Properties

Degree 16
Conductor $ 2^{48} \cdot 5^{8} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 320·7-s + 376·9-s − 2.40e3·17-s − 1.76e3·23-s − 2.50e3·25-s − 3.10e4·31-s + 2.15e4·41-s − 4.76e4·47-s + 2.53e4·49-s + 1.20e5·63-s − 2.46e5·71-s + 4.64e4·73-s − 3.25e5·79-s − 2.23e4·81-s − 7.81e4·89-s + 2.49e4·97-s − 3.92e5·103-s + 2.56e5·113-s − 7.68e5·119-s + 4.61e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 9.02e5·153-s + ⋯
L(s)  = 1  + 2.46·7-s + 1.54·9-s − 2.01·17-s − 0.693·23-s − 4/5·25-s − 5.80·31-s + 2.00·41-s − 3.14·47-s + 1.51·49-s + 3.81·63-s − 5.80·71-s + 1.01·73-s − 5.87·79-s − 0.377·81-s − 1.04·89-s + 0.269·97-s − 3.64·103-s + 1.88·113-s − 4.97·119-s + 2.86·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s − 3.11·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{48} \cdot 5^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{320} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{48} \cdot 5^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )\)
\(L(3)\)  \(\approx\)  \(2.647926980\)
\(L(\frac12)\)  \(\approx\)  \(2.647926980\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 + p^{4} T^{2} )^{4} \)
good3 \( 1 - 376 T^{2} + 163684 T^{4} - 4373416 p^{2} T^{6} + 146644294 p^{4} T^{8} - 4373416 p^{12} T^{10} + 163684 p^{20} T^{12} - 376 p^{30} T^{14} + p^{40} T^{16} \)
7 \( ( 1 - 160 T + 25708 T^{2} - 3666960 T^{3} + 699810714 T^{4} - 3666960 p^{5} T^{5} + 25708 p^{10} T^{6} - 160 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
11 \( 1 - 461928 T^{2} + 80897449276 T^{4} - 9751521734727448 T^{6} + \)\(14\!\cdots\!94\)\( T^{8} - 9751521734727448 p^{10} T^{10} + 80897449276 p^{20} T^{12} - 461928 p^{30} T^{14} + p^{40} T^{16} \)
13 \( 1 - 1173304 T^{2} + 912998161852 T^{4} - 514796045425358792 T^{6} + \)\(21\!\cdots\!70\)\( T^{8} - 514796045425358792 p^{10} T^{10} + 912998161852 p^{20} T^{12} - 1173304 p^{30} T^{14} + p^{40} T^{16} \)
17 \( ( 1 + 1200 T + 4842092 T^{2} + 4006103760 T^{3} + 9533675124390 T^{4} + 4006103760 p^{5} T^{5} + 4842092 p^{10} T^{6} + 1200 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
19 \( 1 - 243608 p T^{2} + 15129915669436 T^{4} - 43933554813794877112 T^{6} + \)\(13\!\cdots\!54\)\( T^{8} - 43933554813794877112 p^{10} T^{10} + 15129915669436 p^{20} T^{12} - 243608 p^{31} T^{14} + p^{40} T^{16} \)
23 \( ( 1 + 880 T + 8141852 T^{2} + 31216146720 T^{3} + 19333174339674 T^{4} + 31216146720 p^{5} T^{5} + 8141852 p^{10} T^{6} + 880 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
29 \( 1 - 57762792 T^{2} + 1773990452604028 T^{4} - \)\(43\!\cdots\!44\)\( T^{6} + \)\(95\!\cdots\!70\)\( T^{8} - \)\(43\!\cdots\!44\)\( p^{10} T^{10} + 1773990452604028 p^{20} T^{12} - 57762792 p^{30} T^{14} + p^{40} T^{16} \)
31 \( ( 1 + 15520 T + 133260604 T^{2} + 763025830560 T^{3} + 3983065379572806 T^{4} + 763025830560 p^{5} T^{5} + 133260604 p^{10} T^{6} + 15520 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
37 \( 1 - 301131416 T^{2} + 47822074973756092 T^{4} - \)\(50\!\cdots\!08\)\( T^{6} + \)\(40\!\cdots\!70\)\( T^{8} - \)\(50\!\cdots\!08\)\( p^{10} T^{10} + 47822074973756092 p^{20} T^{12} - 301131416 p^{30} T^{14} + p^{40} T^{16} \)
41 \( ( 1 - 10792 T + 381689636 T^{2} - 2595640517112 T^{3} + 58822716652823334 T^{4} - 2595640517112 p^{5} T^{5} + 381689636 p^{10} T^{6} - 10792 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
43 \( 1 - 542148344 T^{2} + 153567024284830372 T^{4} - \)\(32\!\cdots\!92\)\( T^{6} + \)\(53\!\cdots\!70\)\( T^{8} - \)\(32\!\cdots\!92\)\( p^{10} T^{10} + 153567024284830372 p^{20} T^{12} - 542148344 p^{30} T^{14} + p^{40} T^{16} \)
47 \( ( 1 + 23840 T + 811201148 T^{2} + 13600000745040 T^{3} + 275458842496788474 T^{4} + 13600000745040 p^{5} T^{5} + 811201148 p^{10} T^{6} + 23840 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
53 \( 1 - 984292984 T^{2} + 650040420165524092 T^{4} - \)\(31\!\cdots\!92\)\( T^{6} + \)\(12\!\cdots\!70\)\( T^{8} - \)\(31\!\cdots\!92\)\( p^{10} T^{10} + 650040420165524092 p^{20} T^{12} - 984292984 p^{30} T^{14} + p^{40} T^{16} \)
59 \( 1 - 3042039688 T^{2} + 4949915655184301308 T^{4} - \)\(54\!\cdots\!56\)\( T^{6} + \)\(44\!\cdots\!70\)\( T^{8} - \)\(54\!\cdots\!56\)\( p^{10} T^{10} + 4949915655184301308 p^{20} T^{12} - 3042039688 p^{30} T^{14} + p^{40} T^{16} \)
61 \( 1 - 2582224408 T^{2} + 4867570283316136828 T^{4} - \)\(60\!\cdots\!56\)\( T^{6} + \)\(59\!\cdots\!70\)\( T^{8} - \)\(60\!\cdots\!56\)\( p^{10} T^{10} + 4867570283316136828 p^{20} T^{12} - 2582224408 p^{30} T^{14} + p^{40} T^{16} \)
67 \( 1 - 7537851704 T^{2} + 25204334234273950564 T^{4} - \)\(51\!\cdots\!16\)\( T^{6} + \)\(78\!\cdots\!94\)\( T^{8} - \)\(51\!\cdots\!16\)\( p^{10} T^{10} + 25204334234273950564 p^{20} T^{12} - 7537851704 p^{30} T^{14} + p^{40} T^{16} \)
71 \( ( 1 + 123360 T + 10006413404 T^{2} + 7455069384480 p T^{3} + 24936343678058479206 T^{4} + 7455069384480 p^{6} T^{5} + 10006413404 p^{10} T^{6} + 123360 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
73 \( ( 1 - 23200 T + 7007019628 T^{2} - 115342151623520 T^{3} + 20469715216228479110 T^{4} - 115342151623520 p^{5} T^{5} + 7007019628 p^{10} T^{6} - 23200 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
79 \( ( 1 + 162880 T + 14181869596 T^{2} + 858599705367360 T^{3} + 48361267115636995206 T^{4} + 858599705367360 p^{5} T^{5} + 14181869596 p^{10} T^{6} + 162880 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
83 \( 1 - 29848826136 T^{2} + \)\(39\!\cdots\!24\)\( T^{4} - \)\(30\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!54\)\( T^{8} - \)\(30\!\cdots\!64\)\( p^{10} T^{10} + \)\(39\!\cdots\!24\)\( p^{20} T^{12} - 29848826136 p^{30} T^{14} + p^{40} T^{16} \)
89 \( ( 1 + 39096 T + 18076226204 T^{2} + 489497603288904 T^{3} + \)\(13\!\cdots\!94\)\( T^{4} + 489497603288904 p^{5} T^{5} + 18076226204 p^{10} T^{6} + 39096 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
97 \( ( 1 - 12480 T + 14325999628 T^{2} - 824172284150080 T^{3} + 95392462085617046694 T^{4} - 824172284150080 p^{5} T^{5} + 14325999628 p^{10} T^{6} - 12480 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
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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

