Properties

Degree 64
Conductor $ 3^{32} \cdot 5^{32} \cdot 7^{32} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 162·16-s − 36·25-s + 72·37-s − 328·43-s − 648·67-s − 237·81-s + 1.88e3·121-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 + 227-s + 229-s + ⋯
L(s)  = 1  + 81/8·16-s − 1.43·25-s + 1.94·37-s − 7.62·43-s − 9.67·67-s − 2.92·81-s + 15.5·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{32} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{32} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(64\)
\( N \)  =  \(3^{32} \cdot 5^{32} \cdot 7^{32}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((64,\ 3^{32} \cdot 5^{32} \cdot 7^{32} ,\ ( \ : [1]^{32} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(54.2561\)
\(L(\frac12)\)  \(\approx\)  \(54.2561\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 64. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 63.
$p$$F_p(T)$
bad3 \( 1 + 79 p T^{4} + 14302 p T^{8} + 4865923 T^{12} + 5770186 p^{4} T^{16} + 4865923 p^{8} T^{20} + 14302 p^{17} T^{24} + 79 p^{25} T^{28} + p^{32} T^{32} \)
5 \( ( 1 + 18 T^{2} - 584 T^{4} - 1802 p T^{6} + 1286 p^{3} T^{8} - 1802 p^{5} T^{10} - 584 p^{8} T^{12} + 18 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
7 \( ( 1 - 16 p T^{3} - 1756 T^{4} - 1104 p T^{5} + 128 p^{2} T^{6} + 96 p^{4} T^{7} + 4486 p^{4} T^{8} + 96 p^{6} T^{9} + 128 p^{6} T^{10} - 1104 p^{7} T^{11} - 1756 p^{8} T^{12} - 16 p^{11} T^{13} + p^{16} T^{16} )^{2} \)
good2 \( ( 1 - 81 T^{4} + 3127 T^{8} - 77539 T^{12} + 353401 p^{2} T^{16} - 77539 p^{8} T^{20} + 3127 p^{16} T^{24} - 81 p^{24} T^{28} + p^{32} T^{32} )^{2} \)
11 \( ( 1 - 471 T^{2} + 112740 T^{4} - 18504917 T^{6} + 20182454 p^{2} T^{8} - 18504917 p^{4} T^{10} + 112740 p^{8} T^{12} - 471 p^{12} T^{14} + p^{16} T^{16} )^{4} \)
13 \( ( 1 + 73677 T^{4} + 1792033258 T^{8} + 2436156815843 T^{12} - 623309733871856438 T^{16} + 2436156815843 p^{8} T^{20} + 1792033258 p^{16} T^{24} + 73677 p^{24} T^{28} + p^{32} T^{32} )^{2} \)
17 \( ( 1 - 29967 T^{4} - 5080976746 T^{8} + 472290417963487 T^{12} + 26661734898089857330 T^{16} + 472290417963487 p^{8} T^{20} - 5080976746 p^{16} T^{24} - 29967 p^{24} T^{28} + p^{32} T^{32} )^{2} \)
19 \( ( 1 + 1568 T^{2} + 1292036 T^{4} + 736908752 T^{6} + 308878351366 T^{8} + 736908752 p^{4} T^{10} + 1292036 p^{8} T^{12} + 1568 p^{12} T^{14} + p^{16} T^{16} )^{4} \)
23 \( ( 1 - 61368 T^{4} - 276561061412 T^{8} + 3838383845564408 T^{12} + 59720466768549250918 p^{2} T^{16} + 3838383845564408 p^{8} T^{20} - 276561061412 p^{16} T^{24} - 61368 p^{24} T^{28} + p^{32} T^{32} )^{2} \)
29 \( ( 1 + 3093 T^{2} + 4842258 T^{4} + 5040685387 T^{6} + 4391538466202 T^{8} + 5040685387 p^{4} T^{10} + 4842258 p^{8} T^{12} + 3093 p^{12} T^{14} + p^{16} T^{16} )^{4} \)
