Properties

Degree 24
Conductor $ 2^{72} \cdot 5^{12} $
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  − 268·7-s + 1.24e3·9-s + 2.40e3·17-s − 8.10e3·23-s − 3.75e3·25-s − 7.97e3·31-s − 5.64e4·41-s − 2.11e4·47-s − 6.77e4·49-s − 3.33e5·63-s − 3.67e5·71-s + 5.87e4·73-s − 2.61e4·79-s + 8.50e5·81-s + 8.76e4·89-s − 1.72e5·97-s − 4.35e5·103-s − 4.46e5·113-s − 6.43e5·119-s + 1.00e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.98e6·153-s + ⋯
L(s)  = 1  − 2.06·7-s + 5.11·9-s + 2.01·17-s − 3.19·23-s − 6/5·25-s − 1.49·31-s − 5.24·41-s − 1.39·47-s − 4.02·49-s − 10.5·63-s − 8.65·71-s + 1.29·73-s − 0.472·79-s + 14.4·81-s + 1.17·89-s − 1.85·97-s − 4.04·103-s − 3.28·113-s − 4.16·119-s + 6.22·121-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 + 10.3·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{72} \cdot 5^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{320} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((24,\ 2^{72} \cdot 5^{12} ,\ ( \ : [5/2]^{12} ),\ 1 )\)
\(L(3)\)  \(\approx\)  \(0.0005503956811\)
\(L(\frac12)\)  \(\approx\)  \(0.0005503956811\)
\(L(\frac{7}{2})\)   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 24. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 + p^{4} T^{2} )^{6} \)
good3 \( 1 - 1244 T^{2} + 232282 p T^{4} - 230065580 T^{6} + 47479013407 T^{8} - 640563641560 p^{2} T^{10} + 7836105400132 p^{4} T^{12} - 640563641560 p^{12} T^{14} + 47479013407 p^{20} T^{16} - 230065580 p^{30} T^{18} + 232282 p^{41} T^{20} - 1244 p^{50} T^{22} + p^{60} T^{24} \)
7 \( ( 1 + 134 T + 60784 T^{2} + 9725794 T^{3} + 1872794971 T^{4} + 305623921700 T^{5} + 37504003751552 T^{6} + 305623921700 p^{5} T^{7} + 1872794971 p^{10} T^{8} + 9725794 p^{15} T^{9} + 60784 p^{20} T^{10} + 134 p^{25} T^{11} + p^{30} T^{12} )^{2} \)
11 \( 1 - 1002172 T^{2} + 493219533810 T^{4} - 14825028358228548 p T^{6} + \)\(41\!\cdots\!27\)\( T^{8} - \)\(86\!\cdots\!44\)\( T^{10} + \)\(15\!\cdots\!36\)\( T^{12} - \)\(86\!\cdots\!44\)\( p^{10} T^{14} + \)\(41\!\cdots\!27\)\( p^{20} T^{16} - 14825028358228548 p^{31} T^{18} + 493219533810 p^{40} T^{20} - 1002172 p^{50} T^{22} + p^{60} T^{24} \)
13 \( 1 - 2253844 T^{2} + 2609423635634 T^{4} - 2090949123359826660 T^{6} + \)\(12\!\cdots\!95\)\( T^{8} - \)\(64\!\cdots\!44\)\( T^{10} + \)\(26\!\cdots\!16\)\( T^{12} - \)\(64\!\cdots\!44\)\( p^{10} T^{14} + \)\(12\!\cdots\!95\)\( p^{20} T^{16} - 2090949123359826660 p^{30} T^{18} + 2609423635634 p^{40} T^{20} - 2253844 p^{50} T^{22} + p^{60} T^{24} \)
17 \( ( 1 - 1200 T + 5829762 T^{2} - 7194221040 T^{3} + 17474250252687 T^{4} - 17877633531638880 T^{5} + 31833151505919656476 T^{6} - 17877633531638880 p^{5} T^{7} + 17474250252687 p^{10} T^{8} - 7194221040 p^{15} T^{9} + 5829762 p^{20} T^{10} - 1200 p^{25} T^{11} + p^{30} T^{12} )^{2} \)