−4.23760458531656687663261283452, −4.23230973143594276699849673685, −4.23195767457465943759198793638, −4.21849711060272298870995726461, −3.73420903188091062458525203115, −3.41595835980484164835800013252, −3.35053245874803016550331255476, −3.12511708912254951296189678641, −3.00438125201692724597275028982, −2.88629840241886848215338095833, −2.81432699574371159402605802485, −2.48773562678238364308668426915, −2.04381161048398273226901902166, −1.90376796228761126259659413166, −1.85821344456188905099223676832, −1.79984889486962031638245859047, −1.72860699056619363049462756238, −1.70949622896718845685421297548, −1.31610092647774825131966385008, −1.31116634097681017148781427033, −1.06991871032352085552239382970, −0.61345512477212894681872597705, −0.31597524006430416689807754983, −0.30206689174511111334824196239, −0.14598088814944597392378694741, 0.14598088814944597392378694741, 0.30206689174511111334824196239, 0.31597524006430416689807754983, 0.61345512477212894681872597705, 1.06991871032352085552239382970, 1.31116634097681017148781427033, 1.31610092647774825131966385008, 1.70949622896718845685421297548, 1.72860699056619363049462756238, 1.79984889486962031638245859047, 1.85821344456188905099223676832, 1.90376796228761126259659413166, 2.04381161048398273226901902166, 2.48773562678238364308668426915, 2.81432699574371159402605802485, 2.88629840241886848215338095833, 3.00438125201692724597275028982, 3.12511708912254951296189678641, 3.35053245874803016550331255476, 3.41595835980484164835800013252, 3.73420903188091062458525203115, 4.21849711060272298870995726461, 4.23195767457465943759198793638, 4.23230973143594276699849673685, 4.23760458531656687663261283452

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

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