Properties

Label 16-595e8-1.1-c0e8-0-1
Degree $16$
Conductor $1.571\times 10^{22}$
Sign $1$
Analytic cond. $6.04495\times 10^{-5}$
Root an. cond. $0.544925$
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  − 2·4-s + 2·9-s + 16-s + 25-s − 4·36-s − 4·49-s + 81-s − 2·100-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 8·196-s + 197-s + ⋯
L(s)  = 1  − 2·4-s + 2·9-s + 16-s + 25-s − 4·36-s − 4·49-s + 81-s − 2·100-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 8·196-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{8} \cdot 7^{8} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(6.04495\times 10^{-5}\)
Root analytic conductor: \(0.544925\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{8} \cdot 7^{8} \cdot 17^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2659127280\)
\(L(\frac12)\) \(\approx\) \(0.2659127280\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
7 \( ( 1 + T^{2} )^{4} \)
17 \( ( 1 + T^{2} )^{4} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
3 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
11 \( ( 1 - T )^{8}( 1 + T )^{8} \)
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 \( ( 1 + T^{2} )^{8} \)
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 \( ( 1 + T^{2} )^{8} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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 \( ( 1 - T )^{8}( 1 + T )^{8} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
79 \( ( 1 - T )^{8}( 1 + T )^{8} \)
83 \( ( 1 + T^{2} )^{8} \)
89 \( ( 1 - T )^{8}( 1 + T )^{8} \)
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

−4.88525627028872056246986318221, −4.72549066111621973516151632524, −4.70488887014552577821441399093, −4.59917460442326889549399134182, −4.41344168131367716501653997964, −4.20880031637745118566691709463, −4.19559719899659002439716997438, −4.04179783920250744786937520009, −3.88479226252338813219282300852, −3.74503393369097399676145947786, −3.60604730450667242401418910635, −3.29685507612423088611875851878, −3.18011851165176019413257849082, −3.11460186069803654669947098457, −3.08777722823703857913177828321, −2.80079153919391625472958966902, −2.65436942874941726731857832039, −2.31779773827848594370283607151, −1.92323250903807836762478498063, −1.88100127907639550668405239851, −1.85398317211527482765458946232, −1.79079941191697797368039226207, −1.26116887579317186780080196876, −0.972231739966072897882836686047, −0.910485245674178209221107723569, 0.910485245674178209221107723569, 0.972231739966072897882836686047, 1.26116887579317186780080196876, 1.79079941191697797368039226207, 1.85398317211527482765458946232, 1.88100127907639550668405239851, 1.92323250903807836762478498063, 2.31779773827848594370283607151, 2.65436942874941726731857832039, 2.80079153919391625472958966902, 3.08777722823703857913177828321, 3.11460186069803654669947098457, 3.18011851165176019413257849082, 3.29685507612423088611875851878, 3.60604730450667242401418910635, 3.74503393369097399676145947786, 3.88479226252338813219282300852, 4.04179783920250744786937520009, 4.19559719899659002439716997438, 4.20880031637745118566691709463, 4.41344168131367716501653997964, 4.59917460442326889549399134182, 4.70488887014552577821441399093, 4.72549066111621973516151632524, 4.88525627028872056246986318221

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

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