50314665263146546899600447676, 0.57624787825082764553617777649, 0.67620212561380455741458993931, 0.806271150537698766524519076454, 1.03921693088365904270909527807, 1.07084058956806449728231798869, 1.32579090498005799832329908452, 1.54508478891026504594513082786, 1.64063992526293053522789000613, 1.78649613979293731993243945337, 1.88837341919346447752905858732, 1.95590941314841440997262091269, 1.95921825406442540299731609109, 1.96520035562596015841153736045, 2.18248130541583922713924155289, 2.29620108616098144077442724052, 2.61910101344009414843902186909, 2.78314527451617909408218542862, 2.83316748512958257491657654022, 2.84635270026773653283050701686, 2.84854939083634840573830341900, 2.87099715177283997738541561912, 2.97174111818743424554710859370, 3.04637826612850192862577202271

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

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