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Properties

Label 16-495e8-1.1-c2e8-0-0
Degree $16$
Conductor $3.604\times 10^{21}$
Sign $1$
Analytic cond. $1.09527\times 10^{9}$
Root an. cond. $3.67257$
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  + 2·4-s − 8·11-s − 4·16-s − 8·23-s + 20·25-s + 36·31-s − 88·37-s − 16·44-s + 8·47-s + 282·49-s + 152·53-s + 16·59-s − 50·64-s − 88·67-s − 276·71-s − 444·89-s − 16·92-s + 312·97-s + 40·100-s + 448·103-s + 272·113-s + 16·121-s + 72·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.727·11-s − 1/4·16-s − 0.347·23-s + 4/5·25-s + 1.16·31-s − 2.37·37-s − 0.363·44-s + 8/47·47-s + 5.75·49-s + 2.86·53-s + 0.271·59-s − 0.781·64-s − 1.31·67-s − 3.88·71-s − 4.98·89-s − 0.173·92-s + 3.21·97-s + 2/5·100-s + 4.34·103-s + 2.40·113-s + 0.132·121-s + 0.580·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 5^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(1.09527\times 10^{9}\)
Root analytic conductor: \(3.67257\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(12.39580495\)
\(L(\frac12)\) \(\approx\) \(12.39580495\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 - p T^{2} )^{4} \)
11 \( 1 + 8 T + 48 T^{2} + 1816 T^{3} + 2170 p T^{4} + 1816 p^{2} T^{5} + 48 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \)
good2 \( 1 - p T^{2} + p^{3} T^{4} + 13 p T^{6} + 175 T^{8} + 13 p^{5} T^{10} + p^{11} T^{12} - p^{13} T^{14} + p^{16} T^{16} \)
7 \( 1 - 282 T^{2} + 39113 T^{4} - 484042 p T^{6} + 199414620 T^{8} - 484042 p^{5} T^{10} + 39113 p^{8} T^{12} - 282 p^{12} T^{14} + p^{16} T^{16} \)
13 \( 1 - 732 T^{2} + 280628 T^{4} - 74620324 T^{6} + 14654347350 T^{8} - 74620324 p^{4} T^{10} + 280628 p^{8} T^{12} - 732 p^{12} T^{14} + p^{16} T^{16} \)
17 \( 1 - 1042 T^{2} + 438593 T^{4} - 106418574 T^{6} + 24599395500 T^{8} - 106418574 p^{4} T^{10} + 438593 p^{8} T^{12} - 1042 p^{12} T^{14} + p^{16} T^{16} \)
19 \( 1 - 42 p T^{2} + 321833 T^{4} - 3361714 p T^{6} + 7282824180 T^{8} - 3361714 p^{5} T^{10} + 321833 p^{8} T^{12} - 42 p^{13} T^{14} + p^{16} T^{16} \)
23 \( ( 1 + 4 T + 64 p T^{2} - 2628 T^{3} + 970110 T^{4} - 2628 p^{2} T^{5} + 64 p^{5} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
29 \( 1 - 1878 T^{2} + 1240433 T^{4} - 1331346766 T^{6} + 1718347166340 T^{8} - 1331346766 p^{4} T^{10} + 1240433 p^{8} T^{12} - 1878 p^{12} T^{14} + p^{16} T^{16} \)
31 \( ( 1 - 18 T + 2893 T^{2} - 39406 T^{3} + 3891380 T^{4} - 39406 p^{2} T^{5} + 2893 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( ( 1 + 44 T + 4047 T^{2} + 92212 T^{3} + 6332160 T^{4} + 92212 p^{2} T^{5} + 4047 p^{4} T^{6} + 44 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( 1 - 3928 T^{2} + 8657788 T^{4} - 13161706216 T^{6} + 22087580007430 T^{8} - 13161706216 p^{4} T^{10} + 8657788 p^{8} T^{12} - 3928 p^{12} T^{14} + p^{16} T^{16} \)
43 \( 1 - 10612 T^{2} + 49946708 T^{4} - 144004303244 T^{6} + 301800503326870 T^{8} - 144004303244 p^{4} T^{10} + 49946708 p^{8} T^{12} - 10612 p^{12} T^{14} + p^{16} T^{16} \)
47 \( ( 1 - 4 T + 7952 T^{2} - 34012 T^{3} + 25394910 T^{4} - 34012 p^{2} T^{5} + 7952 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
53 \( ( 1 - 76 T + 4567 T^{2} + 53092 T^{3} - 4224800 T^{4} + 53092 p^{2} T^{5} + 4567 p^{4} T^{6} - 76 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( ( 1 - 8 T + 5608 T^{2} - 363016 T^{3} + 14342830 T^{4} - 363016 p^{2} T^{5} + 5608 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
61 \( 1 - 12118 T^{2} + 72935393 T^{4} - 278857308126 T^{6} + 965571255463620 T^{8} - 278857308126 p^{4} T^{10} + 72935393 p^{8} T^{12} - 12118 p^{12} T^{14} + p^{16} T^{16} \)
67 \( ( 1 + 44 T + 7172 T^{2} + 297652 T^{3} + 27585430 T^{4} + 297652 p^{2} T^{5} + 7172 p^{4} T^{6} + 44 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( ( 1 + 138 T + 23333 T^{2} + 2101146 T^{3} + 184397020 T^{4} + 2101146 p^{2} T^{5} + 23333 p^{4} T^{6} + 138 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
73 \( 1 - 28732 T^{2} + 396348148 T^{4} - 3478062610564 T^{6} + 21656687866372630 T^{8} - 3478062610564 p^{4} T^{10} + 396348148 p^{8} T^{12} - 28732 p^{12} T^{14} + p^{16} T^{16} \)
79 \( 1 - 22168 T^{2} + 277375868 T^{4} - 2588893088936 T^{6} + 18518155319277190 T^{8} - 2588893088936 p^{4} T^{10} + 277375868 p^{8} T^{12} - 22168 p^{12} T^{14} + p^{16} T^{16} \)
83 \( 1 - 24412 T^{2} + 319401268 T^{4} - 3205385375524 T^{6} + 25331444058804310 T^{8} - 3205385375524 p^{4} T^{10} + 319401268 p^{8} T^{12} - 24412 p^{12} T^{14} + p^{16} T^{16} \)
89 \( ( 1 + 222 T + 537 p T^{2} + 5639874 T^{3} + 630383340 T^{4} + 5639874 p^{2} T^{5} + 537 p^{5} T^{6} + 222 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( ( 1 - 156 T + 30372 T^{2} - 3221028 T^{3} + 373391670 T^{4} - 3221028 p^{2} T^{5} + 30372 p^{4} T^{6} - 156 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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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

