Properties

Degree 16
Conductor $ 2^{16} \cdot 3^{8} \cdot 23^{8} \cdot 29^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 8

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s − 5·5-s − 4·7-s + 36·9-s − 5·11-s − 4·13-s − 40·15-s − 3·17-s − 5·19-s − 32·21-s + 8·23-s − 10·25-s + 120·27-s + 8·29-s − 2·31-s − 40·33-s + 20·35-s − 10·37-s − 32·39-s − 11·41-s − 7·43-s − 180·45-s − 14·47-s − 29·49-s − 24·51-s − 15·53-s + 25·55-s + ⋯
L(s)  = 1  + 4.61·3-s − 2.23·5-s − 1.51·7-s + 12·9-s − 1.50·11-s − 1.10·13-s − 10.3·15-s − 0.727·17-s − 1.14·19-s − 6.98·21-s + 1.66·23-s − 2·25-s + 23.0·27-s + 1.48·29-s − 0.359·31-s − 6.96·33-s + 3.38·35-s − 1.64·37-s − 5.12·39-s − 1.71·41-s − 1.06·43-s − 26.8·45-s − 2.04·47-s − 4.14·49-s − 3.36·51-s − 2.06·53-s + 3.37·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 23^{8} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 23^{8} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{16} \cdot 3^{8} \cdot 23^{8} \cdot 29^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8004} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  8
Selberg data  =  $(16,\ 2^{16} \cdot 3^{8} \cdot 23^{8} \cdot 29^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;23,\;29\}$, \(F_p\) is a polynomial of degree 16. If $p \in \{2,\;3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 15.
$p$$F_p$
bad2 \( 1 \)
3 \( ( 1 - T )^{8} \)
23 \( ( 1 - T )^{8} \)
29 \( ( 1 - T )^{8} \)
good5 \( 1 + p T + 7 p T^{2} + 123 T^{3} + 103 p T^{4} + 1432 T^{5} + 4547 T^{6} + 2098 p T^{7} + 27216 T^{8} + 2098 p^{2} T^{9} + 4547 p^{2} T^{10} + 1432 p^{3} T^{11} + 103 p^{5} T^{12} + 123 p^{5} T^{13} + 7 p^{7} T^{14} + p^{8} T^{15} + p^{8} T^{16} \)
7 \( 1 + 4 T + 45 T^{2} + 155 T^{3} + 138 p T^{4} + 2785 T^{5} + 12546 T^{6} + 30168 T^{7} + 107038 T^{8} + 30168 p T^{9} + 12546 p^{2} T^{10} + 2785 p^{3} T^{11} + 138 p^{5} T^{12} + 155 p^{5} T^{13} + 45 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 + 5 T + 76 T^{2} + 294 T^{3} + 2545 T^{4} + 7973 T^{5} + 50715 T^{6} + 11960 p T^{7} + 672774 T^{8} + 11960 p^{2} T^{9} + 50715 p^{2} T^{10} + 7973 p^{3} T^{11} + 2545 p^{4} T^{12} + 294 p^{5} T^{13} + 76 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 + 4 T + 90 T^{2} + 319 T^{3} + 3697 T^{4} + 11374 T^{5} + 90515 T^{6} + 235529 T^{7} + 1442141 T^{8} + 235529 p T^{9} + 90515 p^{2} T^{10} + 11374 p^{3} T^{11} + 3697 p^{4} T^{12} + 319 p^{5} T^{13} + 90 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 + 3 T + 6 p T^{2} + 230 T^{3} + 4831 T^{4} + 8730 T^{5} + 143335 T^{6} + 216129 T^{7} + 2915821 T^{8} + 216129 p T^{9} + 143335 p^{2} T^{10} + 8730 p^{3} T^{11} + 4831 p^{4} T^{12} + 230 p^{5} T^{13} + 6 p^{7} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 + 5 T + 107 T^{2} + 478 T^{3} + 5359 T^{4} + 21939 T^{5} + 169074 T^{6} + 622661 T^{7} + 3759380 T^{8} + 622661 p T^{9} + 169074 p^{2} T^{10} + 21939 p^{3} T^{11} + 5359 p^{4} T^{12} + 478 p^{5} T^{13} + 107 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 + 2 T + 104 T^{2} + 429 T^{3} + 5928 T^{4} + 32756 T^{5} + 261266 T^{6} + 1458383 T^{7} + 9221706 T^{8} + 1458383 p T^{9} + 261266 p^{2} T^{10} + 32756 p^{3} T^{11} + 5928 p^{4} T^{12} + 429 p^{5} T^{13} + 104 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 + 10 T + 251 T^{2} + 1913 T^{3} + 27402 T^{4} + 167849 T^{5} + 1781631 T^{6} + 9069749 T^{7} + 78490558 T^{8} + 9069749 p T^{9} + 1781631 p^{2} T^{10} + 167849 p^{3} T^{11} + 27402 p^{4} T^{12} + 1913 p^{5} T^{13} + 251 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 + 11 T + 217 T^{2} + 2015 T^{3} + 24755 T^{4} + 190244 T^{5} + 1764337 T^{6} + 11438832 T^{7} + 86678840 T^{8} + 11438832 p T^{9} + 1764337 p^{2} T^{10} + 190244 p^{3} T^{11} + 24755 p^{4} T^{12} + 2015 p^{5} T^{13} + 217 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 + 7 T + 6 p T^{2} + 1386 T^{3} + 30118 T^{4} + 127901 T^{5} + 2161035 T^{6} + 7532761 T^{7} + 108671872 T^{8} + 7532761 p T^{9} + 2161035 p^{2} T^{10} + 127901 p^{3} T^{11} + 30118 p^{4} T^{12} + 1386 p^{5} T^{13} + 6 p^{7} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 14 T + 330 T^{2} + 3314 T^{3} + 42963 T^{4} + 335929 T^{5} + 3171577 T^{6} + 20936253 T^{7} + 167081468 T^{8} + 20936253 p T^{9} + 3171577 p^{2} T^{10} + 335929 p^{3} T^{11} + 42963 p^{4} T^{12} + 3314 p^{5} T^{13} + 330 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 15 T + 297 T^{2} + 2791 T^{3} + 29831 T^{4} + 178666 T^{5} + 1370849 T^{6} + 5070700 T^{7} + 50414860 T^{8} + 5070700 p T^{9} + 1370849 p^{2} T^{10} + 178666 p^{3} T^{11} + 29831 p^{4} T^{12} + 2791 p^{5} T^{13} + 297 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 4 T + 245 T^{2} - 1104 T^{3} + 32870 T^{4} - 142016 T^{5} + 3010387 T^{6} - 11639812 T^{7} + 205110130 T^{8} - 11639812 p T^{9} + 3010387 p^{2} T^{10} - 142016 p^{3} T^{11} + 32870 p^{4} T^{12} - 1104 p^{5} T^{13} + 245 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - T + 282 T^{2} - 746 T^{3} + 40915 T^{4} - 129123 T^{5} + 4080190 T^{6} - 11958870 T^{7} + 292906840 T^{8} - 11958870 p T^{9} + 4080190 p^{2} T^{10} - 129123 p^{3} T^{11} + 40915 p^{4} T^{12} - 746 p^{5} T^{13} + 282 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 + 5 T + 367 T^{2} + 2067 T^{3} + 65226 T^{4} + 359097 T^{5} + 7477626 T^{6} + 36309399 T^{7} + 596860648 T^{8} + 36309399 p T^{9} + 7477626 p^{2} T^{10} + 359097 p^{3} T^{11} + 65226 p^{4} T^{12} + 2067 p^{5} T^{13} + 367 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 + T + 309 T^{2} - 403 T^{3} + 50633 T^{4} - 107515 T^{5} + 5709802 T^{6} - 13531994 T^{7} + 465330730 T^{8} - 13531994 p T^{9} + 5709802 p^{2} T^{10} - 107515 p^{3} T^{11} + 50633 p^{4} T^{12} - 403 p^{5} T^{13} + 309 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 21 T + 479 T^{2} + 6828 T^{3} + 103348 T^{4} + 1162895 T^{5} + 13471003 T^{6} + 125077854 T^{7} + 1189125482 T^{8} + 125077854 p T^{9} + 13471003 p^{2} T^{10} + 1162895 p^{3} T^{11} + 103348 p^{4} T^{12} + 6828 p^{5} T^{13} + 