Properties

Degree 32
Conductor $ 2^{48} \cdot 3^{16} \cdot 7^{16} $
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  + 4·7-s + 9-s − 6·19-s + 11·25-s − 48·31-s − 2·37-s + 20·43-s − 6·49-s + 36·61-s + 4·63-s + 14·67-s + 30·73-s + 28·79-s + 8·81-s + 6·103-s − 46·109-s − 49·121-s + 127-s + 131-s − 24·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1.51·7-s + 1/3·9-s − 1.37·19-s + 11/5·25-s − 8.62·31-s − 0.328·37-s + 3.04·43-s − 6/7·49-s + 4.60·61-s + 0.503·63-s + 1.71·67-s + 3.51·73-s + 3.15·79-s + 8/9·81-s + 0.591·103-s − 4.40·109-s − 4.45·121-s + 0.0887·127-s + 0.0873·131-s − 2.08·133-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{48} \cdot 3^{16} \cdot 7^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{168} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(32,\ 2^{48} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )$
$L(1)$  $\approx$  $2.29805$
$L(\frac12)$  $\approx$  $2.29805$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T^{2} - 7 T^{4} - 8 p T^{5} + 22 T^{6} + 16 p T^{7} + 10 T^{8} + 16 p^{2} T^{9} + 22 p^{2} T^{10} - 8 p^{4} T^{11} - 7 p^{4} T^{12} - p^{6} T^{14} + p^{8} T^{16} \)
7 \( ( 1 - 2 T + 9 T^{2} - 10 T^{3} + 44 T^{4} - 10 p T^{5} + 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
good5 \( ( 1 - 13 T^{2} + 93 T^{4} - 434 T^{6} + 1886 T^{8} - 434 p^{2} T^{10} + 93 p^{4} T^{12} - 13 p^{6} T^{14} + p^{8} T^{16} )( 1 + 2 T^{2} - 39 T^{4} - 38 T^{6} + 836 T^{8} - 38 p^{2} T^{10} - 39 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} ) \)
11 \( 1 + 49 T^{2} + 1300 T^{4} + 22265 T^{6} + 252641 T^{8} + 126848 p T^{10} - 12534274 T^{12} - 440430082 T^{14} - 6222779240 T^{16} - 440430082 p^{2} T^{18} - 12534274 p^{4} T^{20} + 126848 p^{7} T^{22} + 252641 p^{8} T^{24} + 22265 p^{10} T^{26} + 1300 p^{12} T^{28} + 49 p^{14} T^{30} + p^{16} T^{32} \)
13 \( ( 1 - 49 T^{2} + 1278 T^{4} - 23495 T^{6} + 341186 T^{8} - 23495 p^{2} T^{10} + 1278 p^{4} T^{12} - 49 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( 1 - 42 T^{2} + 1059 T^{4} - 6894 T^{6} - 204407 T^{8} + 7559412 T^{10} - 49293810 T^{12} - 1407253056 T^{14} + 49740922386 T^{16} - 1407253056 p^{2} T^{18} - 49293810 p^{4} T^{20} + 7559412 p^{6} T^{22} - 204407 p^{8} T^{24} - 6894 p^{10} T^{26} + 1059 p^{12} T^{28} - 42 p^{14} T^{30} + p^{16} T^{32} \)
19 \( ( 1 + 3 T + 62 T^{2} + 177 T^{3} + 2049 T^{4} + 6000 T^{5} + 55390 T^{6} + 152826 T^{7} + 1217108 T^{8} + 152826 p T^{9} + 55390 p^{2} T^{10} + 6000 p^{3} T^{11} + 2049 p^{4} T^{12} + 177 p^{5} T^{13} + 62 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
23 \( 1 + 102 T^{2} + 6323 T^{4} + 248898 T^{6} + 6664873 T^{8} + 89132724 T^{10} - 1162854034 T^{12} - 108042764448 T^{14} - 3248548802990 T^{16} - 108042764448 p^{2} T^{18} - 1162854034 p^{4} T^{20} + 89132724 p^{6} T^{22} + 6664873 p^{8} T^{24} + 248898 p^{10} T^{26} + 6323 p^{12} T^{28} + 102 p^{14} T^{30} + p^{16} T^{32} \)
