Properties

Degree 32
Conductor $ 2^{32} \cdot 3^{48} $
Sign $1$
Motivic weight 4
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 7·4-s − 352·13-s − 27·16-s − 4.18e3·25-s + 3.20e3·37-s + 1.87e4·49-s + 2.46e3·52-s − 2.75e3·61-s + 417·64-s + 8.24e3·73-s − 6.92e3·97-s + 2.92e4·100-s − 1.69e4·109-s + 9.60e4·121-s + 127-s + 131-s + 137-s + 139-s − 2.24e4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.76e5·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 0.437·4-s − 2.08·13-s − 0.105·16-s − 6.69·25-s + 2.33·37-s + 7.81·49-s + 0.911·52-s − 0.739·61-s + 0.101·64-s + 1.54·73-s − 0.736·97-s + 2.92·100-s − 1.42·109-s + 6.56·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 1.02·148-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 6.18·169-s + 3.34e−5·173-s + 3.12e−5·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{32} \cdot 3^{48}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((32,\ 2^{32} \cdot 3^{48} ,\ ( \ : [2]^{16} ),\ 1 )\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.29603\)
\(L(\frac12)\)  \(\approx\)  \(1.29603\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( 1 + 7 T^{2} + 19 p^{2} T^{4} + 19 p^{4} T^{6} + 185 p^{9} T^{8} + 19 p^{12} T^{10} + 19 p^{18} T^{12} + 7 p^{24} T^{14} + p^{32} T^{16} \)
3 \( 1 \)
good5 \( ( 1 + 2092 T^{2} + 2149522 T^{4} + 1622274208 T^{6} + 1066877108443 T^{8} + 1622274208 p^{8} T^{10} + 2149522 p^{16} T^{12} + 2092 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
7 \( ( 1 - 1340 p T^{2} + 36105250 T^{4} - 10452636224 p T^{6} + 129372615567787 T^{8} - 10452636224 p^{9} T^{10} + 36105250 p^{16} T^{12} - 1340 p^{25} T^{14} + p^{32} T^{16} )^{2} \)
11 \( ( 1 - 48044 T^{2} + 1167571642 T^{4} - 20151290866448 T^{6} + 304446182867375107 T^{8} - 20151290866448 p^{8} T^{10} + 1167571642 p^{16} T^{12} - 48044 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
13 \( ( 1 + 88 T + 63508 T^{2} + 390568 p T^{3} + 2650180198 T^{4} + 390568 p^{5} T^{5} + 63508 p^{8} T^{6} + 88 p^{12} T^{7} + p^{16} T^{8} )^{4} \)
17 \( ( 1 + 239464 T^{2} + 41438060380 T^{4} + 5059782297489112 T^{6} + \)\(47\!\cdots\!02\)\( T^{8} + 5059782297489112 p^{8} T^{10} + 41438060380 p^{16} T^{12} + 239464 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
19 \( ( 1 - 680120 T^{2} + 646972732 p^{2} T^{4} - 51394020863660552 T^{6} + \)\(79\!\cdots\!98\)\( T^{8} - 51394020863660552 p^{8} T^{10} + 646972732 p^{18} T^{12} - 680120 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
23 \( ( 1 - 675608 T^{2} + 314533671388 T^{4} - 115630955446337192 T^{6} + \)\(35\!\cdots\!18\)\( T^{8} - 115630955446337192 p^{8} T^{10} + 314533671388 p^{16} T^{12} - 675608 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
29 \( ( 1 + 2998744 T^{2} + 4863922457692 T^{4} + 5416371986440228072 T^{6} + \)\(44\!\cdots\!14\)\( T^{8} + 5416371986440228072 p^{8} T^{10} + 4863922457692 p^{16} T^{12} + 2998744 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
31 \( ( 1 - 1328756 T^{2} + 2683287954034 T^{4} - 2424947225563311392 T^{6} + \)\(29\!\cdots\!59\)\( T^{8} - 2424947225563311392 p^{8} T^{10} + 2683287954034 p^{16} T^{12} - 1328756 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
37 \( ( 1 - 800 T + 1852732 T^{2} - 3784975328 T^{3} + 1716048721222 T^{4} - 3784975328 p^{4} T^{5} + 1852732 p^{8} T^{6} - 800 p^{12} T^{7} + p^{16} T^{8} )^{4} \)
41 \( ( 1 + 14045080 T^{2} + 100482710328412 T^{4} + \)\(46\!\cdots\!84\)\( T^{6} + \)\(15\!\cdots\!54\)\( T^{8} + \)\(46\!\cdots\!84\)\( p^{8} T^{10} + 100482710328412 p^{16} T^{12} + 14045080 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
43 \( ( 1 - 11990648 T^{2} + 69115070774812 T^{4} - \)\(29\!\cdots\!24\)\( T^{6} + \)\(10\!\cdots\!18\)\( T^{8} - \)\(29\!\cdots\!24\)\( p^{8} T^{10} + 69115070774812 p^{16} T^{12} - 11990648 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
47 \( ( 1 - 27701336 T^{2} + 374201000701276 T^{4} - \)\(31\!\cdots\!48\)\( T^{6} + \)\(18\!\cdots\!14\)\( T^{8} - \)\(31\!\cdots\!48\)\( p^{8} T^{10} + 374201000701276 p^{16} T^{12} - 27701336 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
53 \( ( 1 + 35837884 T^{2} + 582188420839618 T^{4} + \)\(60\!\cdots\!16\)\( T^{6} + \)\(50\!\cdots\!79\)\( T^{8} + \)\(60\!\cdots\!16\)\( p^{8} T^{10} + 582188420839618 p^{16} T^{12} + 35837884 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
59 \( ( 1 - 70312136 T^{2} + 2419955263154716 T^{4} - \)\(51\!\cdots\!68\)\( T^{6} + \)\(75\!\cdots\!74\)\( T^{8} - \)\(51\!\cdots\!68\)\( p^{8} T^{10} + 2419955263154716 p^{16} T^{12} - 70312136 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
61 \( ( 1 + 688 T + 35342980 T^{2} + 28994874640 T^{3} + 632810106882118 T^{4} + 28994874640 p^{4} T^{5} + 35342980 p^{8} T^{6} + 688 p^{12} T^{7} + p^{16} T^{8} )^{4} \)
67 \( ( 1 - 29579000 T^{2} + 1274223737389084 T^{4} - \)\(30\!\cdots\!28\)\( T^{6} + \)\(73\!\cdots\!58\)\( T^{8} - \)\(30\!\cdots\!28\)\( p^{8} T^{10} + 1274223737389084 p^{16} T^{12} - 29579000 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
71 \( ( 1 - 86649128 T^{2} + 3567527319446236 T^{4} - \)\(11\!\cdots\!36\)\( T^{6} + \)\(32\!\cdots\!90\)\( T^{8} - \)\(11\!\cdots\!36\)\( p^{8} T^{10} + 3567527319446236 p^{16} T^{12} - 86649128 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
73 \( ( 1 - 2060 T + 55108594 T^{2} - 101224244576 T^{3} + 2192760100160827 T^{4} - 101224244576 p^{4} T^{5} + 55108594 p^{8} T^{6} - 2060 p^{12} T^{7} + p^{16} T^{8} )^{4} \)
79 \( ( 1 - 162885800 T^{2} + 12409200314583772 T^{4} - \)\(63\!\cdots\!36\)\( T^{6} + \)\(26\!\cdots\!06\)\( T^{8} - \)\(63\!\cdots\!36\)\( p^{8} T^{10} + 12409200314583772 p^{16} T^{12} - 162885800 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
83 \( ( 1 - 140155436 T^{2} + 8287556309932090 T^{4} - \)\(42\!\cdots\!92\)\( T^{6} + \)\(22\!\cdots\!03\)\( T^{8} - \)\(42\!\cdots\!92\)\( p^{8} T^{10} + 8287556309932090 p^{16} T^{12} - 140155436 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
89 \( ( 1 + 257253400 T^{2} + 38619499284608860 T^{4} + \)\(38\!\cdots\!44\)\( T^{6} + \)\(28\!\cdots\!58\)\( T^{8} + \)\(38\!\cdots\!44\)\( p^{8} T^{10} + 38619499284608860 p^{16} T^{12} + 257253400 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
97 \( ( 1 + 1732 T + 297359818 T^{2} + 428249450896 T^{3} + 37002869299122643 T^{4} + 428249450896 p^{4} T^{5} + 297359818 p^{8} T^{6} + 1732 p^{12} T^{7} + p^{16} T^{8} )^{4} \)
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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

