Properties

Degree 32
Conductor $ 2^{32} \cdot 3^{32} \cdot 7^{32} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·13-s − 16-s + 28·25-s − 8·37-s − 16·61-s + 10·64-s − 8·73-s + 88·97-s + 24·109-s − 60·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 68·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 2.21·13-s − 1/4·16-s + 28/5·25-s − 1.31·37-s − 2.04·61-s + 5/4·64-s − 0.936·73-s + 8.93·97-s + 2.29·109-s − 5.45·121-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 − 5.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{32} \cdot 3^{32} \cdot 7^{32}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1764} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(10.49628234\)
\(L(\frac12)\)  \(\approx\)  \(10.49628234\)
\(L(\frac{3}{2})\)   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 32. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( 1 + T^{4} - 5 p T^{6} + p^{2} T^{8} - 5 p^{3} T^{10} + p^{4} T^{12} + p^{8} T^{16} \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 - 14 T^{2} + 117 T^{4} - 698 T^{6} + 3576 T^{8} - 698 p^{2} T^{10} + 117 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( ( 1 + 30 T^{2} + 757 T^{4} + 11666 T^{6} + 154360 T^{8} + 11666 p^{2} T^{10} + 757 p^{4} T^{12} + 30 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 + 2 T + 27 T^{2} + 38 T^{3} + 378 T^{4} + 38 p T^{5} + 27 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
17 \( ( 1 - 76 T^{2} + 3096 T^{4} - 83940 T^{6} + 1656750 T^{8} - 83940 p^{2} T^{10} + 3096 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 - 34 T^{2} + 1041 T^{4} - 25110 T^{6} + 610032 T^{8} - 25110 p^{2} T^{10} + 1041 p^{4} T^{12} - 34 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 + 116 T^{2} + 6936 T^{4} + 269340 T^{6} + 7319790 T^{8} + 269340 p^{2} T^{10} + 6936 p^{4} T^{12} + 116 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 146 T^{2} + 10849 T^{4} - 18010 p T^{6} + 17802004 T^{8} - 18010 p^{3} T^{10} + 10849 p^{4} T^{12} - 146 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 132 T^{2} + 6430 T^{4} - 130264 T^{6} + 1736719 T^{8} - 130264 p^{2} T^{10} + 6430 p^{4} T^{12} - 132 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 + 2 T + 115 T^{2} + 282 T^{3} + 5758 T^{4} + 282 p T^{5} + 115 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
41 \( ( 1 - 188 T^{2} + 15912 T^{4} - 846644 T^{6} + 36194574 T^{8} - 846644 p^{2} T^{10} + 15912 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 - 202 T^{2} + 17289 T^{4} - 881742 T^{6} + 37179072 T^{8} - 881742 p^{2} T^{10} + 17289 p^{4} T^{12} - 202 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( ( 1 + 200 T^{2} + 22212 T^{4} + 1666392 T^{6} + 91084806 T^{8} + 1666392 p^{2} T^{10} + 22212 p^{4} T^{12} + 200 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 282 T^{2} + 34177 T^{4} - 2497906 T^{6} + 140957380 T^{8} - 2497906 p^{2} T^{10} + 34177 p^{4} T^{12} - 282 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( ( 1 + 186 T^{2} + 18161 T^{4} + 1259962 T^{6} + 77631844 T^{8} + 1259962 p^{2} T^{10} + 18161 p^{4} T^{12} + 186 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 + 4 T + 162 T^{2} + 588 T^{3} + 12546 T^{4} + 588 p T^{5} + 162 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
67 \( ( 1 - 254 T^{2} + 32925 T^{4} - 3201102 T^{6} + 246095804 T^{8} - 3201102 p^{2} T^{10} + 32925 p^{4} T^{12} - 254 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
71 \( ( 1 + 360 T^{2} + 62596 T^{4} + 7074488 T^{6} + 580384198 T^{8} + 7074488 p^{2} T^{10} + 62596 p^{4} T^{12} + 360 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 + 2 T + 141 T^{2} - 58 T^{3} + 12336 T^{4} - 58 p T^{5} + 141 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
79 \( ( 1 - 332 T^{2} + 56998 T^{4} - 6691240 T^{6} + 595616087 T^{8} - 6691240 p^{2} T^{10} + 56998 p^{4} T^{12} - 332 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( ( 1 + 110 T^{2} + 7989 T^{4} + 466194 T^{6} + 66686616 T^{8} + 466194 p^{2} T^{10} + 7989 p^{4} T^{12} + 110 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 - 352 T^{2} + 72316 T^{4} - 10162848 T^{6} + 1047015622 T^{8} - 10162848 p^{2} T^{10} + 72316 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 - 22 T + 373 T^{2} - 3806 T^{3} + 41336 T^{4} - 3806 p T^{5} + 373 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.28648554425463789002679714797, −2.28260199480671556929414364391, −2.27810031855356558072781439945, −2.20412504242820188849183128493, −2.04578449131867744961355689480, −2.01911930666568090348988630781, −2.01253190379537618604101417863, −1.75354352526080615316113641475, −1.64913835470040613686119683269, −1.60464812140013274917322500917, −1.54372403257566561120672951710, −1.42166601275010832952675103490, −1.41948091194728482721637279522, −1.31734600330951595961819512338, −1.21478243964803465166301796354, −1.18383846034101734490577184769, −1.11156910859636149200449404898, −0.823827823122380737496298896036, −0.820981595397107216075105430025, −0.64136548603388047988013169383, −0.59221119386588057815291559640, −0.54799028107928627436316676702, −0.43093380708457703407678366733, −0.21789538365148923974296670302, −0.15651073945978104141489081203, 0.15651073945978104141489081203, 0.21789538365148923974296670302, 0.43093380708457703407678366733, 0.54799028107928627436316676702, 0.59221119386588057815291559640, 0.64136548603388047988013169383, 0.820981595397107216075105430025, 0.823827823122380737496298896036, 1.11156910859636149200449404898, 1.18383846034101734490577184769, 1.21478243964803465166301796354, 1.31734600330951595961819512338, 1.41948091194728482721637279522, 1.42166601275010832952675103490, 1.54372403257566561120672951710, 1.60464812140013274917322500917, 1.64913835470040613686119683269, 1.75354352526080615316113641475, 2.01253190379537618604101417863, 2.01911930666568090348988630781, 2.04578449131867744961355689480, 2.20412504242820188849183128493, 2.27810031855356558072781439945, 2.28260199480671556929414364391, 2.28648554425463789002679714797

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

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