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Properties

Label 16-325e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.245\times 10^{20}$
Sign $1$
Analytic cond. $4.78987\times 10^{-7}$
Root an. cond. $0.402735$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·11-s − 16-s + 4·41-s − 4·61-s − 4·71-s − 81-s − 4·101-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 4·11-s − 16-s + 4·41-s − 4·61-s − 4·71-s − 81-s − 4·101-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{16} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(4.78987\times 10^{-7}\)
Root analytic conductor: \(0.402735\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{16} \cdot 13^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1149756023\)
\(L(\frac12)\) \(\approx\) \(0.1149756023\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( ( 1 + T^{4} )^{2} \)
good2 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
3 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
7 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
11 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
17 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
19 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
23 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
29 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
31 \( ( 1 + T^{2} )^{8} \)
37 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
41 \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \)
43 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
47 \( ( 1 + T^{4} )^{4} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
61 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
67 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
71 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
73 \( ( 1 + T^{4} )^{4} \)
79 \( ( 1 - T )^{8}( 1 + T )^{8} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
97 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
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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

−5.65959708059753166873739952448, −5.45502813956234140769918045969, −4.97428864036434284197320686042, −4.96342122420024193220576377386, −4.92200513007803894603375605824, −4.89396078126619507131035953431, −4.50121901249456491606044007286, −4.49912473471969393278016859786, −4.32208143504889964677943462055, −4.20004412422117194119859308834, −4.14829383122256941801871407459, −3.85243823344154307243914425099, −3.52621030662695976480044763924, −3.47397931275578530169078746211, −3.05729677438406405111249475696, −2.97143085763957833404969973555, −2.87059337366895051463651295716, −2.81826998948620157783634034250, −2.56928329808188051150328876854, −2.46645413990232415928067373333, −2.32021644006484557558852550951, −1.82800012338587602132834440354, −1.77247027103046769908399431599, −1.50316105702453525715934316855, −1.01143469305777508276367511293, 1.01143469305777508276367511293, 1.50316105702453525715934316855, 1.77247027103046769908399431599, 1.82800012338587602132834440354, 2.32021644006484557558852550951, 2.46645413990232415928067373333, 2.56928329808188051150328876854, 2.81826998948620157783634034250, 2.87059337366895051463651295716, 2.97143085763957833404969973555, 3.05729677438406405111249475696, 3.47397931275578530169078746211, 3.52621030662695976480044763924, 3.85243823344154307243914425099, 4.14829383122256941801871407459, 4.20004412422117194119859308834, 4.32208143504889964677943462055, 4.49912473471969393278016859786, 4.50121901249456491606044007286, 4.89396078126619507131035953431, 4.92200513007803894603375605824, 4.96342122420024193220576377386, 4.97428864036434284197320686042, 5.45502813956234140769918045969, 5.65959708059753166873739952448

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

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