Properties

Label 16-448e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.623\times 10^{21}$
Sign $1$
Analytic cond. $6.24426\times 10^{-6}$
Root an. cond. $0.472843$
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  − 8·67-s + 8·107-s − 8·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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  − 8·67-s + 8·107-s − 8·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(6.24426\times 10^{-6}\)
Root analytic conductor: \(0.472843\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{448} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2575713407\)
\(L(\frac12)\) \(\approx\) \(0.2575713407\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{8} \)
7 \( 1 + T^{8} \)
good3 \( 1 + T^{16} \)
5 \( 1 + T^{16} \)
11 \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \)
13 \( 1 + T^{16} \)
17 \( ( 1 + T^{4} )^{4} \)
19 \( 1 + T^{16} \)
23 \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \)
29 \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \)
31 \( ( 1 + T^{2} )^{8} \)
37 \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \)
41 \( ( 1 + T^{8} )^{2} \)
43 \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \)
47 \( ( 1 + T^{4} )^{4} \)
53 \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \)
59 \( 1 + T^{16} \)
61 \( 1 + T^{16} \)
67 \( ( 1 + T )^{8}( 1 + T^{8} ) \)
71 \( ( 1 + T^{8} )^{2} \)
73 \( ( 1 + T^{8} )^{2} \)
79 \( ( 1 + T^{8} )^{2} \)
83 \( 1 + T^{16} \)
89 \( ( 1 + T^{8} )^{2} \)
97 \( ( 1 + T^{2} )^{8} \)
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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

−5.17860359059096523011820989178, −5.12654783881691722145337481504, −4.77629470737715626755322504908, −4.63911563441289088642065602172, −4.60794267644005067742792906239, −4.51482934039070637162074868623, −4.45594372290695617134425682395, −4.13307241291556885672749716365, −4.07931343259714838553717245036, −3.85016667805516508285764558040, −3.69589403950151495664122088322, −3.66916154241648517205810318620, −3.29279849857578592394624745439, −3.08755972594004015755799895107, −3.01039583882603933707881056376, −2.98859299644020148231431846203, −2.94726948009999986478979072023, −2.51143894717075304928801970058, −2.30170510873943331393055199239, −2.21115274458941097455014775859, −1.89517604791095561611950382313, −1.80633320806324638069443438443, −1.37792746542501400546364602219, −1.27416153341040169943284816662, −1.12583501653622014575534085598, 1.12583501653622014575534085598, 1.27416153341040169943284816662, 1.37792746542501400546364602219, 1.80633320806324638069443438443, 1.89517604791095561611950382313, 2.21115274458941097455014775859, 2.30170510873943331393055199239, 2.51143894717075304928801970058, 2.94726948009999986478979072023, 2.98859299644020148231431846203, 3.01039583882603933707881056376, 3.08755972594004015755799895107, 3.29279849857578592394624745439, 3.66916154241648517205810318620, 3.69589403950151495664122088322, 3.85016667805516508285764558040, 4.07931343259714838553717245036, 4.13307241291556885672749716365, 4.45594372290695617134425682395, 4.51482934039070637162074868623, 4.60794267644005067742792906239, 4.63911563441289088642065602172, 4.77629470737715626755322504908, 5.12654783881691722145337481504, 5.17860359059096523011820989178

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

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