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Properties

Label 32-3332e16-1.1-c0e16-0-2
Degree 3232
Conductor 2.308×10562.308\times 10^{56}
Sign 11
Analytic cond. 3418.173418.17
Root an. cond. 1.289521.28952
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·13-s + 8·37-s − 16·41-s − 8·61-s + 16·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-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 + ⋯
L(s)  = 1  + 16·13-s + 8·37-s − 16·41-s − 8·61-s + 16·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-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 + ⋯

Functional equation

Λ(s)=((232⋅732⋅1716)s/2 ΓC(s)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{32} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((232⋅732⋅1716)s/2 ΓC(s)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{32} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 3232
Conductor: 232⋅732⋅17162^{32} \cdot 7^{32} \cdot 17^{16}
Sign: 11
Analytic conductor: 3418.173418.17
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 232⋅732⋅1716, ( :[0]16), 1)(32,\ 2^{32} \cdot 7^{32} \cdot 17^{16} ,\ ( \ : [0]^{16} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 17.2337473917.23374739
L(12)L(\frac12) ≈\approx 17.2337473917.23374739
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−T8+T16 1 - T^{8} + T^{16}
7 1 1
17 1−T8+T16 1 - T^{8} + T^{16}
good3 1−T16+T32 1 - T^{16} + T^{32}
5 (1−T4+T8)2(1−T8+T16) ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} )
11 1−T16+T32 1 - T^{16} + T^{32}
13 (1−T)16(1+T2)8 ( 1 - T )^{16}( 1 + T^{2} )^{8}
19 (1−T8+T16)2 ( 1 - T^{8} + T^{16} )^{2}
23 1−T16+T32 1 - T^{16} + T^{32}
29 (1+T4)4(1+T8)2 ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2}
31 1−T16+T32 1 - T^{16} + T^{32}
37 (1−T+T2)8(1−T8+T16) ( 1 - T + T^{2} )^{8}( 1 - T^{8} + T^{16} )
41 (1+T)16(1+T8)2 ( 1 + T )^{16}( 1 + T^{8} )^{2}
43 (1+T8)4 ( 1 + T^{8} )^{4}
47 (1−T4+T8)4 ( 1 - T^{4} + T^{8} )^{4}
53 (1−T2+T4)4(1−T4+T8)2 ( 1 - T^{2} + T^{4} )^{4}( 1 - T^{4} + T^{8} )^{2}
59 (1−T8+T16)2 ( 1 - T^{8} + T^{16} )^{2}
61 (1+T+T2)8(1−T8+T16) ( 1 + T + T^{2} )^{8}( 1 - T^{8} + T^{16} )
67 (1−T2+T4)8 ( 1 - T^{2} + T^{4} )^{8}
71 (1+T16)2 ( 1 + T^{16} )^{2}
73 (1−T4+T8)2(1−T8+T16) ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} )
79 1−T16+T32 1 - T^{16} + T^{32}
83 (1+T8)4 ( 1 + T^{8} )^{4}
89 (1−T4+T8)4 ( 1 - T^{4} + T^{8} )^{4}
97 (1+T4)4(1+T8)2 ( 1 + T^{4} )^{4}( 1 + T^{8} )^{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

−2.20955539447899786613584365787, −2.13531648790975266488702703715, −2.06778309368915818642433088642, −2.01359671180098971698926585273, −1.97529684919302614056206822078, −1.92084045888218602138750094691, −1.83811639308141362873532887781, −1.78933252098970345433658068612, −1.71624125994323266946686127440, −1.67117605416994708389889069628, −1.53242877488487546743621774221, −1.51458560716832772343097917130, −1.42732585888155369331131663231, −1.40339488239027514640298440148, −1.26242599526066894770155284304, −1.23082012389885787418452093199, −1.20835503897156078869665729075, −1.15359923355206541403725485313, −1.13033797369965664055026025301, −1.12261177834117106416987761777, −0.857556904332426372242545584209, −0.845438977943866266081153557075, −0.78485564838477890439651193231, −0.56080221372735873235642345457, −0.39221201625782327861581524558, 0.39221201625782327861581524558, 0.56080221372735873235642345457, 0.78485564838477890439651193231, 0.845438977943866266081153557075, 0.857556904332426372242545584209, 1.12261177834117106416987761777, 1.13033797369965664055026025301, 1.15359923355206541403725485313, 1.20835503897156078869665729075, 1.23082012389885787418452093199, 1.26242599526066894770155284304, 1.40339488239027514640298440148, 1.42732585888155369331131663231, 1.51458560716832772343097917130, 1.53242877488487546743621774221, 1.67117605416994708389889069628, 1.71624125994323266946686127440, 1.78933252098970345433658068612, 1.83811639308141362873532887781, 1.92084045888218602138750094691, 1.97529684919302614056206822078, 2.01359671180098971698926585273, 2.06778309368915818642433088642, 2.13531648790975266488702703715, 2.20955539447899786613584365787

Graph of the ZZ-function along the critical line

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