Properties

Label 16-1792e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.063\times 10^{26}$
Sign $1$
Analytic cond. $0.409223$
Root an. cond. $0.945687$
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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{64} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(0.409223\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1792} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{64} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2458258707\)
\(L(\frac12)\) \(\approx\) \(0.2458258707\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
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

−4.08806435357031403792119358873, −3.87812391465162544664653689493, −3.85622581559046011263019762048, −3.68730877566549624047420401110, −3.67337314230797719050101698519, −3.57737563456813007123505827355, −3.54206089392569076294207002983, −3.47249608988627339979921558485, −3.12242116436875192285188029185, −3.04477361268724157054064952584, −2.65546481801332640547621893441, −2.62283527555943498779730812947, −2.52620547517382661211322628620, −2.45673267128541477511201942361, −2.36996828376079843955614681352, −2.34785295187339635611399767298, −2.28594345944014048133979888164, −1.97821723851981163573984527693, −1.56911931094038941806226569205, −1.35273966865603683291178777174, −1.33490287060637897343076651866, −1.16337105204809241708173268299, −1.15533648557837094571871009355, −1.07440512623192178225494739388, −0.17913457193175150975863816429, 0.17913457193175150975863816429, 1.07440512623192178225494739388, 1.15533648557837094571871009355, 1.16337105204809241708173268299, 1.33490287060637897343076651866, 1.35273966865603683291178777174, 1.56911931094038941806226569205, 1.97821723851981163573984527693, 2.28594345944014048133979888164, 2.34785295187339635611399767298, 2.36996828376079843955614681352, 2.45673267128541477511201942361, 2.52620547517382661211322628620, 2.62283527555943498779730812947, 2.65546481801332640547621893441, 3.04477361268724157054064952584, 3.12242116436875192285188029185, 3.47249608988627339979921558485, 3.54206089392569076294207002983, 3.57737563456813007123505827355, 3.67337314230797719050101698519, 3.68730877566549624047420401110, 3.85622581559046011263019762048, 3.87812391465162544664653689493, 4.08806435357031403792119358873

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

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