Properties

Label 32-72e32-1.1-c1e16-0-6
Degree $32$
Conductor $2.720\times 10^{59}$
Sign $1$
Analytic cond. $7.43145\times 10^{25}$
Root an. cond. $6.43385$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 24·19-s − 32·25-s + 24·43-s + 32·49-s + 120·67-s + 16·73-s − 16·97-s + 104·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 112·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 5.50·19-s − 6.39·25-s + 3.65·43-s + 32/7·49-s + 14.6·67-s + 1.87·73-s − 1.62·97-s + 9.45·121-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 + 8.61·169-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{64}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{96} \cdot 3^{64}\)
Sign: $1$
Analytic conductor: \(7.43145\times 10^{25}\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5184} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{96} \cdot 3^{64} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1099.188208\)
\(L(\frac12)\) \(\approx\) \(1099.188208\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 + 16 T^{2} + 118 T^{4} + 544 T^{6} + 2299 T^{8} + 544 p^{2} T^{10} + 118 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
7 \( ( 1 - 16 T^{2} + 232 T^{4} - 2272 T^{6} + 17950 T^{8} - 2272 p^{2} T^{10} + 232 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( ( 1 - 52 T^{2} + 1312 T^{4} - 21676 T^{6} + 268654 T^{8} - 21676 p^{2} T^{10} + 1312 p^{4} T^{12} - 52 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 - 56 T^{2} + 10 p^{2} T^{4} - 34832 T^{6} + 523435 T^{8} - 34832 p^{2} T^{10} + 10 p^{6} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( ( 1 - 32 T^{2} + 807 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
19 \( ( 1 - 6 T + 70 T^{2} - 306 T^{3} + 1974 T^{4} - 306 p T^{5} + 70 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
23 \( ( 1 + 76 T^{2} + 2608 T^{4} + 61108 T^{6} + 1353070 T^{8} + 61108 p^{2} T^{10} + 2608 p^{4} T^{12} + 76 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 + 88 T^{2} + 2974 T^{4} + 26368 T^{6} - 471821 T^{8} + 26368 p^{2} T^{10} + 2974 p^{4} T^{12} + 88 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 112 T^{2} + 7804 T^{4} - 378256 T^{6} + 13414150 T^{8} - 378256 p^{2} T^{10} + 7804 p^{4} T^{12} - 112 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 - 176 T^{2} + 15202 T^{4} - 876512 T^{6} + 37334035 T^{8} - 876512 p^{2} T^{10} + 15202 p^{4} T^{12} - 176 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
41 \( ( 1 - 184 T^{2} + 15316 T^{4} - 815272 T^{6} + 35082982 T^{8} - 815272 p^{2} T^{10} + 15316 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 - 6 T + 82 T^{2} - 18 T^{3} + 1950 T^{4} - 18 p T^{5} + 82 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
47 \( ( 1 + 232 T^{2} + 26860 T^{4} + 2036248 T^{6} + 111340966 T^{8} + 2036248 p^{2} T^{10} + 26860 p^{4} T^{12} + 232 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 + 232 T^{2} + 29236 T^{4} + 2477560 T^{6} + 152663494 T^{8} + 2477560 p^{2} T^{10} + 29236 p^{4} T^{12} + 232 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( ( 1 - 184 T^{2} + 11500 T^{4} + 13304 T^{6} - 30101498 T^{8} + 13304 p^{2} T^{10} + 11500 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 - 296 T^{2} + 43690 T^{4} - 4325744 T^{6} + 309323707 T^{8} - 4325744 p^{2} T^{10} + 43690 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 30 T + 550 T^{2} - 6858 T^{3} + 65190 T^{4} - 6858 p T^{5} + 550 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
71 \( ( 1 + 388 T^{2} + 67168 T^{4} + 7187452 T^{6} + 570357646 T^{8} + 7187452 p^{2} T^{10} + 67168 p^{4} T^{12} + 388 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 - 4 T + 202 T^{2} - 400 T^{3} + 18487 T^{4} - 400 p T^{5} + 202 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
79 \( ( 1 - 280 T^{2} + 40216 T^{4} - 4462648 T^{6} + 402660286 T^{8} - 4462648 p^{2} T^{10} + 40216 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( ( 1 - 184 T^{2} + 21724 T^{4} - 2413768 T^{6} + 243251686 T^{8} - 2413768 p^{2} T^{10} + 21724 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 - 280 T^{2} + 49366 T^{4} - 6284032 T^{6} + 626620651 T^{8} - 6284032 p^{2} T^{10} + 49366 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 + 4 T + 280 T^{2} + 1516 T^{3} + 35326 T^{4} + 1516 p T^{5} + 280 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
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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

−1.97554219773220097136611558736, −1.96933485590991223777156987876, −1.85291591291755721933988230716, −1.78297127471191404526243665226, −1.75763749613619184942337450237, −1.71897256245016825940984367244, −1.70281048922341112274259226157, −1.69655962746198192582876284979, −1.40528597619435326292783077572, −1.20648744714710260857257497019, −1.20404778432676945936821468410, −1.08758120490554610050782233685, −1.00553005088229000165310067622, −0.897695623420779653313694258016, −0.880363202891032442017401122184, −0.834227260030314766597297739188, −0.800987921106885069141632984949, −0.67097078866003187244621622611, −0.62891944940105190531701394501, −0.61856657291801898081342233182, −0.55620694563555298156621512930, −0.52059823497113147149600092691, −0.42777993559747970056986657984, −0.34174167272872266964324437966, −0.29648270441128732308810374175, 0.29648270441128732308810374175, 0.34174167272872266964324437966, 0.42777993559747970056986657984, 0.52059823497113147149600092691, 0.55620694563555298156621512930, 0.61856657291801898081342233182, 0.62891944940105190531701394501, 0.67097078866003187244621622611, 0.800987921106885069141632984949, 0.834227260030314766597297739188, 0.880363202891032442017401122184, 0.897695623420779653313694258016, 1.00553005088229000165310067622, 1.08758120490554610050782233685, 1.20404778432676945936821468410, 1.20648744714710260857257497019, 1.40528597619435326292783077572, 1.69655962746198192582876284979, 1.70281048922341112274259226157, 1.71897256245016825940984367244, 1.75763749613619184942337450237, 1.78297127471191404526243665226, 1.85291591291755721933988230716, 1.96933485590991223777156987876, 1.97554219773220097136611558736

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

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