Properties

Degree 24
Conductor $ 3^{12} \cdot 5^{12} \cdot 7^{12} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 6·4-s − 8·7-s + 4·8-s − 18·9-s − 16·11-s + 32·14-s − 51·16-s + 72·18-s + 64·22-s − 64·23-s − 30·25-s − 48·28-s + 104·29-s + 80·32-s − 108·36-s + 32·37-s + 152·43-s − 96·44-s + 256·46-s + 62·49-s + 120·50-s + 176·53-s − 32·56-s − 416·58-s + 144·63-s + 8·64-s + ⋯
L(s)  = 1  − 2·2-s + 3/2·4-s − 8/7·7-s + 1/2·8-s − 2·9-s − 1.45·11-s + 16/7·14-s − 3.18·16-s + 4·18-s + 2.90·22-s − 2.78·23-s − 6/5·25-s − 1.71·28-s + 3.58·29-s + 5/2·32-s − 3·36-s + 0.864·37-s + 3.53·43-s − 2.18·44-s + 5.56·46-s + 1.26·49-s + 12/5·50-s + 3.32·53-s − 4/7·56-s − 7.17·58-s + 16/7·63-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(3^{12} \cdot 5^{12} \cdot 7^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((24,\ 3^{12} \cdot 5^{12} \cdot 7^{12} ,\ ( \ : [1]^{12} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.395514\)
\(L(\frac12)\)  \(\approx\)  \(0.395514\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 24. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad3 \( ( 1 + p T^{2} )^{6} \)
5 \( ( 1 + p T^{2} )^{6} \)
7 \( 1 + 8 T + 2 T^{2} + 312 T^{3} + 4255 T^{4} + 1984 p T^{5} + 892 p^{2} T^{6} + 1984 p^{3} T^{7} + 4255 p^{4} T^{8} + 312 p^{6} T^{9} + 2 p^{8} T^{10} + 8 p^{10} T^{11} + p^{12} T^{12} \)
good2 \( ( 1 + p T + 3 T^{2} + 7 T^{4} + 19 p T^{5} + 109 T^{6} + 19 p^{3} T^{7} + 7 p^{4} T^{8} + 3 p^{8} T^{10} + p^{11} T^{11} + p^{12} T^{12} )^{2} \)
11 \( ( 1 + 8 T + 18 p T^{2} + 1416 T^{3} + 30895 T^{4} + 264944 T^{5} + 5610292 T^{6} + 264944 p^{2} T^{7} + 30895 p^{4} T^{8} + 1416 p^{6} T^{9} + 18 p^{9} T^{10} + 8 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
13 \( 1 - 852 T^{2} + 436674 T^{4} - 159409348 T^{6} + 45042421839 T^{8} - 10253858128680 T^{10} + 1907667001388316 T^{12} - 10253858128680 p^{4} T^{14} + 45042421839 p^{8} T^{16} - 159409348 p^{12} T^{18} + 436674 p^{16} T^{20} - 852 p^{20} T^{22} + p^{24} T^{24} \)
17 \( 1 - 2172 T^{2} + 141282 p T^{4} - 1757595148 T^{6} + 55392204927 p T^{8} - 387997181982840 T^{10} + 125896935467491356 T^{12} - 387997181982840 p^{4} T^{14} + 55392204927 p^{9} T^{16} - 1757595148 p^{12} T^{18} + 141282 p^{17} T^{20} - 2172 p^{20} T^{22} + p^{24} T^{24} \)
19 \( 1 - 1404 T^{2} + 1342050 T^{4} - 896154316 T^{6} + 72919437 p^{3} T^{8} - 11983720660392 p T^{10} + 90028611036772572 T^{12} - 11983720660392 p^{5} T^{14} + 72919437 p^{11} T^{16} - 896154316 p^{12} T^{18} + 1342050 p^{16} T^{20} - 1404 p^{20} T^{22} + p^{24} T^{24} \)
23 \( ( 1 + 32 T + 2418 T^{2} + 60144 T^{3} + 2694751 T^{4} + 53735792 T^{5} + 1782680284 T^{6} + 53735792 p^{2} T^{7} + 2694751 p^{4} T^{8} + 60144 p^{6} T^{9} + 2418 p^{8} T^{10} + 32 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
29 \( ( 1 - 52 T + 3990 T^{2} - 132324 T^{3} + 6311983 T^{4} - 164304136 T^{5} + 6334525012 T^{6} - 164304136 p^{2} T^{7} + 6311983 p^{4} T^{8} - 132324 p^{6} T^{9} + 3990 p^{8} T^{10} - 52 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
