Properties

Degree $32$
Conductor $3.418\times 10^{34}$
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  − 4·16-s + 16·19-s − 32·43-s − 48·49-s − 32·61-s − 8·64-s − 16·67-s + 96·103-s − 32·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 16-s + 3.67·19-s − 4.87·43-s − 6.85·49-s − 4.09·61-s − 64-s − 1.95·67-s + 9.45·103-s − 3.06·109-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 + 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 + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{64} \cdot 3^{32}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{144} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{64} \cdot 3^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.778923\)
\(L(\frac12)\) \(\approx\) \(0.778923\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{2} T^{4} + p^{3} T^{6} + p^{2} T^{8} + p^{5} T^{10} + p^{6} T^{12} + p^{8} T^{16} \)
3 \( 1 \)
good5 \( 1 - 8 T^{4} - 804 T^{8} + 2376 T^{12} + 502406 T^{16} + 2376 p^{4} T^{20} - 804 p^{8} T^{24} - 8 p^{12} T^{28} + p^{16} T^{32} \)
7 \( ( 1 + 12 T^{2} + 16 T^{3} + 74 T^{4} + 16 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
11 \( 1 - 184 T^{4} + 8796 T^{8} + 2616696 T^{12} - 490804602 T^{16} + 2616696 p^{4} T^{20} + 8796 p^{8} T^{24} - 184 p^{12} T^{28} + p^{16} T^{32} \)
13 \( ( 1 + 64 T^{3} - 36 T^{4} - 704 T^{5} + 2048 T^{6} + 384 T^{7} - 43930 T^{8} + 384 p T^{9} + 2048 p^{2} T^{10} - 704 p^{3} T^{11} - 36 p^{4} T^{12} + 64 p^{5} T^{13} + p^{8} T^{16} )^{2} \)
17 \( ( 1 - 64 T^{2} + 1956 T^{4} - 40128 T^{6} + 697542 T^{8} - 40128 p^{2} T^{10} + 1956 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 - 8 T + 32 T^{2} - 184 T^{3} + 388 T^{4} + 1992 T^{5} - 11424 T^{6} + 82616 T^{7} - 538074 T^{8} + 82616 p T^{9} - 11424 p^{2} T^{10} + 1992 p^{3} T^{11} + 388 p^{4} T^{12} - 184 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
23 \( ( 1 - 88 T^{2} + 3004 T^{4} - 46440 T^{6} + 546118 T^{8} - 46440 p^{2} T^{10} + 3004 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( 1 - 1672 T^{4} + 1133916 T^{8} - 595679544 T^{12} + 410245939974 T^{16} - 595679544 p^{4} T^{20} + 1133916 p^{8} T^{24} - 1672 p^{12} T^{28} + p^{16} T^{32} \)
31 \( ( 1 - 56 T^{2} + 3492 T^{4} - 137064 T^{6} + 4924550 T^{8} - 137064 p^{2} T^{10} + 3492 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 - 512 T^{3} + 700 T^{4} + 9728 T^{5} + 131072 T^{6} - 437248 T^{7} - 3354522 T^{8} - 437248 p T^{9} + 131072 p^{2} T^{10} + 9728 p^{3} T^{11} + 700 p^{4} T^{12} - 512 p^{5} T^{13} + p^{8} T^{16} )^{2} \)
41 \( ( 1 + 160 T^{2} + 12772 T^{4} + 745952 T^{6} + 34631942 T^{8} + 745952 p^{2} T^{10} + 12772 p^{4} T^{12} + 160 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 + 16 T + 128 T^{2} + 1200 T^{3} + 6980 T^{4} - 80 T^{5} - 174720 T^{6} - 2568048 T^{7} - 25827098 T^{8} - 2568048 p T^{9} - 174720 p^{2} T^{10} - 80 p^{3} T^{11} + 6980 p^{4} T^{12} + 1200 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
47 \( ( 1 + 216 T^{2} + 24572 T^{4} + 1856872 T^{6} + 101744902 T^{8} + 1856872 p^{2} T^{10} + 24572 p^{4} T^{12} + 216 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( 1 + 2936 T^{4} + 12654300 T^{8} + 22427423688 T^{12} + 91065624180102 T^{16} + 22427423688 p^{4} T^{20} + 12654300 p^{8} T^{24} + 2936 p^{12} T^{28} + p^{16} T^{32} \)
