Properties

Label 16-42e16-1.1-c0e8-0-0
Degree $16$
Conductor $9.375\times 10^{25}$
Sign $1$
Analytic cond. $0.360782$
Root an. cond. $0.938270$
Motivic weight $0$
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  − 4·4-s + 10·16-s − 20·64-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯
L(s)  = 1  − 4·4-s + 10·16-s − 20·64-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(0.360782\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1598330390\)
\(L(\frac12)\) \(\approx\) \(0.1598330390\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{4} \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 + T^{8} )^{2} \)
11 \( ( 1 + T^{2} )^{8} \)
13 \( ( 1 + T^{8} )^{2} \)
17 \( ( 1 + T^{8} )^{2} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 + T^{2} )^{8} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 + T^{2} )^{8} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 + T^{8} )^{2} \)
43 \( ( 1 - T )^{8}( 1 + T )^{8} \)
47 \( ( 1 - T )^{8}( 1 + T )^{8} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 - T )^{8}( 1 + T )^{8} \)
61 \( ( 1 + T^{8} )^{2} \)
67 \( ( 1 - T )^{8}( 1 + T )^{8} \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 + T^{8} )^{2} \)
79 \( ( 1 - T )^{8}( 1 + T )^{8} \)
83 \( ( 1 - T )^{8}( 1 + T )^{8} \)
89 \( ( 1 + T^{8} )^{2} \)
97 \( ( 1 + T^{8} )^{2} \)
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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

−4.21614727837055322269356924867, −4.14767984974902007584685434526, −3.85349063105782053290624673275, −3.79051453753473926564416174066, −3.78761371303869051775689980247, −3.59292354269028910237545565392, −3.57383524391651644464231861577, −3.40488846095353589649218946965, −3.11494875182472100248204663555, −3.08092458944129551683799876104, −2.98548705900688903005765360359, −2.87467717383615755648803791688, −2.67107320136561336442808555938, −2.62139492364066400511973267529, −2.30140401196221761589119658538, −2.22472491269030609607377519732, −1.94654061436474676738336277594, −1.91706041128369932223437314601, −1.52861518672177461537377006355, −1.42032917235380558682707499699, −1.28862775764033900154475398541, −1.11350700702207569549548662926, −1.01789687975060917556622782686, −0.69618160730337280507733034771, −0.26632204724506199773764490625, 0.26632204724506199773764490625, 0.69618160730337280507733034771, 1.01789687975060917556622782686, 1.11350700702207569549548662926, 1.28862775764033900154475398541, 1.42032917235380558682707499699, 1.52861518672177461537377006355, 1.91706041128369932223437314601, 1.94654061436474676738336277594, 2.22472491269030609607377519732, 2.30140401196221761589119658538, 2.62139492364066400511973267529, 2.67107320136561336442808555938, 2.87467717383615755648803791688, 2.98548705900688903005765360359, 3.08092458944129551683799876104, 3.11494875182472100248204663555, 3.40488846095353589649218946965, 3.57383524391651644464231861577, 3.59292354269028910237545565392, 3.78761371303869051775689980247, 3.79051453753473926564416174066, 3.85349063105782053290624673275, 4.14767984974902007584685434526, 4.21614727837055322269356924867

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

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