−3.15925243047587622804431985635, −3.09638534976922843669367291848, −3.07648674690865398081628016939, −3.05017417807237682813802031279, −2.61422801154241349163862214094, −2.57480643095888072848729294451, −2.47933650845999230631295996225, −2.46363680760086784172211193320, −2.38321031305154646931017364456, −2.31732638318030288464284135470, −2.16461540014128999808426889621, −2.01806744814795316124852776432, −1.91189709319388451590974474890, −1.86256621160982315710344715938, −1.61806399770637892483946823285, −1.51219220201182143335381569914, −1.35206977739720704030771945884, −0.925895872414073322722767301993, −0.912140374598141559900755921253, −0.889730015685631032152157297536, −0.861566627392220923948893808175, −0.48189235055705601045811559503, −0.33288481037541383349117351049, −0.15863656552820087026755668749, −0.14354681859381375727616163243, 0.14354681859381375727616163243, 0.15863656552820087026755668749, 0.33288481037541383349117351049, 0.48189235055705601045811559503, 0.861566627392220923948893808175, 0.889730015685631032152157297536, 0.912140374598141559900755921253, 0.925895872414073322722767301993, 1.35206977739720704030771945884, 1.51219220201182143335381569914, 1.61806399770637892483946823285, 1.86256621160982315710344715938, 1.91189709319388451590974474890, 2.01806744814795316124852776432, 2.16461540014128999808426889621, 2.31732638318030288464284135470, 2.38321031305154646931017364456, 2.46363680760086784172211193320, 2.47933650845999230631295996225, 2.57480643095888072848729294451, 2.61422801154241349163862214094, 3.05017417807237682813802031279, 3.07648674690865398081628016939, 3.09638534976922843669367291848, 3.15925243047587622804431985635

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

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