19 \( 1 - 1047044 p T^{2} + 197127587113970 T^{4} - \)\(12\!\cdots\!52\)\( T^{6} + \)\(60\!\cdots\!07\)\( T^{8} - \)\(21\!\cdots\!16\)\( T^{10} + \)\(60\!\cdots\!52\)\( T^{12} - \)\(21\!\cdots\!16\)\( p^{10} T^{14} + \)\(60\!\cdots\!07\)\( p^{20} T^{16} - \)\(12\!\cdots\!52\)\( p^{30} T^{18} + 197127587113970 p^{40} T^{20} - 1047044 p^{51} T^{22} + p^{60} T^{24} \)
23 \( ( 1 + 4054 T + 24767560 T^{2} + 85869926290 T^{3} + 327319542066779 T^{4} + 869764424865766180 T^{5} + \)\(26\!\cdots\!28\)\( T^{6} + 869764424865766180 p^{5} T^{7} + 327319542066779 p^{10} T^{8} + 85869926290 p^{15} T^{9} + 24767560 p^{20} T^{10} + 4054 p^{25} T^{11} + p^{30} T^{12} )^{2} \)
29 \( 1 - 118766236 T^{2} + 8086517092300050 T^{4} - \)\(38\!\cdots\!40\)\( T^{6} + \)\(13\!\cdots\!91\)\( T^{8} - \)\(38\!\cdots\!40\)\( T^{10} + \)\(88\!\cdots\!84\)\( T^{12} - \)\(38\!\cdots\!40\)\( p^{10} T^{14} + \)\(13\!\cdots\!91\)\( p^{20} T^{16} - \)\(38\!\cdots\!40\)\( p^{30} T^{18} + 8086517092300050 p^{40} T^{20} - 118766236 p^{50} T^{22} + p^{60} T^{24} \)
31 \( ( 1 + 3988 T + 3534574 p T^{2} + 302364450332 T^{3} + 5668865865566767 T^{4} + 12320621141215714888 T^{5} + \)\(19\!\cdots\!04\)\( T^{6} + 12320621141215714888 p^{5} T^{7} + 5668865865566767 p^{10} T^{8} + 302364450332 p^{15} T^{9} + 3534574 p^{21} T^{10} + 3988 p^{25} T^{11} + p^{30} T^{12} )^{2} \)
37 \( 1 - 298070820 T^{2} + 53469062376708978 T^{4} - \)\(69\!\cdots\!56\)\( T^{6} + \)\(71\!\cdots\!03\)\( T^{8} - \)\(61\!\cdots\!04\)\( T^{10} + \)\(45\!\cdots\!00\)\( T^{12} - \)\(61\!\cdots\!04\)\( p^{10} T^{14} + \)\(71\!\cdots\!03\)\( p^{20} T^{16} - \)\(69\!\cdots\!56\)\( p^{30} T^{18} + 53469062376708978 p^{40} T^{20} - 298070820 p^{50} T^{22} + p^{60} T^{24} \)
41 \( ( 1 + 28204 T + 645547054 T^{2} + 9891921342268 T^{3} + 144344267066611295 T^{4} + \)\(17\!\cdots\!96\)\( T^{5} + \)\(20\!\cdots\!08\)\( T^{6} + \)\(17\!\cdots\!96\)\( p^{5} T^{7} + 144344267066611295 p^{10} T^{8} + 9891921342268 p^{15} T^{9} + 645547054 p^{20} T^{10} + 28204 p^{25} T^{11} + p^{30} T^{12} )^{2} \)
43 \( 1 - 854218268 T^{2} + 379569874740918830 T^{4} - \)\(11\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!71\)\( T^{8} - \)\(50\!\cdots\!20\)\( T^{10} + \)\(79\!\cdots\!96\)\( T^{12} - \)\(50\!\cdots\!20\)\( p^{10} T^{14} + \)\(26\!\cdots\!71\)\( p^{20} T^{16} - \)\(11\!\cdots\!80\)\( p^{30} T^{18} + 379569874740918830 p^{40} T^{20} - 854218268 p^{50} T^{22} + p^{60} T^{24} \)
47 \( ( 1 + 10586 T + 559351288 T^{2} + 141995130734 T^{3} + 85696824820019915 T^{4} - \)\(20\!\cdots\!52\)\( T^{5} + \)\(41\!\cdots\!80\)\( T^{6} - \)\(20\!\cdots\!52\)\( p^{5} T^{7} + 85696824820019915 p^{10} T^{8} + 141995130734 p^{15} T^{9} + 559351288 p^{20} T^{10} + 10586 p^{25} T^{11} + p^{30} T^{12} )^{2} \)
