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Properties

Label 16-90e8-1.1-c1e8-0-1
Degree $16$
Conductor $4.305\times 10^{15}$
Sign $1$
Analytic cond. $0.0711470$
Root an. cond. $0.847734$
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  + 4·3-s + 12·5-s − 8·7-s + 8·9-s − 24·11-s + 48·15-s + 16-s − 32·21-s + 74·25-s + 8·27-s + 8·31-s − 96·33-s − 96·35-s − 24·37-s + 12·41-s + 96·45-s + 4·48-s + 32·49-s − 288·55-s + 12·61-s − 64·63-s + 4·67-s − 16·73-s + 296·75-s + 192·77-s + 12·80-s − 7·81-s + ⋯
L(s)  = 1  + 2.30·3-s + 5.36·5-s − 3.02·7-s + 8/3·9-s − 7.23·11-s + 12.3·15-s + 1/4·16-s − 6.98·21-s + 74/5·25-s + 1.53·27-s + 1.43·31-s − 16.7·33-s − 16.2·35-s − 3.94·37-s + 1.87·41-s + 14.3·45-s + 0.577·48-s + 32/7·49-s − 38.8·55-s + 1.53·61-s − 8.06·63-s + 0.488·67-s − 1.87·73-s + 34.1·75-s + 21.8·77-s + 1.34·80-s − 7/9·81-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(0.0711470\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.902347850\)
\(L(\frac12)\) \(\approx\) \(1.902347850\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
3 \( 1 - 4 T + 8 T^{2} - 8 T^{3} + 7 T^{4} - 8 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
5 \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
good7 \( 1 + 8 T + 32 T^{2} + 64 T^{3} - p T^{4} - 464 T^{5} - 1440 T^{6} - 2472 T^{7} - 4016 T^{8} - 2472 p T^{9} - 1440 p^{2} T^{10} - 464 p^{3} T^{11} - p^{5} T^{12} + 64 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
11 \( ( 1 + 12 T + 74 T^{2} + 312 T^{3} + 1083 T^{4} + 312 p T^{5} + 74 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 + 142 T^{4} - 8397 T^{8} + 142 p^{4} T^{12} + p^{8} T^{16} \)
17 \( 1 + 188 T^{4} - 45306 T^{8} + 188 p^{4} T^{12} + p^{8} T^{16} \)
19 \( ( 1 - 32 T^{2} + 594 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( 1 - 967 T^{4} + 655248 T^{8} - 967 p^{4} T^{12} + p^{8} T^{16} \)
29 \( 1 - 106 T^{2} + 6769 T^{4} - 295210 T^{6} + 9797332 T^{8} - 295210 p^{2} T^{10} + 6769 p^{4} T^{12} - 106 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 - 4 T - 44 T^{2} + 8 T^{3} + 2143 T^{4} + 8 p T^{5} - 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 6 T + 65 T^{2} - 318 T^{3} + 1620 T^{4} - 318 p T^{5} + 65 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 1778 T^{4} - 257517 T^{8} - 1778 p^{4} T^{12} + p^{8} T^{16} \)
47 \( 1 + 4249 T^{4} + 13174320 T^{8} + 4249 p^{4} T^{12} + p^{8} T^{16} \)
53 \( 1 - 4900 T^{4} + 13820838 T^{8} - 4900 p^{4} T^{12} + p^{8} T^{16} \)
59 \( 1 - 16 T^{2} - 5906 T^{4} + 12800 T^{6} + 24961747 T^{8} + 12800 p^{2} T^{10} - 5906 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 - 6 T - 89 T^{2} - 18 T^{3} + 9708 T^{4} - 18 p T^{5} - 89 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 4 T + 8 T^{2} + 304 T^{3} - 3511 T^{4} + 10624 T^{5} + 31800 T^{6} - 1654308 T^{7} - 4277600 T^{8} - 1654308 p T^{9} + 31800 p^{2} T^{10} + 10624 p^{3} T^{11} - 3511 p^{4} T^{12} + 304 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
71 \( ( 1 - 244 T^{2} + 24582 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 8 T + 32 T^{2} + 264 T^{3} + 578 T^{4} + 264 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 152 T^{2} + 16863 T^{4} + 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( 1 - 12 T + 72 T^{2} - 288 T^{3} - 8375 T^{4} + 56640 T^{5} - 35208 T^{6} - 3433884 T^{7} + 82789344 T^{8} - 3433884 p T^{9} - 35208 p^{2} T^{10} + 56640 p^{3} T^{11} - 8375 p^{4} T^{12} - 288 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 + 286 T^{2} + 35427 T^{4} + 286 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( 1 - 12 T + 72 T^{2} + 744 T^{3} - 12542 T^{4} + 57996 T^{5} + 483840 T^{6} - 14345892 T^{7} + 128096019 T^{8} - 14345892 p T^{9} + 483840 p^{2} T^{10} + 57996 p^{3} T^{11} - 12542 p^{4} T^{12} + 744 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
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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

−6.79174937133085905782558528668, −6.37784630758992714555097048811, −6.36111826115680485824442115002, −5.83392848652815130925357164227, −5.82559861955877040318123449483, −5.80845260597834299632988121465, −5.74882234926733652262896832916, −5.71272174359680223929437924893, −5.64038994306734883677943244563, −4.95278368525193385784889916190, −4.89563015544137714844409378790, −4.85865416446699825093350710311, −4.80515888137985132870733706611, −4.69661118540076006162861913531, −3.66838064547409474418430929729, −3.48634924429859908949961157423, −3.45443420619949451364967755639, −3.21760879772961501756350908526, −2.85018954803007261721190497747, −2.73670241054487457635893304344, −2.55617330786342676836139657284, −2.31256053256784956475585516413, −2.25654140609901535549447980407, −2.24765046214370218406016682020, −1.74013925689599066407174594820, 1.74013925689599066407174594820, 2.24765046214370218406016682020, 2.25654140609901535549447980407, 2.31256053256784956475585516413, 2.55617330786342676836139657284, 2.73670241054487457635893304344, 2.85018954803007261721190497747, 3.21760879772961501756350908526, 3.45443420619949451364967755639, 3.48634924429859908949961157423, 3.66838064547409474418430929729, 4.69661118540076006162861913531, 4.80515888137985132870733706611, 4.85865416446699825093350710311, 4.89563015544137714844409378790, 4.95278368525193385784889916190, 5.64038994306734883677943244563, 5.71272174359680223929437924893, 5.74882234926733652262896832916, 5.80845260597834299632988121465, 5.82559861955877040318123449483, 5.83392848652815130925357164227, 6.36111826115680485824442115002, 6.37784630758992714555097048811, 6.79174937133085905782558528668

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

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