Properties

Degree 16
Conductor $ 2^{40} \cdot 7^{8} $
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  + 8·3-s + 20·9-s + 32·11-s − 80·17-s − 56·19-s + 92·25-s − 24·27-s + 256·33-s + 128·41-s − 28·49-s − 640·51-s − 448·57-s − 104·59-s − 304·67-s − 112·73-s + 736·75-s − 344·81-s − 72·83-s − 512·89-s + 64·97-s + 640·99-s − 688·107-s + 136·113-s + 112·121-s + 1.02e3·123-s + 127-s + 131-s + ⋯
L(s)  = 1  + 8/3·3-s + 20/9·9-s + 2.90·11-s − 4.70·17-s − 2.94·19-s + 3.67·25-s − 8/9·27-s + 7.75·33-s + 3.12·41-s − 4/7·49-s − 12.5·51-s − 7.85·57-s − 1.76·59-s − 4.53·67-s − 1.53·73-s + 9.81·75-s − 4.24·81-s − 0.867·83-s − 5.75·89-s + 0.659·97-s + 6.46·99-s − 6.42·107-s + 1.20·113-s + 0.925·121-s + 8.32·123-s + 0.00787·127-s + 0.00763·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{40} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{224} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{40} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.59153\)
\(L(\frac12)\)  \(\approx\)  \(1.59153\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( ( 1 + p T^{2} )^{4} \)
good3 \( ( 1 - 4 T + 14 T^{2} - 28 T^{3} + 98 T^{4} - 28 p^{2} T^{5} + 14 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
5 \( 1 - 92 T^{2} + 5464 T^{4} - 211956 T^{6} + 6231214 T^{8} - 211956 p^{4} T^{10} + 5464 p^{8} T^{12} - 92 p^{12} T^{14} + p^{16} T^{16} \)
11 \( ( 1 - 16 T + 328 T^{2} - 4944 T^{3} + 49230 T^{4} - 4944 p^{2} T^{5} + 328 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
13 \( 1 - 444 T^{2} + 142936 T^{4} - 35361044 T^{6} + 6575436334 T^{8} - 35361044 p^{4} T^{10} + 142936 p^{8} T^{12} - 444 p^{12} T^{14} + p^{16} T^{16} \)
17 \( ( 1 + 40 T + 1308 T^{2} + 31512 T^{3} + 588230 T^{4} + 31512 p^{2} T^{5} + 1308 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
19 \( ( 1 + 28 T + 90 p T^{2} + 31332 T^{3} + 975266 T^{4} + 31332 p^{2} T^{5} + 90 p^{5} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
23 \( 1 - 1744 T^{2} + 1272156 T^{4} - 412076080 T^{6} + 99307893702 T^{8} - 412076080 p^{4} T^{10} + 1272156 p^{8} T^{12} - 1744 p^{12} T^{14} + p^{16} T^{16} \)
29 \( 1 - 3384 T^{2} + 6555580 T^{4} - 8754768776 T^{6} + 8490907402822 T^{8} - 8754768776 p^{4} T^{10} + 6555580 p^{8} T^{12} - 3384 p^{12} T^{14} + p^{16} T^{16} \)
31 \( 1 - 3944 T^{2} + 8438620 T^{4} - 12447428312 T^{6} + 13694235978694 T^{8} - 12447428312 p^{4} T^{10} + 8438620 p^{8} T^{12} - 3944 p^{12} T^{14} + p^{16} T^{16} \)
37 \( 1 - 3512 T^{2} + 9188668 T^{4} - 18622781448 T^{6} + 27544347275206 T^{8} - 18622781448 p^{4} T^{10} + 9188668 p^{8} T^{12} - 3512 p^{12} T^{14} + p^{16} T^{16} \)
41 \( ( 1 - 64 T + 4956 T^{2} - 221760 T^{3} + 11848326 T^{4} - 221760 p^{2} T^{5} + 4956 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
43 \( ( 1 + 4680 T^{2} + 58016 T^{3} + 10251086 T^{4} + 58016 p^{2} T^{5} + 4680 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( 1 - 8392 T^{2} + 39566748 T^{4} - 127207295352 T^{6} + 316693927920198 T^{8} - 127207295352 p^{4} T^{10} + 39566748 p^{8} T^{12} - 8392 p^{12} T^{14} + p^{16} T^{16} \)
53 \( 1 - 18920 T^{2} + 162796828 T^{4} - 840091728600 T^{6} + 2864724835962118 T^{8} - 840091728600 p^{4} T^{10} + 162796828 p^{8} T^{12} - 18920 p^{12} T^{14} + p^{16} T^{16} \)
59 \( ( 1 + 52 T + 2254 T^{2} - 207508 T^{3} - 19795230 T^{4} - 207508 p^{2} T^{5} + 2254 p^{4} T^{6} + 52 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
61 \( 1 - 16316 T^{2} + 140172120 T^{4} - 816942037524 T^{6} + 3499102878259502 T^{8} - 816942037524 p^{4} T^{10} + 140172120 p^{8} T^{12} - 16316 p^{12} T^{14} + p^{16} T^{16} \)
67 \( ( 1 + 152 T + 22224 T^{2} + 2037320 T^{3} + 158433022 T^{4} + 2037320 p^{2} T^{5} + 22224 p^{4} T^{6} + 152 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( 1 - 9864 T^{2} + 51888284 T^{4} - 294531431096 T^{6} + 1789421441990854 T^{8} - 294531431096 p^{4} T^{10} + 51888284 p^{8} T^{12} - 9864 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 + 56 T + 18460 T^{2} + 736008 T^{3} + 138223494 T^{4} + 736008 p^{2} T^{5} + 18460 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 - 24968 T^{2} + 330869788 T^{4} - 3106127956152 T^{6} + 22189846569597766 T^{8} - 3106127956152 p^{4} T^{10} + 330869788 p^{8} T^{12} - 24968 p^{12} T^{14} + p^{16} T^{16} \)
83 \( ( 1 + 36 T + 16478 T^{2} + 177884 T^{3} + 135298114 T^{4} + 177884 p^{2} T^{5} + 16478 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
89 \( ( 1 + 256 T + 48252 T^{2} + 6269952 T^{3} + 638304966 T^{4} + 6269952 p^{2} T^{5} + 48252 p^{4} T^{6} + 256 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( ( 1 - 32 T + 19484 T^{2} - 1437536 T^{3} + 199130566 T^{4} - 1437536 p^{2} T^{5} + 19484 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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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

