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Properties

Label 32-1020e16-1.1-c0e16-0-0
Degree $32$
Conductor $1.373\times 10^{48}$
Sign $1$
Analytic cond. $2.03290\times 10^{-5}$
Root an. cond. $0.713474$
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  + 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 + 257-s + 263-s + 269-s + ⋯
L(s)  = 1  + 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 + 257-s + 263-s + 269-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 17^{16}\)
Sign: $1$
Analytic conductor: \(2.03290\times 10^{-5}\)
Root analytic conductor: \(0.713474\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 17^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1269532336\)
\(L(\frac12)\) \(\approx\) \(0.1269532336\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{16} \)
3 \( 1 + T^{16} \)
5 \( 1 + T^{16} \)
17 \( 1 + T^{16} \)
good7 \( ( 1 + T^{16} )^{2} \)
11 \( ( 1 + T^{16} )^{2} \)
13 \( ( 1 + T^{4} )^{8} \)
19 \( ( 1 + T^{2} )^{8}( 1 + T^{4} )^{4} \)
23 \( ( 1 + T^{16} )^{2} \)
29 \( ( 1 + T^{16} )^{2} \)
31 \( ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2} \)
37 \( ( 1 + T^{16} )^{2} \)
41 \( ( 1 + T^{16} )^{2} \)
43 \( ( 1 + T^{8} )^{4} \)
47 \( ( 1 + T^{16} )^{2} \)
53 \( ( 1 + T^{16} )^{2} \)
59 \( ( 1 + T^{8} )^{4} \)
61 \( ( 1 + T^{2} )^{8}( 1 + T^{8} )^{2} \)
67 \( ( 1 - T )^{16}( 1 + T )^{16} \)
71 \( ( 1 + T^{16} )^{2} \)
73 \( ( 1 + T^{16} )^{2} \)
79 \( ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2} \)
83 \( ( 1 + T^{16} )^{2} \)
89 \( ( 1 + T^{4} )^{8} \)
97 \( ( 1 + T^{16} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.81110309351978217735781622570, −2.80570627200465257618672620433, −2.58260026392467077265575628889, −2.53754969307520753550051482809, −2.53168305925684701834046944554, −2.43412095366924676402637839236, −2.41897542541334512804867164128, −2.40579435070972329083106504992, −2.26724841138352773518928903509, −2.20292262920009732387955782383, −2.18181223939938870383164631483, −2.12957255821904708995357952197, −2.01165920898450086482298397064, −1.79193683910883543721680266992, −1.62994738404096938322916247199, −1.55763646793779086736871781000, −1.48671161032624166855222918889, −1.29793675977589172519717675666, −1.25255579459130105101455211483, −1.19101894754925677556364277854, −1.16327607194070429750335500061, −1.15961611182617140704787241528, −1.12030872322884848244546111926, −0.848533627719318610801011031217, −0.21669656063834619983073197022, 0.21669656063834619983073197022, 0.848533627719318610801011031217, 1.12030872322884848244546111926, 1.15961611182617140704787241528, 1.16327607194070429750335500061, 1.19101894754925677556364277854, 1.25255579459130105101455211483, 1.29793675977589172519717675666, 1.48671161032624166855222918889, 1.55763646793779086736871781000, 1.62994738404096938322916247199, 1.79193683910883543721680266992, 2.01165920898450086482298397064, 2.12957255821904708995357952197, 2.18181223939938870383164631483, 2.20292262920009732387955782383, 2.26724841138352773518928903509, 2.40579435070972329083106504992, 2.41897542541334512804867164128, 2.43412095366924676402637839236, 2.53168305925684701834046944554, 2.53754969307520753550051482809, 2.58260026392467077265575628889, 2.80570627200465257618672620433, 2.81110309351978217735781622570

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

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