31 \( 1 - 6228 T^{2} + 19622274 T^{4} - 42341564932 T^{6} + 69647410996719 T^{8} - 91080823295899560 T^{10} + 96735996796679061276 T^{12} - 91080823295899560 p^{4} T^{14} + 69647410996719 p^{8} T^{16} - 42341564932 p^{12} T^{18} + 19622274 p^{16} T^{20} - 6228 p^{20} T^{22} + p^{24} T^{24} \)
37 \( ( 1 - 16 T + 4622 T^{2} + 32784 T^{3} + 7136719 T^{4} + 278279680 T^{5} + 7429424740 T^{6} + 278279680 p^{2} T^{7} + 7136719 p^{4} T^{8} + 32784 p^{6} T^{9} + 4622 p^{8} T^{10} - 16 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
41 \( 1 - 7524 T^{2} + 32464386 T^{4} - 93240551380 T^{6} + 206850565610415 T^{8} - 379921653079011144 T^{10} + \)\(65\!\cdots\!56\)\( T^{12} - 379921653079011144 p^{4} T^{14} + 206850565610415 p^{8} T^{16} - 93240551380 p^{12} T^{18} + 32464386 p^{16} T^{20} - 7524 p^{20} T^{22} + p^{24} T^{24} \)
43 \( ( 1 - 76 T + 9590 T^{2} - 631788 T^{3} + 42146383 T^{4} - 2188827368 T^{5} + 102971644948 T^{6} - 2188827368 p^{2} T^{7} + 42146383 p^{4} T^{8} - 631788 p^{6} T^{9} + 9590 p^{8} T^{10} - 76 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
47 \( 1 - 6132 T^{2} + 27456354 T^{4} - 92403901348 T^{6} + 277727516070639 T^{8} - 754801362269230440 T^{10} + \)\(17\!\cdots\!96\)\( T^{12} - 754801362269230440 p^{4} T^{14} + 277727516070639 p^{8} T^{16} - 92403901348 p^{12} T^{18} + 27456354 p^{16} T^{20} - 6132 p^{20} T^{22} + p^{24} T^{24} \)
53 \( ( 1 - 88 T + 10434 T^{2} - 574440 T^{3} + 50272015 T^{4} - 2575931008 T^{5} + 181692199804 T^{6} - 2575931008 p^{2} T^{7} + 50272015 p^{4} T^{8} - 574440 p^{6} T^{9} + 10434 p^{8} T^{10} - 88 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
59 \( 1 - 15948 T^{2} + 137050242 T^{4} - 864152710108 T^{6} + 4419517662043791 T^{8} - 18954945757384385304 T^{10} + \)\(70\!\cdots\!16\)\( T^{12} - 18954945757384385304 p^{4} T^{14} + 4419517662043791 p^{8} T^{16} - 864152710108 p^{12} T^{18} + 137050242 p^{16} T^{20} - 15948 p^{20} T^{22} + p^{24} T^{24} \)
61 \( 1 - 13716 T^{2} + 109224834 T^{4} - 648593370628 T^{6} + 3203041286245839 T^{8} - 13994290873734287016 T^{10} + \)\(55\!\cdots\!56\)\( T^{12} - 13994290873734287016 p^{4} T^{14} + 3203041286245839 p^{8} T^{16} - 648593370628 p^{12} T^{18} + 109224834 p^{16} T^{20} - 13716 p^{20} T^{22} + p^{24} T^{24} \)
67 \( ( 1 - 84 T + 15366 T^{2} - 763540 T^{3} + 102124911 T^{4} - 4033313976 T^{5} + 514098700788 T^{6} - 4033313976 p^{2} T^{7} + 102124911 p^{4} T^{8} - 763540 p^{6} T^{9} + 15366 p^{8} T^{10} - 84 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
71 \( ( 1 - 16 T + 16698 T^{2} - 42336 T^{3} + 131112895 T^{4} + 1041328304 T^{5} + 716701578892 T^{6} + 1041328304 p^{2} T^{7} + 131112895 p^{4} T^{8} - 42336 p^{6} T^{9} + 16698 p^{8} T^{10} - 16 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
73 \( 1 - 19284 T^{2} + 2959410 p T^{4} - 1973102936452 T^{6} + 14880325491672495 T^{8} - 96746229105564519336 T^{10} + \)\(55\!\cdots\!08\)\( T^{12} - 96746229105564519336 p^{4} T^{14} + 14880325491672495 p^{8} T^{16} - 1973102936452 p^{12} T^{18} + 2959410 p^{17} T^{20} - 19284 p^{20} T^{22} + p^{24} T^{24} \)
