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Properties

Label 16-3072e8-1.1-c0e8-0-1
Degree $16$
Conductor $7.932\times 10^{27}$
Sign $1$
Analytic cond. $30.5229$
Root an. cond. $1.23819$
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  − 8·37-s + 8·61-s + 8·73-s − 8·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + ⋯
L(s)  = 1  − 8·37-s + 8·61-s + 8·73-s − 8·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{80} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(30.5229\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{80} \cdot 3^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03955935551\)
\(L(\frac12)\) \(\approx\) \(0.03955935551\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T^{8} \)
good5 \( ( 1 + T^{8} )^{2} \)
7 \( ( 1 + T^{8} )^{2} \)
11 \( ( 1 + T^{8} )^{2} \)
13 \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \)
17 \( ( 1 + T^{2} )^{8} \)
19 \( ( 1 + T^{8} )^{2} \)
23 \( ( 1 + T^{4} )^{4} \)
29 \( ( 1 + T^{8} )^{2} \)
31 \( ( 1 + T^{8} )^{2} \)
37 \( ( 1 + T )^{8}( 1 + T^{4} )^{2} \)
41 \( ( 1 + T^{4} )^{4} \)
43 \( ( 1 + T^{8} )^{2} \)
47 \( ( 1 + T^{2} )^{8} \)
53 \( ( 1 + T^{8} )^{2} \)
59 \( ( 1 + T^{8} )^{2} \)
61 \( ( 1 - T )^{8}( 1 + T^{4} )^{2} \)
67 \( ( 1 + T^{8} )^{2} \)
71 \( ( 1 + T^{4} )^{4} \)
73 \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \)
79 \( ( 1 + T^{8} )^{2} \)
83 \( ( 1 + T^{8} )^{2} \)
89 \( ( 1 + T^{4} )^{4} \)
97 \( ( 1 + T^{4} )^{4} \)
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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

−3.88312981101868358593417281620, −3.62959717849181295420610447950, −3.52941823085183219732624520022, −3.51262110243495553771533600252, −3.44566655506188047664935395100, −3.38978873110789633835030681482, −3.28026118524072269078701162743, −3.01485152115033405575705328468, −2.96690950421897460765673965633, −2.60193172671760953410885215598, −2.55093837591814356978759055346, −2.41021240239924764305888627731, −2.37346222347245491448588230473, −2.27845352128515187543827568310, −2.17579258925687479251584308114, −2.00643615845684064520767421890, −1.80085573098790808429731454552, −1.79215098797286723014965572334, −1.69060692272440831890103879441, −1.32973211895075168880117986978, −1.19240539893880441963275024960, −0.980619162143275163864283279250, −0.881784627542399476360778456216, −0.873965147225082793822240325414, −0.05041344215138490145195803139, 0.05041344215138490145195803139, 0.873965147225082793822240325414, 0.881784627542399476360778456216, 0.980619162143275163864283279250, 1.19240539893880441963275024960, 1.32973211895075168880117986978, 1.69060692272440831890103879441, 1.79215098797286723014965572334, 1.80085573098790808429731454552, 2.00643615845684064520767421890, 2.17579258925687479251584308114, 2.27845352128515187543827568310, 2.37346222347245491448588230473, 2.41021240239924764305888627731, 2.55093837591814356978759055346, 2.60193172671760953410885215598, 2.96690950421897460765673965633, 3.01485152115033405575705328468, 3.28026118524072269078701162743, 3.38978873110789633835030681482, 3.44566655506188047664935395100, 3.51262110243495553771533600252, 3.52941823085183219732624520022, 3.62959717849181295420610447950, 3.88312981101868358593417281620

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

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