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Properties

Label 16-3332e8-1.1-c0e8-0-3
Degree $16$
Conductor $1.519\times 10^{28}$
Sign $1$
Analytic cond. $58.4651$
Root an. cond. $1.28952$
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  − 2·2-s + 4-s + 2·9-s − 4·18-s + 2·25-s + 2·32-s + 2·36-s − 4·50-s − 4·53-s − 4·64-s + 81-s + 2·100-s + 8·106-s − 8·121-s + 127-s + 2·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 2·162-s + 163-s + 167-s − 8·169-s + 173-s + ⋯
L(s)  = 1  − 2·2-s + 4-s + 2·9-s − 4·18-s + 2·25-s + 2·32-s + 2·36-s − 4·50-s − 4·53-s − 4·64-s + 81-s + 2·100-s + 8·106-s − 8·121-s + 127-s + 2·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 2·162-s + 163-s + 167-s − 8·169-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 7^{16} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(58.4651\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 7^{16} \cdot 17^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2448983451\)
\(L(\frac12)\) \(\approx\) \(0.2448983451\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
7 \( 1 \)
17 \( ( 1 + T^{2} )^{4} \)
good3 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
5 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
11 \( ( 1 + T^{2} )^{8} \)
13 \( ( 1 + T^{2} )^{8} \)
19 \( ( 1 - T )^{8}( 1 + T )^{8} \)
23 \( ( 1 + T^{2} )^{8} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
37 \( ( 1 - T )^{8}( 1 + T )^{8} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
47 \( ( 1 - T )^{8}( 1 + T )^{8} \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
59 \( ( 1 - T )^{8}( 1 + T )^{8} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
79 \( ( 1 + T^{2} )^{8} \)
83 \( ( 1 - T )^{8}( 1 + T )^{8} \)
89 \( ( 1 + T^{2} )^{8} \)
97 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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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

−3.77635662762562764657124836406, −3.75827565334714500279100061670, −3.71170158658171454974450518195, −3.42849716697419220056990482517, −3.35844822315962800705005349867, −3.01440043748345871667891083389, −2.97023712705228218343347982686, −2.93468934792038077756834811203, −2.92359918632938164209394546321, −2.63665263051126010664007305395, −2.48010584139227378246523420455, −2.47523039157240254767747154150, −2.46470683101953460461444464079, −2.19560185736846533075566238329, −2.08121405490845365240587063667, −1.87578178707204606551755902552, −1.54086825200268911134098121986, −1.42701142835708391599590223461, −1.39310252268959699561528607274, −1.31954123019773204741349016328, −1.25850729758525555311915982887, −1.15708969489113269222637803072, −0.925432228531517379927853493944, −0.41153908255671764986959645631, −0.31735805659882902828967360592, 0.31735805659882902828967360592, 0.41153908255671764986959645631, 0.925432228531517379927853493944, 1.15708969489113269222637803072, 1.25850729758525555311915982887, 1.31954123019773204741349016328, 1.39310252268959699561528607274, 1.42701142835708391599590223461, 1.54086825200268911134098121986, 1.87578178707204606551755902552, 2.08121405490845365240587063667, 2.19560185736846533075566238329, 2.46470683101953460461444464079, 2.47523039157240254767747154150, 2.48010584139227378246523420455, 2.63665263051126010664007305395, 2.92359918632938164209394546321, 2.93468934792038077756834811203, 2.97023712705228218343347982686, 3.01440043748345871667891083389, 3.35844822315962800705005349867, 3.42849716697419220056990482517, 3.71170158658171454974450518195, 3.75827565334714500279100061670, 3.77635662762562764657124836406

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

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