−5.41290231415776988915866673513, −5.02333227321736714061387694455, −4.85660935491023157703044268636, −4.36719874430586989441543684022, −4.35430646577268367463356695720, −4.31757122362489525776129829974, −4.26393930904899217452118623932, −4.25416233653326295398420032189, −4.19803804955054356522641160768, −4.19608644914000489959271259883, −3.52368282682146355357071823154, −3.34567381800936061227059017084, −3.07344323878881009943488113422, −3.00725620442309465079541705051, −2.94545863117885836015063369937, −2.68833386320037761770043497563, −2.57297291506444161396090485057, −2.38859923264360069341101217852, −2.14560546054751703566061367445, −1.90320483923375796969158310344, −1.51183478155550317557141775287, −1.44845735155733057308628993800, −1.39260911693242028043536374185, −0.57262552769252382739705012423, −0.12962795379553296829796567929, 0.12962795379553296829796567929, 0.57262552769252382739705012423, 1.39260911693242028043536374185, 1.44845735155733057308628993800, 1.51183478155550317557141775287, 1.90320483923375796969158310344, 2.14560546054751703566061367445, 2.38859923264360069341101217852, 2.57297291506444161396090485057, 2.68833386320037761770043497563, 2.94545863117885836015063369937, 3.00725620442309465079541705051, 3.07344323878881009943488113422, 3.34567381800936061227059017084, 3.52368282682146355357071823154, 4.19608644914000489959271259883, 4.19803804955054356522641160768, 4.25416233653326295398420032189, 4.26393930904899217452118623932, 4.31757122362489525776129829974, 4.35430646577268367463356695720, 4.36719874430586989441543684022, 4.85660935491023157703044268636, 5.02333227321736714061387694455, 5.41290231415776988915866673513

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

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