29 \( ( 1 - 103 T^{2} + 6606 T^{4} - 293969 T^{6} + 9810626 T^{8} - 293969 p^{2} T^{10} + 6606 p^{4} T^{12} - 103 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 + 24 T + 350 T^{2} + 3792 T^{3} + 33057 T^{4} + 243840 T^{5} + 1597150 T^{6} + 9593544 T^{7} + 54548420 T^{8} + 9593544 p T^{9} + 1597150 p^{2} T^{10} + 243840 p^{3} T^{11} + 33057 p^{4} T^{12} + 3792 p^{5} T^{13} + 350 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
37 \( ( 1 + T - 60 T^{2} + 689 T^{3} + 2741 T^{4} - 33360 T^{5} + 198298 T^{6} + 1145242 T^{7} - 8840520 T^{8} + 1145242 p T^{9} + 198298 p^{2} T^{10} - 33360 p^{3} T^{11} + 2741 p^{4} T^{12} + 689 p^{5} T^{13} - 60 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2} \)
41 \( ( 1 + 240 T^{2} + 27996 T^{4} + 2035728 T^{6} + 100303238 T^{8} + 2035728 p^{2} T^{10} + 27996 p^{4} T^{12} + 240 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 - 5 T + 76 T^{2} - 341 T^{3} + 2710 T^{4} - 341 p T^{5} + 76 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
47 \( 1 - 158 T^{2} + 13171 T^{4} - 396034 T^{6} - 16747399 T^{8} + 2338864468 T^{10} - 73391270338 T^{12} - 1850417430616 T^{14} + 239524986298546 T^{16} - 1850417430616 p^{2} T^{18} - 73391270338 p^{4} T^{20} + 2338864468 p^{6} T^{22} - 16747399 p^{8} T^{24} - 396034 p^{10} T^{26} + 13171 p^{12} T^{28} - 158 p^{14} T^{30} + p^{16} T^{32} \)
53 \( 1 + 5 p T^{2} + 32968 T^{4} + 3168029 T^{6} + 288393293 T^{8} + 22350648928 T^{10} + 1460664537746 T^{12} + 90342579430370 T^{14} + 5160533915433520 T^{16} + 90342579430370 p^{2} T^{18} + 1460664537746 p^{4} T^{20} + 22350648928 p^{6} T^{22} + 288393293 p^{8} T^{24} + 3168029 p^{10} T^{26} + 32968 p^{12} T^{28} + 5 p^{15} T^{30} + p^{16} T^{32} \)
59 \( 1 - 187 T^{2} + 11904 T^{4} - 267479 T^{6} + 18518765 T^{8} - 3131679552 T^{10} + 215589458578 T^{12} - 149572503026 p T^{14} + 112996047744 p^{2} T^{16} - 149572503026 p^{3} T^{18} + 215589458578 p^{4} T^{20} - 3131679552 p^{6} T^{22} + 18518765 p^{8} T^{24} - 267479 p^{10} T^{26} + 11904 p^{12} T^{28} - 187 p^{14} T^{30} + p^{16} T^{32} \)
61 \( ( 1 - 18 T + 331 T^{2} - 4014 T^{3} + 46777 T^{4} - 446148 T^{5} + 4203790 T^{6} - 34615152 T^{7} + 285617722 T^{8} - 34615152 p T^{9} + 4203790 p^{2} T^{10} - 446148 p^{3} T^{11} + 46777 p^{4} T^{12} - 4014 p^{5} T^{13} + 331 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 7 T - 200 T^{2} + 865 T^{3} + 28259 T^{4} - 67468 T^{5} - 2804524 T^{6} + 1482940 T^{7} + 222922288 T^{8} + 1482940 p T^{9} - 2804524 p^{2} T^{10} - 67468 p^{3} T^{11} + 28259 p^{4} T^{12} + 865 p^{5} T^{13} - 200 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
71 \( ( 1 - 224 T^{2} + 21244 T^{4} - 988448 T^{6} + 37865158 T^{8} - 988448 p^{2} T^{10} + 21244 p^{4} T^{12} - 224 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 - 15 T + 312 T^{2} - 3555 T^{3} + 48909 T^{4} - 547104 T^{5} + 5639826 T^{6} - 55083990 T^{7} + 457851344 T^{8} - 55083990 p T^{9} + 5639826 p^{2} T^{10} - 547104 p^{3} T^{11} + 48909 p^{4} T^{12} - 3555 p^{5} T^{13} + 312 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 14 T - 152 T^{2} + 1448 T^{3} + 34763 T^{4} - 170492 T^{5} - 3816772 T^{6} + 43058 p T^{7} + 382142944 T^{8} + 43058 p^{2} T^{9} - 3816772 p^{2} T^{10} - 170492 p^{3} T^{11} + 34763 p^{4} T^{12} + 1448 p^{5} T^{13} - 152 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
