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Properties

Label 16-726e8-1.1-c1e8-0-12
Degree $16$
Conductor $7.718\times 10^{22}$
Sign $1$
Analytic cond. $1.27558\times 10^{6}$
Root an. cond. $2.40772$
Motivic weight $1$
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  + 8·2-s − 3·3-s + 36·4-s − 24·6-s + 120·8-s + 8·9-s − 108·12-s + 330·16-s + 10·17-s + 64·18-s − 360·24-s + 27·25-s − 20·27-s + 2·29-s − 22·31-s + 792·32-s + 80·34-s + 288·36-s − 24·37-s − 2·41-s − 990·48-s + 39·49-s + 216·50-s − 30·51-s − 160·54-s + 16·58-s − 176·62-s + ⋯
L(s)  = 1  + 5.65·2-s − 1.73·3-s + 18·4-s − 9.79·6-s + 42.4·8-s + 8/3·9-s − 31.1·12-s + 82.5·16-s + 2.42·17-s + 15.0·18-s − 73.4·24-s + 27/5·25-s − 3.84·27-s + 0.371·29-s − 3.95·31-s + 140.·32-s + 13.7·34-s + 48·36-s − 3.94·37-s − 0.312·41-s − 142.·48-s + 39/7·49-s + 30.5·50-s − 4.20·51-s − 21.7·54-s + 2.10·58-s − 22.3·62-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(1.27558\times 10^{6}\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(328.9775589\)
\(L(\frac12)\) \(\approx\) \(328.9775589\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{8} \)
3 \( 1 + p T + T^{2} - T^{3} + 4 T^{4} - p T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
11 \( 1 \)
good5 \( 1 - 27 T^{2} + 359 T^{4} - 3053 T^{6} + 18096 T^{8} - 3053 p^{2} T^{10} + 359 p^{4} T^{12} - 27 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 - 39 T^{2} + 727 T^{4} - 8557 T^{6} + 70520 T^{8} - 8557 p^{2} T^{10} + 727 p^{4} T^{12} - 39 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 - 4 p T^{2} + 1460 T^{4} - 2204 p T^{6} + 426934 T^{8} - 2204 p^{3} T^{10} + 1460 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \)
17 \( ( 1 - 5 T + 33 T^{2} - 5 T^{3} + 164 T^{4} - 5 p T^{5} + 33 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 - 67 T^{2} + 2303 T^{4} - 60089 T^{6} + 1279480 T^{8} - 60089 p^{2} T^{10} + 2303 p^{4} T^{12} - 67 p^{6} T^{14} + p^{8} T^{16} \)
23 \( 1 - 96 T^{2} + 5212 T^{4} - 191648 T^{6} + 5108870 T^{8} - 191648 p^{2} T^{10} + 5212 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - T + 37 T^{2} + 217 T^{3} + 340 T^{4} + 217 p T^{5} + 37 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 11 T + 105 T^{2} + 559 T^{3} + 3504 T^{4} + 559 p T^{5} + 105 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 12 T + 152 T^{2} + 1140 T^{3} + 8686 T^{4} + 1140 p T^{5} + 152 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + T + 105 T^{2} + 29 T^{3} + 5244 T^{4} + 29 p T^{5} + 105 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 231 T^{2} + 27167 T^{4} - 2024653 T^{6} + 103981560 T^{8} - 2024653 p^{2} T^{10} + 27167 p^{4} T^{12} - 231 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 164 T^{2} + 15332 T^{4} - 1066012 T^{6} + 57811510 T^{8} - 1066012 p^{2} T^{10} + 15332 p^{4} T^{12} - 164 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 239 T^{2} + 30907 T^{4} - 2622557 T^{6} + 162048160 T^{8} - 2622557 p^{2} T^{10} + 30907 p^{4} T^{12} - 239 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 - 270 T^{2} + 36259 T^{4} - 3206780 T^{6} + 213479381 T^{8} - 3206780 p^{2} T^{10} + 36259 p^{4} T^{12} - 270 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 - 276 T^{2} + 30260 T^{4} - 1762444 T^{6} + 87429174 T^{8} - 1762444 p^{2} T^{10} + 30260 p^{4} T^{12} - 276 p^{6} T^{14} + p^{8} T^{16} \)
67 \( ( 1 + T + 119 T^{2} - 83 T^{3} + 10044 T^{4} - 83 p T^{5} + 119 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 184 T^{2} + 16126 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( 1 - 494 T^{2} + 111827 T^{4} - 15171772 T^{6} + 1350802885 T^{8} - 15171772 p^{2} T^{10} + 111827 p^{4} T^{12} - 494 p^{6} T^{14} + p^{8} T^{16} \)
79 \( 1 - 475 T^{2} + 104239 T^{4} - 14117865 T^{6} + 1320054056 T^{8} - 14117865 p^{2} T^{10} + 104239 p^{4} T^{12} - 475 p^{6} T^{14} + p^{8} T^{16} \)
83 \( ( 1 + 12 T + 301 T^{2} + 2436 T^{3} + 35329 T^{4} + 2436 p T^{5} + 301 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 - 199 T^{2} + 19855 T^{4} - 1662361 T^{6} + 149152624 T^{8} - 1662361 p^{2} T^{10} + 19855 p^{4} T^{12} - 199 p^{6} T^{14} + p^{8} T^{16} \)
97 \( ( 1 + 8 T + 257 T^{2} + 1400 T^{3} + 29601 T^{4} + 1400 p T^{5} + 257 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−4.61099506270485685729187165177, −4.35331098569277353352717944262, −4.31436241867709474988136694276, −4.16532871752659799812924172134, −4.07728250722650091309049147791, −3.87556160608518483312448653294, −3.80353357746365190393671499639, −3.61575239278200694378618781046, −3.34279108470773028754301936867, −3.33337201170104550173741203552, −3.27673357439540325245337484878, −3.27569002298397889986025205785, −3.06004005621262161440130314929, −2.72605839960397382238700184203, −2.65544240340962970931578478285, −2.63432389226631725122018819944, −2.10785082707966588082275502888, −1.99134677491700843486930906959, −1.96539019258681578999464644157, −1.89787412843579363360305401235, −1.46281983040940634085033616522, −1.38299680249037258922503359948, −1.00150344755662047388026479959, −0.996058649031943055801998937590, −0.61799895706812870963643006890, 0.61799895706812870963643006890, 0.996058649031943055801998937590, 1.00150344755662047388026479959, 1.38299680249037258922503359948, 1.46281983040940634085033616522, 1.89787412843579363360305401235, 1.96539019258681578999464644157, 1.99134677491700843486930906959, 2.10785082707966588082275502888, 2.63432389226631725122018819944, 2.65544240340962970931578478285, 2.72605839960397382238700184203, 3.06004005621262161440130314929, 3.27569002298397889986025205785, 3.27673357439540325245337484878, 3.33337201170104550173741203552, 3.34279108470773028754301936867, 3.61575239278200694378618781046, 3.80353357746365190393671499639, 3.87556160608518483312448653294, 4.07728250722650091309049147791, 4.16532871752659799812924172134, 4.31436241867709474988136694276, 4.35331098569277353352717944262, 4.61099506270485685729187165177

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

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