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Properties

Label 16-50e16-1.1-c0e8-0-0
Degree $16$
Conductor $1.526\times 10^{27}$
Sign $1$
Analytic cond. $5.87187$
Root an. cond. $1.11698$
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-s + 2·9-s − 6·29-s + 2·36-s + 6·41-s − 8·49-s + 6·61-s + 81-s + 4·89-s − 4·101-s − 6·109-s − 6·116-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 6·164-s + 167-s − 3·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 4-s + 2·9-s − 6·29-s + 2·36-s + 6·41-s − 8·49-s + 6·61-s + 81-s + 4·89-s − 4·101-s − 6·109-s − 6·116-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 6·164-s + 167-s − 3·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(5.87187\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 5^{32} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.410237948\)
\(L(\frac12)\) \(\approx\) \(1.410237948\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.91606213651111719508366354945, −3.87972867293138964735737782127, −3.69611484380375774875207772922, −3.68240626458517311079111153570, −3.68114378799668163629428273763, −3.29322901309741937730363633878, −3.16087552154813809890806011833, −3.11482969979624526931686863808, −2.96097112052332881223790415217, −2.82982317526351385012324958370, −2.81201456969045353424007726141, −2.51156753555057922610181773711, −2.42315183133040485919879929671, −2.19477488653037964372345342282, −2.13760148543456719917480459789, −2.10960781444890083213953858315, −1.85697905770701417153395102395, −1.82712676092231839045379489486, −1.70052029853804210968613347936, −1.49698339769286531086402846576, −1.37358505941803403156758882708, −1.10799447974058672047951640002, −1.05011273037368801703507147097, −0.827590008240657589550111364329, −0.31092466616925168089455889371, 0.31092466616925168089455889371, 0.827590008240657589550111364329, 1.05011273037368801703507147097, 1.10799447974058672047951640002, 1.37358505941803403156758882708, 1.49698339769286531086402846576, 1.70052029853804210968613347936, 1.82712676092231839045379489486, 1.85697905770701417153395102395, 2.10960781444890083213953858315, 2.13760148543456719917480459789, 2.19477488653037964372345342282, 2.42315183133040485919879929671, 2.51156753555057922610181773711, 2.81201456969045353424007726141, 2.82982317526351385012324958370, 2.96097112052332881223790415217, 3.11482969979624526931686863808, 3.16087552154813809890806011833, 3.29322901309741937730363633878, 3.68114378799668163629428273763, 3.68240626458517311079111153570, 3.69611484380375774875207772922, 3.87972867293138964735737782127, 3.91606213651111719508366354945

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

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