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Properties

Label 16-3072e8-1.1-c0e8-0-3
Degree $16$
Conductor $7.932\times 10^{27}$
Sign $1$
Analytic cond. $30.5229$
Root an. cond. $1.23819$
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  + 8·13-s − 8·73-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-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 + 251-s + ⋯
L(s)  = 1  + 8·13-s − 8·73-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-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 + 251-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{80} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(30.5229\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{80} \cdot 3^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.330169980\)
\(L(\frac12)\) \(\approx\) \(5.330169980\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.77517641070543661383595041347, −3.66951290686587161244252711823, −3.60372134981667828154407606084, −3.52984534232785803063335439413, −3.38655850740737544284869700820, −3.16539024848146181297138486773, −3.12299953769093333479382334896, −3.03634823113941890618423380034, −2.98714584168916666603824810032, −2.91434562437788430552093691004, −2.78796803851887883902450408014, −2.72379420524008302338984415808, −2.31842780243654169438872265015, −2.14296035011328333502575124918, −1.92764796764071932154030066015, −1.81251205300347804146669034936, −1.75808958739044999023835972064, −1.66989907582111133712343168679, −1.52827055133606254980378963983, −1.43064534959063924852575815594, −1.36232259967031122454416315607, −1.02538880827801870979427363888, −0.860012202235964090556655836363, −0.74707294803367714676339572193, −0.71570560363981475023299691122, 0.71570560363981475023299691122, 0.74707294803367714676339572193, 0.860012202235964090556655836363, 1.02538880827801870979427363888, 1.36232259967031122454416315607, 1.43064534959063924852575815594, 1.52827055133606254980378963983, 1.66989907582111133712343168679, 1.75808958739044999023835972064, 1.81251205300347804146669034936, 1.92764796764071932154030066015, 2.14296035011328333502575124918, 2.31842780243654169438872265015, 2.72379420524008302338984415808, 2.78796803851887883902450408014, 2.91434562437788430552093691004, 2.98714584168916666603824810032, 3.03634823113941890618423380034, 3.12299953769093333479382334896, 3.16539024848146181297138486773, 3.38655850740737544284869700820, 3.52984534232785803063335439413, 3.60372134981667828154407606084, 3.66951290686587161244252711823, 3.77517641070543661383595041347

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

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