31 \( ( 1 - 4564 T^{2} + 11409148 T^{4} - 18354206412 T^{6} + 20921932458934 T^{8} - 18354206412 p^{4} T^{10} + 11409148 p^{8} T^{12} - 4564 p^{12} T^{14} + p^{16} T^{16} )^{4} \)
37 \( ( 1 - 18 T + 162 T^{2} + 55190 T^{3} + 9584 T^{4} - 113564166 T^{5} + 3565570430 T^{6} - 19214177022 T^{7} - 5278828701794 T^{8} - 19214177022 p^{2} T^{9} + 3565570430 p^{4} T^{10} - 113564166 p^{6} T^{11} + 9584 p^{8} T^{12} + 55190 p^{10} T^{13} + 162 p^{12} T^{14} - 18 p^{14} T^{15} + p^{16} T^{16} )^{4} \)
41 \( ( 1 + 8402 T^{2} + 35731576 T^{4} + 98961156078 T^{6} + 194774826167086 T^{8} + 98961156078 p^{4} T^{10} + 35731576 p^{8} T^{12} + 8402 p^{12} T^{14} + p^{16} T^{16} )^{4} \)
43 \( ( 1 + 82 T + 3362 T^{2} + 157602 T^{3} + 5262644 T^{4} + 138218758 T^{5} + 6060124230 T^{6} + 349007434806 T^{7} + 19292580703846 T^{8} + 349007434806 p^{2} T^{9} + 6060124230 p^{4} T^{10} + 138218758 p^{6} T^{11} + 5262644 p^{8} T^{12} + 157602 p^{10} T^{13} + 3362 p^{12} T^{14} + 82 p^{14} T^{15} + p^{16} T^{16} )^{4} \)
47 \( ( 1 + 8840509 T^{4} + 77995000864282 T^{8} + \)\(52\!\cdots\!51\)\( T^{12} + \)\(27\!\cdots\!54\)\( T^{16} + \)\(52\!\cdots\!51\)\( p^{8} T^{20} + 77995000864282 p^{16} T^{24} + 8840509 p^{24} T^{28} + p^{32} T^{32} )^{2} \)
53 \( ( 1 - 347948 T^{4} + 214420611906468 T^{8} - 43356302968478412372 T^{12} + \)\(19\!\cdots\!62\)\( T^{16} - 43356302968478412372 p^{8} T^{20} + 214420611906468 p^{16} T^{24} - 347948 p^{24} T^{28} + p^{32} T^{32} )^{2} \)
59 \( ( 1 - 12202 T^{2} + 72803328 T^{4} - 319961508630 T^{6} + 1204283503693566 T^{8} - 319961508630 p^{4} T^{10} + 72803328 p^{8} T^{12} - 12202 p^{12} T^{14} + p^{16} T^{16} )^{4} \)
61 \( ( 1 - 19252 T^{2} + 172954996 T^{4} - 993191665452 T^{6} + 4202794643071894 T^{8} - 993191665452 p^{4} T^{10} + 172954996 p^{8} T^{12} - 19252 p^{12} T^{14} + p^{16} T^{16} )^{4} \)
67 \( 1 + 648T + 2.09e5T^{2} + 4.72e7T^{3} + 8.74e9T^{4} + 1.44e12T^{5} + 2.21e14T^{6} + 3.12e16T^{7} + 4.12e18T^{8} + 5.16e20T^{9} + 6.17e22T^{10} + 7.07e24T^{11} + 7.78e26T^{12} + 8.26e28T^{13} + 8.49e30T^{14} + 8.45e32T^{15} + 8.17e34T^{16} + 7.68e36T^{17} + 7.03e38T^{18} + 6.27e40T^{19} + 5.46e42T^{20} + 4.65e44T^{21} + 3.86e46T^{22} + 3.14e48T^{23} + 2.50e50T^{24} + 1.94e52T^{25} + 1.48e54T^{26} + 1.11e56T^{27} + 8.13e57T^{28} + 5.84e59T^{29} + 4.11e61T^{30} + 2.83e63T^{31}+O(T^{32}) \)
71 \( 1 - 1.08e5T^{2} + 5.87e9T^{4} - 2.09e14T^{6} + 5.55e18T^{8} - 1.16e23T^{10} + 2.01e27T^{12} - 2.95e31T^{14} + 3.73e35T^{16} - 4.13e39T^{18} + 4.04e43T^{20} - 3.53e47T^{22} + 2.77e51T^{24} - 1.96e55T^{26} + 1.25e59T^{28} - 7.32e62T^{30}+O(T^{32}) \)
