Properties

Label 16-1148e8-1.1-c0e8-0-1
Degree $16$
Conductor $3.017\times 10^{24}$
Sign $1$
Analytic cond. $0.0116089$
Root an. cond. $0.756919$
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  − 2·7-s + 4·9-s + 4·11-s − 4·23-s − 25-s − 6·29-s − 2·37-s − 2·43-s + 49-s − 8·63-s − 8·77-s + 8·79-s + 6·81-s + 16·99-s + 6·107-s + 4·109-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 8·161-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2·7-s + 4·9-s + 4·11-s − 4·23-s − 25-s − 6·29-s − 2·37-s − 2·43-s + 49-s − 8·63-s − 8·77-s + 8·79-s + 6·81-s + 16·99-s + 6·107-s + 4·109-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 8·161-s + 163-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 7^{8} \cdot 41^{8}\)
Sign: $1$
Analytic conductor: \(0.0116089\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1148} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 7^{8} \cdot 41^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9281073344\)
\(L(\frac12)\) \(\approx\) \(0.9281073344\)
\(L(1)\) 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 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
good3 \( ( 1 - T^{2} + T^{4} )^{4} \)
5 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
13 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
19 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
23 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
29 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
31 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
37 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
43 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
79 \( ( 1 - T + T^{2} )^{8} \)
83 \( ( 1 - T^{2} + T^{4} )^{4} \)
89 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
97 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
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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

−4.33502516858585723191863004225, −4.19860332853286735152006471386, −4.10348167578830668093049217838, −3.92701632411256822637741222932, −3.84204859307483786844571902384, −3.76989916261071248229868798099, −3.64648862564269952157790599424, −3.63110088906770882665329424562, −3.46689244697828483856567413030, −3.44922150668446790820367215838, −3.33727996604478259022845221442, −3.21071544385479668746564437921, −3.14445561347690398015634708095, −2.42617748776200514945094350410, −2.37589993944782810652193623903, −2.12432984687920650746321530622, −2.04776698668677107267462544796, −1.98748383566477736721312006568, −1.90442345253304963982786053466, −1.73059664554291718060909411395, −1.68063720069178119651450381867, −1.45851542342228874076480281989, −1.29128018708751784883325961845, −0.913343631389713829830884171142, −0.59816032920170460789373094072, 0.59816032920170460789373094072, 0.913343631389713829830884171142, 1.29128018708751784883325961845, 1.45851542342228874076480281989, 1.68063720069178119651450381867, 1.73059664554291718060909411395, 1.90442345253304963982786053466, 1.98748383566477736721312006568, 2.04776698668677107267462544796, 2.12432984687920650746321530622, 2.37589993944782810652193623903, 2.42617748776200514945094350410, 3.14445561347690398015634708095, 3.21071544385479668746564437921, 3.33727996604478259022845221442, 3.44922150668446790820367215838, 3.46689244697828483856567413030, 3.63110088906770882665329424562, 3.64648862564269952157790599424, 3.76989916261071248229868798099, 3.84204859307483786844571902384, 3.92701632411256822637741222932, 4.10348167578830668093049217838, 4.19860332853286735152006471386, 4.33502516858585723191863004225

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

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