Properties

Degree $16$
Conductor $1.362\times 10^{25}$
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  + 2·4-s + 8·5-s + 16-s − 4·17-s + 24·19-s + 16·20-s + 24·23-s + 40·25-s + 12·31-s + 12·37-s + 16·41-s − 32·43-s + 48·53-s + 16·59-s − 2·64-s − 24·67-s − 8·68-s − 24·73-s + 48·76-s + 40·79-s + 8·80-s + 72·83-s − 32·85-s − 16·89-s + 48·92-s + 192·95-s + 80·100-s + ⋯
L(s)  = 1  + 4-s + 3.57·5-s + 1/4·16-s − 0.970·17-s + 5.50·19-s + 3.57·20-s + 5.00·23-s + 8·25-s + 2.15·31-s + 1.97·37-s + 2.49·41-s − 4.87·43-s + 6.59·53-s + 2.08·59-s − 1/4·64-s − 2.93·67-s − 0.970·68-s − 2.80·73-s + 5.50·76-s + 4.50·79-s + 0.894·80-s + 7.90·83-s − 3.47·85-s − 1.69·89-s + 5.00·92-s + 19.6·95-s + 8·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 11^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1386} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(125.7704172\)
\(L(\frac12)\) \(\approx\) \(125.7704172\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{2} \)
3 \( 1 \)
7 \( 1 + 71 T^{4} + p^{4} T^{8} \)
11 \( ( 1 - T^{2} + T^{4} )^{2} \)
good5 \( 1 - 8 T + 24 T^{2} - 48 T^{3} + 161 T^{4} - 504 T^{5} + 1016 T^{6} - 2336 T^{7} + 6096 T^{8} - 2336 p T^{9} + 1016 p^{2} T^{10} - 504 p^{3} T^{11} + 161 p^{4} T^{12} - 48 p^{5} T^{13} + 24 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
13 \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \)
17 \( 1 + 4 T - 18 T^{2} - 120 T^{3} + 89 T^{4} + 2088 T^{5} + 8462 T^{6} - 17972 T^{7} - 230892 T^{8} - 17972 p T^{9} + 8462 p^{2} T^{10} + 2088 p^{3} T^{11} + 89 p^{4} T^{12} - 120 p^{5} T^{13} - 18 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 - 24 T + 284 T^{2} - 2208 T^{3} + 12394 T^{4} - 50808 T^{5} + 139952 T^{6} - 178344 T^{7} - 68669 T^{8} - 178344 p T^{9} + 139952 p^{2} T^{10} - 50808 p^{3} T^{11} + 12394 p^{4} T^{12} - 2208 p^{5} T^{13} + 284 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 24 T + 344 T^{2} - 3648 T^{3} + 31425 T^{4} - 229104 T^{5} + 1458616 T^{6} - 8233512 T^{7} + 41664800 T^{8} - 8233512 p T^{9} + 1458616 p^{2} T^{10} - 229104 p^{3} T^{11} + 31425 p^{4} T^{12} - 3648 p^{5} T^{13} + 344 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 88 T^{2} + 5028 T^{4} - 216200 T^{6} + 6970406 T^{8} - 216200 p^{2} T^{10} + 5028 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \)
31 \( 1 - 12 T + 156 T^{2} - 1296 T^{3} + 11254 T^{4} - 84084 T^{5} + 570528 T^{6} - 3614628 T^{7} + 20158851 T^{8} - 3614628 p T^{9} + 570528 p^{2} T^{10} - 84084 p^{3} T^{11} + 11254 p^{4} T^{12} - 1296 p^{5} T^{13} + 156 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 - 12 T + 60 T^{2} - 312 T^{3} + 1798 T^{4} - 16140 T^{5} + 105696 T^{6} - 201156 T^{7} - 255885 T^{8} - 201156 p T^{9} + 105696 p^{2} T^{10} - 16140 p^{3} T^{11} + 1798 p^{4} T^{12} - 312 p^{5} T^{13} + 60 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
41 \( ( 1 - 8 T + 118 T^{2} - 736 T^{3} + 6891 T^{4} - 736 p T^{5} + 118 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 + 16 T + 120 T^{2} + 560 T^{3} + 2978 T^{4} + 560 p T^{5} + 120 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - 112 T^{2} + 6313 T^{4} - 203056 T^{6} + 6235984 T^{8} - 203056 p^{2} T^{10} + 6313 p^{4} T^{12} - 112 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 48 T + 1224 T^{2} - 21888 T^{3} + 307006 T^{4} - 3578640 T^{5} + 35805312 T^{6} - 313299312 T^{7} + 2422296723 T^{8} - 313299312 p T^{9} + 35805312 p^{2} T^{10} - 3578640 p^{3} T^{11} + 307006 p^{4} T^{12} - 21888 p^{5} T^{13} + 1224 p^{6} T^{14} - 48 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 16 T - 48 T^{2} + 864 T^{3} + 17774 T^{4} - 117456 T^{5} - 1145344 T^{6} - 14992 T^{7} + 117866451 T^{8} - 14992 p T^{9} - 1145344 