53 \( 1 - 2472111540 T^{2} + 3272109664551065298 T^{4} - \)\(29\!\cdots\!96\)\( T^{6} + \)\(20\!\cdots\!23\)\( T^{8} - \)\(11\!\cdots\!44\)\( T^{10} + \)\(51\!\cdots\!20\)\( T^{12} - \)\(11\!\cdots\!44\)\( p^{10} T^{14} + \)\(20\!\cdots\!23\)\( p^{20} T^{16} - \)\(29\!\cdots\!96\)\( p^{30} T^{18} + 3272109664551065298 p^{40} T^{20} - 2472111540 p^{50} T^{22} + p^{60} T^{24} \)
59 \( 1 - 3285269100 T^{2} + 3759201793217414610 T^{4} - \)\(13\!\cdots\!44\)\( T^{6} + \)\(12\!\cdots\!35\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(52\!\cdots\!56\)\( p^{2} T^{12} - \)\(13\!\cdots\!20\)\( p^{10} T^{14} + \)\(12\!\cdots\!35\)\( p^{20} T^{16} - \)\(13\!\cdots\!44\)\( p^{30} T^{18} + 3759201793217414610 p^{40} T^{20} - 3285269100 p^{50} T^{22} + p^{60} T^{24} \)
61 \( 1 - 9275968036 T^{2} + 40037810273961512978 T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(19\!\cdots\!03\)\( T^{8} - \)\(25\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!52\)\( T^{12} - \)\(25\!\cdots\!00\)\( p^{10} T^{14} + \)\(19\!\cdots\!03\)\( p^{20} T^{16} - \)\(10\!\cdots\!60\)\( p^{30} T^{18} + 40037810273961512978 p^{40} T^{20} - 9275968036 p^{50} T^{22} + p^{60} T^{24} \)
67 \( 1 - 7932747836 T^{2} + 31498024738605168590 T^{4} - \)\(85\!\cdots\!12\)\( T^{6} + \)\(17\!\cdots\!55\)\( T^{8} - \)\(30\!\cdots\!40\)\( T^{10} + \)\(44\!\cdots\!12\)\( T^{12} - \)\(30\!\cdots\!40\)\( p^{10} T^{14} + \)\(17\!\cdots\!55\)\( p^{20} T^{16} - \)\(85\!\cdots\!12\)\( p^{30} T^{18} + 31498024738605168590 p^{40} T^{20} - 7932747836 p^{50} T^{22} + p^{60} T^{24} \)
71 \( ( 1 + 183852 T + 22719949218 T^{2} + 1935102444145668 T^{3} + \)\(13\!\cdots\!79\)\( T^{4} + \)\(73\!\cdots\!52\)\( T^{5} + \)\(34\!\cdots\!16\)\( T^{6} + \)\(73\!\cdots\!52\)\( p^{5} T^{7} + \)\(13\!\cdots\!79\)\( p^{10} T^{8} + 1935102444145668 p^{15} T^{9} + 22719949218 p^{20} T^{10} + 183852 p^{25} T^{11} + p^{30} T^{12} )^{2} \)
73 \( ( 1 - 29368 T + 61112786 p T^{2} - 258540669925656 T^{3} + 16168785780639245407 T^{4} - \)\(68\!\cdots\!32\)\( T^{5} + \)\(45\!\cdots\!60\)\( T^{6} - \)\(68\!\cdots\!32\)\( p^{5} T^{7} + 16168785780639245407 p^{10} T^{8} - 258540669925656 p^{15} T^{9} + 61112786 p^{21} T^{10} - 29368 p^{25} T^{11} + p^{30} T^{12} )^{2} \)
79 \( ( 1 + 13096 T + 12178064842 T^{2} - 46215677031496 T^{3} + 67636133741719889263 T^{4} - \)\(78\!\cdots\!08\)\( T^{5} + \)\(24\!\cdots\!84\)\( T^{6} - \)\(78\!\cdots\!08\)\( p^{5} T^{7} + 67636133741719889263 p^{10} T^{8} - 46215677031496 p^{15} T^{9} + 12178064842 p^{20} T^{10} + 13096 p^{25} T^{11} + p^{30} T^{12} )^{2} \)
83 \( 1 - 15441152812 T^{2} + \)\(13\!\cdots\!30\)\( T^{4} - \)\(79\!\cdots\!24\)\( T^{6} + \)\(37\!\cdots\!87\)\( T^{8} - \)\(15\!\cdots\!88\)\( T^{10} + \)\(63\!\cdots\!52\)\( T^{12} - \)\(15\!\cdots\!88\)\( p^{10} T^{14} + \)\(37\!\cdots\!87\)\( p^{20} T^{16} - \)\(79\!\cdots\!24\)\( p^{30} T^{18} + \)\(13\!\cdots\!30\)\( p^{40} T^{20} - 15441152812 p^{50} T^{22} + p^{60} T^{24} \)