479 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 + 8 T + 441 T^{2} + 3855 T^{3} + 87828 T^{4} + 828813 T^{5} + 10853967 T^{6} + 103733333 T^{7} + 975633828 T^{8} + 103733333 p T^{9} + 10853967 p^{2} T^{10} + 828813 p^{3} T^{11} + 87828 p^{4} T^{12} + 3855 p^{5} T^{13} + 441 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 - 3 T + 339 T^{2} - 265 T^{3} + 59257 T^{4} + 2766 T^{5} + 7528439 T^{6} + 900678 T^{7} + 722055524 T^{8} + 900678 p T^{9} + 7528439 p^{2} T^{10} + 2766 p^{3} T^{11} + 59257 p^{4} T^{12} - 265 p^{5} T^{13} + 339 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 + 20 T + 264 T^{2} + 3299 T^{3} + 52221 T^{4} + 576266 T^{5} + 5870883 T^{6} + 60574359 T^{7} + 659088331 T^{8} + 60574359 p T^{9} + 5870883 p^{2} T^{10} + 576266 p^{3} T^{11} + 52221 p^{4} T^{12} + 3299 p^{5} T^{13} + 264 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 7 T + 592 T^{2} + 3692 T^{3} + 165275 T^{4} + 908555 T^{5} + 28552838 T^{6} + 135175014 T^{7} + 3332566836 T^{8} + 135175014 p T^{9} + 28552838 p^{2} T^{10} + 908555 p^{3} T^{11} + 165275 p^{4} T^{12} + 3692 p^{5} T^{13} + 592 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.61957744589759668635710845991, −3.34686455140572478766511708091, −3.33751666564754310903987238677, −3.19886378678959776244596207190, −3.14503774895373652153171867080, −3.13224947603627666351764390864, −3.05436786563321344329255638435, −3.04788441250730363541327967364, −2.68732456205018528481332524017, −2.47800045106545242599413016678, −2.45492843636136302291258141979, −2.42014834836775487118337779404, −2.36899913905120411432138558132, −2.33414834708812021740703344094, −2.26887277563603058996729836410, −2.23949710425103708705234735818, −2.15919536834616792503931305142, −1.51664714391710664031849748978, −1.48929284275036063428538481384, −1.43558036318543306773175639961, −1.38608665368054335285387118348, −1.30161477152882774194915738861, −1.28444693408459198439036465212, −1.23806246603826155256980515818, −1.21180081081801366025377704891, 0, 0, 0, 0, 0, 0, 0, 0, 1.21180081081801366025377704891, 1.23806246603826155256980515818, 1.28444693408459198439036465212, 1.30161477152882774194915738861, 1.38608665368054335285387118348, 1.43558036318543306773175639961, 1.48929284275036063428538481384, 1.51664714391710664031849748978, 2.15919536834616792503931305142, 2.23949710425103708705234735818, 2.26887277563603058996729836410, 2.33414834708812021740703344094, 2.36899913905120411432138558132, 2.42014834836775487118337779404, 2.45492843636136302291258141979, 2.47800045106545242599413016678, 2.68732456205018528481332524017, 3.04788441250730363541327967364, 3.05436786563321344329255638435, 3.13224947603627666351764390864, 3.14503774895373652153171867080, 3.19886378678959776244596207190, 3.33751666564754310903987238677, 3.34686455140572478766511708091, 3.61957744589759668635710845991

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

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