p^{2} T^{10} - 117456 p^{3} T^{11} + 17774 p^{4} T^{12} + 864 p^{5} T^{13} - 48 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 64 T^{2} - 2487 T^{4} - 936 T^{5} - 22528 T^{6} - 2053896 T^{7} + 16228160 T^{8} - 2053896 p T^{9} - 22528 p^{2} T^{10} - 936 p^{3} T^{11} - 2487 p^{4} T^{12} + 64 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 + 24 T + 162 T^{2} + 528 T^{3} + 13777 T^{4} + 115920 T^{5} - 234846 T^{6} + 1238904 T^{7} + 74527668 T^{8} + 1238904 p T^{9} - 234846 p^{2} T^{10} + 115920 p^{3} T^{11} + 13777 p^{4} T^{12} + 528 p^{5} T^{13} + 162 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 16 T^{2} + 36 T^{4} - 289328 T^{6} + 20048390 T^{8} - 289328 p^{2} T^{10} + 36 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 + 24 T + 456 T^{2} + 6336 T^{3} + 73870 T^{4} + 719688 T^{5} + 6574464 T^{6} + 54754008 T^{7} + 470730531 T^{8} + 54754008 p T^{9} + 6574464 p^{2} T^{10} + 719688 p^{3} T^{11} + 73870 p^{4} T^{12} + 6336 p^{5} T^{13} + 456 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 40 T + 712 T^{2} - 9632 T^{3} + 135425 T^{4} - 1688128 T^{5} + 17529320 T^{6} - 177393800 T^{7} + 1703306752 T^{8} - 177393800 p T^{9} + 17529320 p^{2} T^{10} - 1688128 p^{3} T^{11} + 135425 p^{4} T^{12} - 9632 p^{5} T^{13} + 712 p^{6} T^{14} - 40 p^{7} T^{15} + p^{8} T^{16} \)
83 \( ( 1 - 36 T + 650 T^{2} - 7992 T^{3} + 79011 T^{4} - 7992 p T^{5} + 650 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 + 16 T + 24 T^{2} - 1056 T^{3} - 11602 T^{4} - 107760 T^{5} - 365440 T^{6} + 7272880 T^{7} + 110341923 T^{8} + 7272880 p T^{9} - 365440 p^{2} T^{10} - 107760 p^{3} T^{11} - 11602 p^{4} T^{12} - 1056 p^{5} T^{13} + 24 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - 452 T^{2} + 104394 T^{4} - 16099984 T^{6} + 1808979155 T^{8} - 16099984 p^{2} T^{10} + 104394 p^{4} T^{12} - 452 p^{6} T^{14} + p^{8} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.10966042269207682219255683872, −3.88757481733624123063091302289, −3.67889107841272697825666750457, −3.53231458111168320101523058175, −3.32386733151507176540267794291, −3.31889641775730839643376172842, −3.30645694263501414407191431435, −3.16943071744283706932018298869, −2.87661057309070680041859703242, −2.76547544665914397952105995894, −2.71316496552743646045531460394, −2.54249615772210774631468221955, −2.46362859258588318191638001736, −2.44926721889607779474111785565, −2.26017035062498667562823716689, −2.17521287269402542179947594059, −1.83118020492141917830317763902, −1.55166826205007776885710591263, −1.48506277649331084175485344498, −1.33662354323557497185627644563, −1.10914939931494547880564605440, −0.966540208665610387484184556035, −0.867751934221223898710504501988, −0.75844179188059928297878762301, −0.75538822869407592697024860327, 0.75538822869407592697024860327, 0.75844179188059928297878762301, 0.867751934221223898710504501988, 0.966540208665610387484184556035, 1.10914939931494547880564605440, 1.33662354323557497185627644563, 1.48506277649331084175485344498, 1.55166826205007776885710591263, 1.83118020492141917830317763902, 2.17521287269402542179947594059, 2.26017035062498667562823716689, 2.44926721889607779474111785565, 2.46362859258588318191638001736, 2.54249615772210774631468221955, 2.71316496552743646045531460394, 2.76547544665914397952105995894, 2.87661057309070680041859703242, 3.16943071744283706932018298869, 3.30645694263501414407191431435, 3.31889641775730839643376172842, 3.32386733151507176540267794291, 3.53231458111168320101523058175, 3.67889107841272697825666750457, 3.88757481733624123063091302289, 4.10966042269207682219255683872

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

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