Properties

Degree 32
Conductor $ 3^{32} \cdot 7^{16} \cdot 11^{32} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·4-s − 16·7-s + 16·16-s − 10·19-s − 18·25-s + 96·28-s + 30·31-s + 38·37-s − 16·43-s + 136·49-s + 28·61-s − 29·64-s + 52·67-s + 14·73-s + 60·76-s − 54·79-s + 108·97-s + 108·100-s + 62·103-s − 40·109-s − 256·112-s − 180·124-s + 127-s + 131-s + 160·133-s + 137-s + 139-s + ⋯
L(s)  = 1  − 3·4-s − 6.04·7-s + 4·16-s − 2.29·19-s − 3.59·25-s + 18.1·28-s + 5.38·31-s + 6.24·37-s − 2.43·43-s + 19.4·49-s + 3.58·61-s − 3.62·64-s + 6.35·67-s + 1.63·73-s + 6.88·76-s − 6.07·79-s + 10.9·97-s + 54/5·100-s + 6.10·103-s − 3.83·109-s − 24.1·112-s − 16.1·124-s + 0.0887·127-s + 0.0873·131-s + 13.8·133-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(3^{32} \cdot 7^{16} \cdot 11^{32}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(32,\ 3^{32} \cdot 7^{16} \cdot 11^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )$
$L(1)$  $\approx$  $36.51580869$
$L(\frac12)$  $\approx$  $36.51580869$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( ( 1 + T )^{16} \)
11 \( 1 \)
good2 \( 1 + 3 p T^{2} + 5 p^{2} T^{4} + 53 T^{6} + 17 p^{3} T^{8} + 321 T^{10} + 371 p T^{12} + 433 p^{2} T^{14} + 3753 T^{16} + 433 p^{4} T^{18} + 371 p^{5} T^{20} + 321 p^{6} T^{22} + 17 p^{11} T^{24} + 53 p^{10} T^{26} + 5 p^{14} T^{28} + 3 p^{15} T^{30} + p^{16} T^{32} \)
5 \( 1 + 18 T^{2} + 187 T^{4} + 288 p T^{6} + 9271 T^{8} + 10188 p T^{10} + 254033 T^{12} + 1194274 T^{14} + 5790136 T^{16} + 1194274 p^{2} T^{18} + 254033 p^{4} T^{20} + 10188 p^{7} T^{22} + 9271 p^{8} T^{24} + 288 p^{11} T^{26} + 187 p^{12} T^{28} + 18 p^{14} T^{30} + p^{16} T^{32} \)
13 \( ( 1 + 23 T^{2} - 60 T^{3} + 430 T^{4} - 1620 T^{5} + 501 p T^{6} - 25120 T^{7} + 110114 T^{8} - 25120 p T^{9} + 501 p^{3} T^{10} - 1620 p^{3} T^{11} + 430 p^{4} T^{12} - 60 p^{5} T^{13} + 23 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( 1 + 83 T^{2} + 3996 T^{4} + 141086 T^{6} + 4032975 T^{8} + 98649908 T^{10} + 125656280 p T^{12} + 41705258555 T^{14} + 742088310776 T^{16} + 41705258555 p^{2} T^{18} + 125656280 p^{5} T^{20} + 98649908 p^{6} T^{22} + 4032975 p^{8} T^{24} + 141086 p^{10} T^{26} + 3996 p^{12} T^{28} + 83 p^{14} T^{30} + p^{16} T^{32} \)
19 \( ( 1 + 5 T + 53 T^{2} + 385 T^{3} + 2238 T^{4} + 13185 T^{5} + 71811 T^{6} + 329805 T^{7} + 1585730 T^{8} + 329805 p T^{9} + 71811 p^{2} T^{10} + 13185 p^{3} T^{11} + 2238 p^{4} T^{12} + 385 p^{5} T^{13} + 53 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
23 \( 1 + 141 T^{2} + 10058 T^{4} + 508504 T^{6} + 20887458 T^{8} + 735324351 T^{10} + 22617452235 T^{12} + 614206741660 T^{14} + 14898756254016 T^{16} + 614206741660 p^{2} T^{18} + 22617452235 p^{4} T^{20} + 735324351 p^{6} T^{22} + 20887458 p^{8} T^{24} + 508504 p^{10} T^{26} + 10058 p^{12} T^{28} + 141 p^{14} T^{30} + p^{16} T^{32} \)
29 \( 1 + 170 T^{2} + 11642 T^{4} + 378370 T^{6} + 3597338 T^{8} - 148553960 T^{10} - 201212699 p T^{12} - 3225886300 p T^{14} - 1409800693980 T^{16} - 3225886300 p^{3} T^{18} - 201212699 p^{5} T^{20} - 148553960 p^{6} T^{22} + 3597338 p^{8} T^{24} + 378370 p^{10} T^{26} + 11642 p^{12} T^{28} + 170 p^{14} T^{30} + p^{16} T^{32} \)
31 \( ( 1 - 15 T + 214 T^{2} - 1830 T^{3} + 15705 T^{4} - 104570 T^{5} + 751866 T^{6} - 4460195 T^{7} + 27640604 T^{8} - 4460195 p T^{9} + 751866 p^{2} T^{10} - 104570 p^{3} T^{11} + 15705 p^{4} T^{12} - 1830 p^{5} T^{13} + 214 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