83 \( ( 1 + 141 T^{2} + 22278 T^{4} + 1798779 T^{6} + 189218258 T^{8} + 1798779 p^{2} T^{10} + 22278 p^{4} T^{12} + 141 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( 1 - 378 T^{2} + 69795 T^{4} - 8481534 T^{6} + 768803689 T^{8} - 53847736620 T^{10} + 2382276614382 T^{12} + 24847322779392 T^{14} - 11761219221673230 T^{16} + 24847322779392 p^{2} T^{18} + 2382276614382 p^{4} T^{20} - 53847736620 p^{6} T^{22} + 768803689 p^{8} T^{24} - 8481534 p^{10} T^{26} + 69795 p^{12} T^{28} - 378 p^{14} T^{30} + p^{16} T^{32} \)
97 \( ( 1 - 429 T^{2} + 87258 T^{4} - 11450019 T^{6} + 1191212138 T^{8} - 11450019 p^{2} T^{10} + 87258 p^{4} T^{12} - 429 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
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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

−3.88520029547800311132316240542, −3.87000755462529653979581323571, −3.82682366036790885042336687141, −3.47095154039627990587849372921, −3.41259078870631410097943477117, −3.29050507828690273397047372033, −3.24001845956244916305295914090, −3.15169494529441115946275292868, −3.12830969473408613093517874603, −2.94124069348704167821615651012, −2.65262409796952525560170474544, −2.64299377343995631968414634571, −2.55279051297150666047897605751, −2.26273609731735888471291721841, −2.15084230867860230027899069378, −2.13815474993861405299726239242, −2.09757629818352275877504583079, −1.99561098480782906180991168269, −1.87291598327957002758365077659, −1.63347844976968252628310885627, −1.54599971558872930799597257565, −1.36971516016772276785733922471, −1.04060118499167186243482119534, −0.817357482609938198999289428208, −0.49503094609902077882147536377, 0.49503094609902077882147536377, 0.817357482609938198999289428208, 1.04060118499167186243482119534, 1.36971516016772276785733922471, 1.54599971558872930799597257565, 1.63347844976968252628310885627, 1.87291598327957002758365077659, 1.99561098480782906180991168269, 2.09757629818352275877504583079, 2.13815474993861405299726239242, 2.15084230867860230027899069378, 2.26273609731735888471291721841, 2.55279051297150666047897605751, 2.64299377343995631968414634571, 2.65262409796952525560170474544, 2.94124069348704167821615651012, 3.12830969473408613093517874603, 3.15169494529441115946275292868, 3.24001845956244916305295914090, 3.29050507828690273397047372033, 3.41259078870631410097943477117, 3.47095154039627990587849372921, 3.82682366036790885042336687141, 3.87000755462529653979581323571, 3.88520029547800311132316240542

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

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