79 \( ( 1 - 60 T + 22242 T^{2} - 662476 T^{3} + 201003183 T^{4} - 1192106616 T^{5} + 1261845815388 T^{6} - 1192106616 p^{2} T^{7} + 201003183 p^{4} T^{8} - 662476 p^{6} T^{9} + 22242 p^{8} T^{10} - 60 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
83 \( 1 - 45420 T^{2} + 1066844514 T^{4} - 17098817034364 T^{6} + 206563644037003983 T^{8} - \)\(19\!\cdots\!08\)\( T^{10} + \)\(15\!\cdots\!04\)\( T^{12} - \)\(19\!\cdots\!08\)\( p^{4} T^{14} + 206563644037003983 p^{8} T^{16} - 17098817034364 p^{12} T^{18} + 1066844514 p^{16} T^{20} - 45420 p^{20} T^{22} + p^{24} T^{24} \)
89 \( 1 - 46020 T^{2} + 1126013634 T^{4} - 19221077499316 T^{6} + 252845241317038383 T^{8} - \)\(26\!\cdots\!72\)\( T^{10} + \)\(23\!\cdots\!84\)\( T^{12} - \)\(26\!\cdots\!72\)\( p^{4} T^{14} + 252845241317038383 p^{8} T^{16} - 19221077499316 p^{12} T^{18} + 1126013634 p^{16} T^{20} - 46020 p^{20} T^{22} + p^{24} T^{24} \)
97 \( 1 - 51924 T^{2} + 1396974978 T^{4} - 26352399780484 T^{6} + 389015604150559215 T^{8} - \)\(47\!\cdots\!84\)\( T^{10} + \)\(48\!\cdots\!52\)\( T^{12} - \)\(47\!\cdots\!84\)\( p^{4} T^{14} + 389015604150559215 p^{8} T^{16} - 26352399780484 p^{12} T^{18} + 1396974978 p^{16} T^{20} - 51924 p^{20} T^{22} + p^{24} T^{24} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.88751123307868311969223732529, −4.52330250120004627104984218749, −4.50033305824677525352445900492, −4.24248050260734257194883325928, −3.99158677212916992690628271758, −3.98858830840453437637118199938, −3.97083947266056149043176909940, −3.81449134162338526634134408736, −3.78747182477589951776794821808, −3.71819608584162075286781847470, −3.16369221078375360571947733378, −2.98845106546030290889479681782, −2.89988737057636046266227934663, −2.79236252205494713049229162049, −2.54524952247243406350548104096, −2.54400904959764824688823624802, −2.50629075228292060907591811509, −2.11229438753656113950831550964, −2.06014887344532381242191703502, −1.76217013701862250769654404008, −1.57722150818967912378539568425, −0.914502784216565748829261743206, −0.74302812743333406581924166557, −0.51063846384969029906319308001, −0.30646226579973574785856289536, 0.30646226579973574785856289536, 0.51063846384969029906319308001, 0.74302812743333406581924166557, 0.914502784216565748829261743206, 1.57722150818967912378539568425, 1.76217013701862250769654404008, 2.06014887344532381242191703502, 2.11229438753656113950831550964, 2.50629075228292060907591811509, 2.54400904959764824688823624802, 2.54524952247243406350548104096, 2.79236252205494713049229162049, 2.89988737057636046266227934663, 2.98845106546030290889479681782, 3.16369221078375360571947733378, 3.71819608584162075286781847470, 3.78747182477589951776794821808, 3.81449134162338526634134408736, 3.97083947266056149043176909940, 3.98858830840453437637118199938, 3.99158677212916992690628271758, 4.24248050260734257194883325928, 4.50033305824677525352445900492, 4.52330250120004627104984218749, 4.88751123307868311969223732529

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

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