89 \( ( 1 - 43836 T + 13171715490 T^{2} - 211972966345068 T^{3} + 95738414377062864255 T^{4} - \)\(28\!\cdots\!20\)\( T^{5} + \)\(69\!\cdots\!56\)\( T^{6} - \)\(28\!\cdots\!20\)\( p^{5} T^{7} + 95738414377062864255 p^{10} T^{8} - 211972966345068 p^{15} T^{9} + 13171715490 p^{20} T^{10} - 43836 p^{25} T^{11} + p^{30} T^{12} )^{2} \)
97 \( ( 1 + 86168 T + 30789268338 T^{2} + 1787720936745144 T^{3} + \)\(45\!\cdots\!83\)\( T^{4} + \)\(19\!\cdots\!32\)\( T^{5} + \)\(44\!\cdots\!92\)\( T^{6} + \)\(19\!\cdots\!32\)\( p^{5} T^{7} + \)\(45\!\cdots\!83\)\( p^{10} T^{8} + 1787720936745144 p^{15} T^{9} + 30789268338 p^{20} T^{10} + 86168 p^{25} T^{11} + p^{30} T^{12} )^{2} \)
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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

−3.18099830583243793210980006241, −3.06856239524724361466792817603, −2.77046799936795646809726219999, −2.75647751806232953961439276016, −2.54998857417611203916032150702, −2.29574393430344780455877861510, −2.29152328936853236888435886443, −2.19128296732849287556235814977, −1.88243691056100744289611501858, −1.81178548793451389068686783146, −1.76546795500751770666797402947, −1.62856870754518905643971614165, −1.62716249252595876025730287151, −1.45012471293659643468251714355, −1.40603603756712659564852137878, −1.36821672993161962648371791319, −1.35244061796468085518440507993, −1.21640516625785591145220218716, −0.918845148119916677974166588374, −0.62044490724986986135648336148, −0.48326786052838275737283059647, −0.43638603285141010746659671203, −0.26395195007646110864636698218, −0.088946133704560300383513461476, −0.00341728204513170318580187366, 0.00341728204513170318580187366, 0.088946133704560300383513461476, 0.26395195007646110864636698218, 0.43638603285141010746659671203, 0.48326786052838275737283059647, 0.62044490724986986135648336148, 0.918845148119916677974166588374, 1.21640516625785591145220218716, 1.35244061796468085518440507993, 1.36821672993161962648371791319, 1.40603603756712659564852137878, 1.45012471293659643468251714355, 1.62716249252595876025730287151, 1.62856870754518905643971614165, 1.76546795500751770666797402947, 1.81178548793451389068686783146, 1.88243691056100744289611501858, 2.19128296732849287556235814977, 2.29152328936853236888435886443, 2.29574393430344780455877861510, 2.54998857417611203916032150702, 2.75647751806232953961439276016, 2.77046799936795646809726219999, 3.06856239524724361466792817603, 3.18099830583243793210980006241

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

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