37 \( ( 1 - 19 T + 294 T^{2} - 88 p T^{3} + 31994 T^{4} - 263281 T^{5} + 1993923 T^{6} - 13472988 T^{7} + 86070388 T^{8} - 13472988 p T^{9} + 1993923 p^{2} T^{10} - 263281 p^{3} T^{11} + 31994 p^{4} T^{12} - 88 p^{6} T^{13} + 294 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
41 \( 1 + 284 T^{2} + 40621 T^{4} + 3925468 T^{6} + 292611087 T^{8} + 18270233504 T^{10} + 1002225656887 T^{12} + 49066240684968 T^{14} + 2136999595406616 T^{16} + 49066240684968 p^{2} T^{18} + 1002225656887 p^{4} T^{20} + 18270233504 p^{6} T^{22} + 292611087 p^{8} T^{24} + 3925468 p^{10} T^{26} + 40621 p^{12} T^{28} + 284 p^{14} T^{30} + p^{16} T^{32} \)
43 \( ( 1 + 8 T + 182 T^{2} + 714 T^{3} + 282 p T^{4} + 18648 T^{5} + 591955 T^{6} + 533592 T^{7} + 28125884 T^{8} + 533592 p T^{9} + 591955 p^{2} T^{10} + 18648 p^{3} T^{11} + 282 p^{5} T^{12} + 714 p^{5} T^{13} + 182 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
47 \( 1 + 189 T^{2} + 16293 T^{4} + 1005791 T^{6} + 62126478 T^{8} + 3858063829 T^{10} + 216619715755 T^{12} + 11267038690895 T^{14} + 550848868000706 T^{16} + 11267038690895 p^{2} T^{18} + 216619715755 p^{4} T^{20} + 3858063829 p^{6} T^{22} + 62126478 p^{8} T^{24} + 1005791 p^{10} T^{26} + 16293 p^{12} T^{28} + 189 p^{14} T^{30} + p^{16} T^{32} \)
53 \( 1 + 351 T^{2} + 64830 T^{4} + 8433028 T^{6} + 868191366 T^{8} + 74745980881 T^{10} + 5523680770247 T^{12} + 355312335854452 T^{14} + 20065793083018168 T^{16} + 355312335854452 p^{2} T^{18} + 5523680770247 p^{4} T^{20} + 74745980881 p^{6} T^{22} + 868191366 p^{8} T^{24} + 8433028 p^{10} T^{26} + 64830 p^{12} T^{28} + 351 p^{14} T^{30} + p^{16} T^{32} \)
59 \( 1 + 543 T^{2} + 139357 T^{4} + 22696633 T^{6} + 2682074118 T^{8} + 251509115123 T^{10} + 19990430200355 T^{12} + 1399738718709125 T^{14} + 87411313846785426 T^{16} + 1399738718709125 p^{2} T^{18} + 19990430200355 p^{4} T^{20} + 251509115123 p^{6} T^{22} + 2682074118 p^{8} T^{24} + 22696633 p^{10} T^{26} + 139357 p^{12} T^{28} + 543 p^{14} T^{30} + p^{16} T^{32} \)
61 \( ( 1 - 14 T + 369 T^{2} - 3982 T^{3} + 63690 T^{4} - 570114 T^{5} + 6849895 T^{6} - 51546370 T^{7} + 501244586 T^{8} - 51546370 p T^{9} + 6849895 p^{2} T^{10} - 570114 p^{3} T^{11} + 63690 p^{4} T^{12} - 3982 p^{5} T^{13} + 369 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 26 T + 589 T^{2} - 8044 T^{3} + 101599 T^{4} - 918934 T^{5} + 8384403 T^{6} - 59948252 T^{7} + 526057568 T^{8} - 59948252 p T^{9} + 8384403 p^{2} T^{10} - 918934 p^{3} T^{11} + 101599 p^{4} T^{12} - 8044 p^{5} T^{13} + 589 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
71 \( 1 + 834 T^{2} + 339866 T^{4} + 89812338 T^{6} + 17223424482 T^{8} + 2541454758064 T^{10} + 298486048517537 T^{12} + 28463160728293788 T^{14} + 2227198780428295156 T^{16} + 28463160728293788 p^{2} T^{18} + 298486048517537 p^{4} T^{20} + 2541454758064 p^{6} T^{22} + 17223424482 p^{8} T^{24} + 89812338 p^{10} T^{26} + 339866 p^{12} T^{28} + 834 p^{14} T^{30} + p^{16} T^{32} \)
73 \( ( 1 - 7 T + 301 T^{2} - 2487 T^{3} + 52814 T^{4} - 420273 T^{5} + 6264227 T^{6} - 45136561 T^{7} + 535074018 T^{8} - 45136561 p T^{9} + 6264227 p^{2} T^{10} - 420273 p^{3} T^{11} + 52814 p^{4} T^{12} - 2487 p^{5} T^{13} + 301 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