−4.53985621002702611629859065574, −4.39237462785816519579392607356, −4.37352892424082976281470255524, −4.20527144780901550505762135830, −4.17581641568749391628854019771, −3.79597775171374147993524280849, −3.62319435334941134350903612329, −3.46489247708967602147962932452, −3.38589554483771870060380198214, −3.23868259524818638796454217931, −3.04878144447615942067049996525, −2.91636668463908913492969995713, −2.73710652915622440563936196918, −2.46039126632138804675757907817, −2.42755460410834865980142613860, −2.31405122229308949955878359175, −2.05376317456041291869428648820, −1.83427188841801982760748063748, −1.64992030666646784730653112112, −1.58848897156566526769056442303, −1.16617965972290259930738647068, −0.873591727806638744069445553638, −0.64248931511288902672818306330, −0.48162209382682485533786848987, −0.40866058028383835137466096013, 0.40866058028383835137466096013, 0.48162209382682485533786848987, 0.64248931511288902672818306330, 0.873591727806638744069445553638, 1.16617965972290259930738647068, 1.58848897156566526769056442303, 1.64992030666646784730653112112, 1.83427188841801982760748063748, 2.05376317456041291869428648820, 2.31405122229308949955878359175, 2.42755460410834865980142613860, 2.46039126632138804675757907817, 2.73710652915622440563936196918, 2.91636668463908913492969995713, 3.04878144447615942067049996525, 3.23868259524818638796454217931, 3.38589554483771870060380198214, 3.46489247708967602147962932452, 3.62319435334941134350903612329, 3.79597775171374147993524280849, 4.17581641568749391628854019771, 4.20527144780901550505762135830, 4.37352892424082976281470255524, 4.39237462785816519579392607356, 4.53985621002702611629859065574

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

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