73 \( 1 - 8.50e7T^{4} + 7.58e15T^{8} - 4.10e23T^{12} + 2.20e31T^{16} - 8.95e38T^{20} + 3.56e46T^{24} - 1.14e54T^{28}+O(T^{31}) \)
79 \( 1 - 1.30e5T^{2} + 8.49e9T^{4} - 3.67e14T^{6} + 1.18e19T^{8} - 3.05e23T^{10} + 6.48e27T^{12} - 1.16e32T^{14} + 1.81e36T^{16} - 2.48e40T^{18} + 3.00e44T^{20} - 3.24e48T^{22} + 3.15e52T^{24} - 2.76e56T^{26} + 2.18e60T^{28} - 1.57e64T^{30}+O(T^{31}) \)
83 \( 1 + 2.18e8T^{4} + 2.97e16T^{8} + 3.08e24T^{12} + 2.61e32T^{16} + 1.89e40T^{20} + 1.20e48T^{24} + 6.77e55T^{28}+O(T^{31}) \)
89 \( 1 - 7.11e4T^{2} + 2.78e9T^{4} - 7.69e13T^{6} + 1.67e18T^{8} - 3.06e22T^{10} + 4.89e26T^{12} - 7.01e30T^{14} + 9.20e34T^{16} - 1.11e39T^{18} + 1.27e43T^{20} - 1.35e47T^{22} + 1.35e51T^{24} - 1.28e55T^{26} + 1.16e59T^{28}+O(T^{30}) \)
97 \( 1 + 2.98e8T^{4} + 5.64e16T^{8} + 7.88e24T^{12} + 9.15e32T^{16} + 1.00e41T^{20} + 1.08e49T^{24} + 1.12e57T^{28}+O(T^{30}) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{64} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.50464403759182809581020598010, −2.34533341253338967305835472565, −2.17363320427576425980906832399, −2.15253824130510070704307976764, −2.14649878153927787694943115360, −2.10776973351888749159279285859, −2.03779857650544937873703411917, −2.03451511104998771911319784382, −1.89711609327720206838822880085, −1.79623594621382108552937919350, −1.78444511111767206639241343531, −1.75444612475016349392683330174, −1.46937766676904354976203739990, −1.42634916627372273614260621228, −1.37930725559908685957762691505, −1.28441409378062444644494788939, −1.25450594003756216512570658190, −1.11545248726907422777713751183, −1.06356229310377854849745520878, −1.05239878435453956070183159159, −0.77775461455980263043742307130, −0.49118799839339151465359996626, −0.48020994996230853930567029042, −0.37285198986103562626621932439, −0.36324139024709615945713561161, 0.36324139024709615945713561161, 0.37285198986103562626621932439, 0.48020994996230853930567029042, 0.49118799839339151465359996626, 0.77775461455980263043742307130, 1.05239878435453956070183159159, 1.06356229310377854849745520878, 1.11545248726907422777713751183, 1.25450594003756216512570658190, 1.28441409378062444644494788939, 1.37930725559908685957762691505, 1.42634916627372273614260621228, 1.46937766676904354976203739990, 1.75444612475016349392683330174, 1.78444511111767206639241343531, 1.79623594621382108552937919350, 1.89711609327720206838822880085, 2.03451511104998771911319784382, 2.03779857650544937873703411917, 2.10776973351888749159279285859, 2.14649878153927787694943115360, 2.15253824130510070704307976764, 2.17363320427576425980906832399, 2.34533341253338967305835472565, 2.50464403759182809581020598010

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

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