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Properties

Label 32-1020e16-1.1-c0e16-0-0
Degree 3232
Conductor 1.373×10481.373\times 10^{48}
Sign 11
Analytic cond. 2.03290×10−52.03290\times 10^{-5}
Root an. cond. 0.7134740.713474
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

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

Λ(s)=((232⋅316⋅516⋅1716)s/2 ΓC(s)16 L(s)=( Λ(1−s)\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}
Λ(s)=((232⋅316⋅516⋅1716)s/2 ΓC(s)16 L(s)=( Λ(1−s)\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: 3232
Conductor: 232⋅316⋅516⋅17162^{32} \cdot 3^{16} \cdot 5^{16} \cdot 17^{16}
Sign: 11
Analytic conductor: 2.03290×10−52.03290\times 10^{-5}
Root analytic conductor: 0.7134740.713474
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 232⋅316⋅516⋅1716, ( :[0]16), 1)(32,\ 2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 17^{16} ,\ ( \ : [0]^{16} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.12695323360.1269532336
L(12)L(\frac12) ≈\approx 0.12695323360.1269532336
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+T16 1 + T^{16}
3 1+T16 1 + T^{16}
5 1+T16 1 + T^{16}
17 1+T16 1 + T^{16}
good7 (1+T16)2 ( 1 + T^{16} )^{2}
11 (1+T16)2 ( 1 + T^{16} )^{2}
13 (1+T4)8 ( 1 + T^{4} )^{8}
19 (1+T2)8(1+T4)4 ( 1 + T^{2} )^{8}( 1 + T^{4} )^{4}
23 (1+T16)2 ( 1 + T^{16} )^{2}
29 (1+T16)2 ( 1 + T^{16} )^{2}
31 (1+T4)4(1+T8)2 ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2}
37 (1+T16)2 ( 1 + T^{16} )^{2}
41 (1+T16)2 ( 1 + T^{16} )^{2}
43 (1+T8)4 ( 1 + T^{8} )^{4}
47 (1+T16)2 ( 1 + T^{16} )^{2}
53 (1+T16)2 ( 1 + T^{16} )^{2}
59 (1+T8)4 ( 1 + T^{8} )^{4}
61 (1+T2)8(1+T8)2 ( 1 + T^{2} )^{8}( 1 + T^{8} )^{2}
67 (1−T)16(1+T)16 ( 1 - T )^{16}( 1 + T )^{16}
71 (1+T16)2 ( 1 + T^{16} )^{2}
73 (1+T16)2 ( 1 + T^{16} )^{2}
79 (1+T4)4(1+T8)2 ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2}
83 (1+T16)2 ( 1 + T^{16} )^{2}
89 (1+T4)8 ( 1 + T^{4} )^{8}
97 (1+T16)2 ( 1 + T^{16} )^{2}
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   L(s)=∏p ∏j=132(1−αj,p p−s)−1L(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 ZZ-function along the critical line

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