Properties

Label 16-42e16-1.1-c2e8-0-5
Degree $16$
Conductor $9.375\times 10^{25}$
Sign $1$
Analytic cond. $2.84884\times 10^{13}$
Root an. cond. $6.93293$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 24·11-s + 104·23-s + 4·25-s − 64·29-s + 128·37-s + 80·43-s + 24·53-s − 112·67-s − 448·71-s − 240·79-s − 40·107-s − 272·109-s − 832·113-s + 504·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 200·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2.18·11-s + 4.52·23-s + 4/25·25-s − 2.20·29-s + 3.45·37-s + 1.86·43-s + 0.452·53-s − 1.67·67-s − 6.30·71-s − 3.03·79-s − 0.373·107-s − 2.49·109-s − 7.36·113-s + 4.16·121-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 − 1.18·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(2.84884\times 10^{13}\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.461968887\)
\(L(\frac12)\) \(\approx\) \(1.461968887\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 4 T^{2} - 846 T^{4} + 1552 T^{6} + 342419 T^{8} + 1552 p^{4} T^{10} - 846 p^{8} T^{12} - 4 p^{12} T^{14} + p^{16} T^{16} \)
11 \( ( 1 + 12 T - 36 T^{2} - 744 T^{3} + 335 T^{4} - 744 p^{2} T^{5} - 36 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
13 \( ( 1 + 100 T^{2} + 59230 T^{4} + 100 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( 1 + 976 T^{2} + 555328 T^{4} + 224681056 T^{6} + 71270115967 T^{8} + 224681056 p^{4} T^{10} + 555328 p^{8} T^{12} + 976 p^{12} T^{14} + p^{16} T^{16} \)
19 \( 1 + 384 T^{2} + 85248 T^{4} - 76198656 T^{6} - 32330341153 T^{8} - 76198656 p^{4} T^{10} + 85248 p^{8} T^{12} + 384 p^{12} T^{14} + p^{16} T^{16} \)
23 \( ( 1 - 26 T + 147 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
29 \( ( 1 + 16 T + 1354 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
31 \( 1 + 2276 T^{2} + 2345418 T^{4} + 2248041616 T^{6} + 2398960561427 T^{8} + 2248041616 p^{4} T^{10} + 2345418 p^{8} T^{12} + 2276 p^{12} T^{14} + p^{16} T^{16} \)
37 \( ( 1 - 32 T - 345 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
41 \( ( 1 - 5984 T^{2} + 14598784 T^{4} - 5984 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 - 20 T - 1004 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
47 \( 1 + 2804 T^{2} + 4041738 T^{4} - 16652069936 T^{6} - 47579788251373 T^{8} - 16652069936 p^{4} T^{10} + 4041738 p^{8} T^{12} + 2804 p^{12} T^{14} + p^{16} T^{16} \)
53 \( ( 1 - 12 T - 5118 T^{2} + 4272 T^{3} + 19393667 T^{4} + 4272 p^{2} T^{5} - 5118 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( 1 + 9168 T^{2} + 44124768 T^{4} + 143870985312 T^{6} + 428880163966751 T^{8} + 143870985312 p^{4} T^{10} + 44124768 p^{8} T^{12} + 9168 p^{12} T^{14} + p^{16} T^{16} \)
61 \( 1 + 10940 T^{2} + 63012210 T^{4} + 317038005520 T^{6} + 1362819035627219 T^{8} + 317038005520 p^{4} T^{10} + 63012210 p^{8} T^{12} + 10940 p^{12} T^{14} + p^{16} T^{16} \)
67 \( ( 1 + 56 T - 3098 T^{2} - 153664 T^{3} + 4634131 T^{4} - 153664 p^{2} T^{5} - 3098 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( ( 1 + 112 T + 11650 T^{2} + 112 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
73 \( 1 + 16032 T^{2} + 141292608 T^{4} + 944860893888 T^{6} + 5288473446720191 T^{8} + 944860893888 p^{4} T^{10} + 141292608 p^{8} T^{12} + 16032 p^{12} T^{14} + p^{16} T^{16} \)
79 \( ( 1 + 120 T - 1290 T^{2} + 384960 T^{3} + 117355619 T^{4} + 384960 p^{2} T^{5} - 1290 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( ( 1 - 2448 T^{2} + 41142720 T^{4} - 2448 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( 1 + 12144 T^{2} - 501408 T^{4} + 273163031328 T^{6} + 8707921383977279 T^{8} + 273163031328 p^{4} T^{10} - 501408 p^{8} T^{12} + 12144 p^{12} T^{14} + p^{16} T^{16} \)
97 \( ( 1 - 2304 T^{2} + 176950848 T^{4} - 2304 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
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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

−3.70306861543160483454049758050, −3.45400745831645335399332480071, −3.45037997143592350313918222040, −3.42774675229964699049481313543, −2.93592081592963131672866423087, −2.90606272389745228064310109459, −2.86271938749811461710077319495, −2.81497009317867984925968753254, −2.75597469461496814350659157393, −2.75230648052172626892430344496, −2.57962012607466835542330736112, −2.34152374075776061188122248568, −2.26659924088150940239175126674, −1.90119443438419738761162314692, −1.88105076923645374814005268833, −1.57793555983774330900960043817, −1.48940946887470746759471129796, −1.43281640718091463804636531235, −1.02352674527469370218644912802, −1.02126962005248285180582017250, −0.961928121107703897143441623122, −0.943725093609386939070907931512, −0.29023280984017384568061675354, −0.19721053128674291811314945057, −0.18375958747905416685140172317, 0.18375958747905416685140172317, 0.19721053128674291811314945057, 0.29023280984017384568061675354, 0.943725093609386939070907931512, 0.961928121107703897143441623122, 1.02126962005248285180582017250, 1.02352674527469370218644912802, 1.43281640718091463804636531235, 1.48940946887470746759471129796, 1.57793555983774330900960043817, 1.88105076923645374814005268833, 1.90119443438419738761162314692, 2.26659924088150940239175126674, 2.34152374075776061188122248568, 2.57962012607466835542330736112, 2.75230648052172626892430344496, 2.75597469461496814350659157393, 2.81497009317867984925968753254, 2.86271938749811461710077319495, 2.90606272389745228064310109459, 2.93592081592963131672866423087, 3.42774675229964699049481313543, 3.45037997143592350313918222040, 3.45400745831645335399332480071, 3.70306861543160483454049758050

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

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