79 \( ( 1 + 27 T + 616 T^{2} + 10464 T^{3} + 155660 T^{4} + 1975347 T^{5} + 22887955 T^{6} + 234448440 T^{7} + 2202874636 T^{8} + 234448440 p T^{9} + 22887955 p^{2} T^{10} + 1975347 p^{3} T^{11} + 155660 p^{4} T^{12} + 10464 p^{5} T^{13} + 616 p^{6} T^{14} + 27 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
83 \( 1 + 692 T^{2} + 253584 T^{4} + 63764956 T^{6} + 12182483532 T^{8} + 1863587574932 T^{10} + 235286312480688 T^{12} + 24958715395566236 T^{14} + 2245860181831703846 T^{16} + 24958715395566236 p^{2} T^{18} + 235286312480688 p^{4} T^{20} + 1863587574932 p^{6} T^{22} + 12182483532 p^{8} T^{24} + 63764956 p^{10} T^{26} + 253584 p^{12} T^{28} + 692 p^{14} T^{30} + p^{16} T^{32} \)
89 \( 1 + 656 T^{2} + 226181 T^{4} + 53166932 T^{6} + 9489558447 T^{8} + 1367605904636 T^{10} + 166086358267207 T^{12} + 17553705254600032 T^{14} + 1650023239040136456 T^{16} + 17553705254600032 p^{2} T^{18} + 166086358267207 p^{4} T^{20} + 1367605904636 p^{6} T^{22} + 9489558447 p^{8} T^{24} + 53166932 p^{10} T^{26} + 226181 p^{12} T^{28} + 656 p^{14} T^{30} + p^{16} T^{32} \)
97 \( ( 1 - 54 T + 1829 T^{2} - 44646 T^{3} + 872014 T^{4} - 14158746 T^{5} + 196508043 T^{6} - 2365703818 T^{7} + 24885324098 T^{8} - 2365703818 p T^{9} + 196508043 p^{2} T^{10} - 14158746 p^{3} T^{11} + 872014 p^{4} T^{12} - 44646 p^{5} T^{13} + 1829 p^{6} T^{14} - 54 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
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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

−1.83788242389732781732054587857, −1.79161169236509887114302560863, −1.73063303236634560439984050281, −1.65758612199463069490165522081, −1.53633250786120851048187376996, −1.50524233469347375472435899247, −1.44874439686589303564898207505, −1.40661473689758627434928155529, −1.18907238581181833300885785711, −1.17188673130087490301791007906, −1.13110758672522219657796484932, −0.835122253052977125304300667495, −0.833526768094489858765390349809, −0.791463000678015970976906404819, −0.78297010862927210340556488729, −0.69082934597137531002280852467, −0.62310000263061230541292800944, −0.56673786017223428580767637651, −0.54534535227894161270423687677, −0.51998562932322066522382746385, −0.42557063951255206299986976594, −0.39313432557612264849238685441, −0.34146157359852475834287067124, −0.28124047761337443790898211727, −0.22108181502561449854573434761, 0.22108181502561449854573434761, 0.28124047761337443790898211727, 0.34146157359852475834287067124, 0.39313432557612264849238685441, 0.42557063951255206299986976594, 0.51998562932322066522382746385, 0.54534535227894161270423687677, 0.56673786017223428580767637651, 0.62310000263061230541292800944, 0.69082934597137531002280852467, 0.78297010862927210340556488729, 0.791463000678015970976906404819, 0.833526768094489858765390349809, 0.835122253052977125304300667495, 1.13110758672522219657796484932, 1.17188673130087490301791007906, 1.18907238581181833300885785711, 1.40661473689758627434928155529, 1.44874439686589303564898207505, 1.50524233469347375472435899247, 1.53633250786120851048187376996, 1.65758612199463069490165522081, 1.73063303236634560439984050281, 1.79161169236509887114302560863, 1.83788242389732781732054587857

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

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