59 \( 1 - 40 p T^{4} + 15662172 T^{8} - 90024948744 T^{12} + 175698504275846 T^{16} - 90024948744 p^{4} T^{20} + 15662172 p^{8} T^{24} - 40 p^{13} T^{28} + p^{16} T^{32} \)
61 \( ( 1 + 16 T + 128 T^{2} + 1392 T^{3} + 16892 T^{4} + 124816 T^{5} + 803712 T^{6} + 7357296 T^{7} + 67536550 T^{8} + 7357296 p T^{9} + 803712 p^{2} T^{10} + 124816 p^{3} T^{11} + 16892 p^{4} T^{12} + 1392 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
67 \( ( 1 + 8 T + 32 T^{2} + 1240 T^{3} + 5540 T^{4} - 65000 T^{5} + 71520 T^{6} - 2339576 T^{7} - 86245658 T^{8} - 2339576 p T^{9} + 71520 p^{2} T^{10} - 65000 p^{3} T^{11} + 5540 p^{4} T^{12} + 1240 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
71 \( ( 1 - 376 T^{2} + 67068 T^{4} - 7636296 T^{6} + 626574150 T^{8} - 7636296 p^{2} T^{10} + 67068 p^{4} T^{12} - 376 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 - 408 T^{2} + 80156 T^{4} - 9970856 T^{6} + 862104454 T^{8} - 9970856 p^{2} T^{10} + 80156 p^{4} T^{12} - 408 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 408 T^{2} + 84004 T^{4} - 11101576 T^{6} + 1033756678 T^{8} - 11101576 p^{2} T^{10} + 84004 p^{4} T^{12} - 408 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( 1 - 5432 T^{4} + 103273180 T^{8} - 984945914120 T^{12} + 5200099763862790 T^{16} - 984945914120 p^{4} T^{20} + 103273180 p^{8} T^{24} - 5432 p^{12} T^{28} + p^{16} T^{32} \)
89 \( ( 1 + 512 T^{2} + 123588 T^{4} + 18716160 T^{6} + 1970104134 T^{8} + 18716160 p^{2} T^{10} + 123588 p^{4} T^{12} + 512 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 + 316 T^{2} - 256 T^{3} + 42310 T^{4} - 256 p T^{5} + 316 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.82445704783862574617619330052, −3.79017462950688121524413506771, −3.61173170693009651097098897032, −3.48396164761634913952741917454, −3.44140390952219715933913615276, −3.42577724117731872585422721826, −3.41999606282690611058903004178, −3.36733806326437601363928003519, −3.13988077070056694786208827310, −3.12570831033476695127540879071, −2.91351473777109356366258258133, −2.78944170829984523088386842377, −2.75380374741549645899048178440, −2.58912882997263270749218077864, −2.52957698623826832091819826160, −2.51819270110812640098923984345, −1.85786522019020539428296435516, −1.85147493123057283421607396291, −1.80752032818315695640285654960, −1.73326008034501168371744012335, −1.60107677076249520353303756797, −1.50493045485174487212830001669, −1.36942597756664329621759116926, −0.947942240431247221091139007976, −0.34622736850952760303374964206, 0.34622736850952760303374964206, 0.947942240431247221091139007976, 1.36942597756664329621759116926, 1.50493045485174487212830001669, 1.60107677076249520353303756797, 1.73326008034501168371744012335, 1.80752032818315695640285654960, 1.85147493123057283421607396291, 1.85786522019020539428296435516, 2.51819270110812640098923984345, 2.52957698623826832091819826160, 2.58912882997263270749218077864, 2.75380374741549645899048178440, 2.78944170829984523088386842377, 2.91351473777109356366258258133, 3.12570831033476695127540879071, 3.13988077070056694786208827310, 3.36733806326437601363928003519, 3.41999606282690611058903004178, 3.42577724117731872585422721826, 3.44140390952219715933913615276, 3.48396164761634913952741917454, 3.61173170693009651097098897032, 3.79017462950688121524413506771, 3.82445704783862574617619330052

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

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