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Properties

Label 32-3332e16-1.1-c0e16-0-2
Degree $32$
Conductor $2.308\times 10^{56}$
Sign $1$
Analytic cond. $3418.17$
Root an. cond. $1.28952$
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  + 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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{32} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{32} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 7^{32} \cdot 17^{16}\)
Sign: $1$
Analytic conductor: \(3418.17\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 7^{32} \cdot 17^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(17.23374739\)
\(L(\frac12)\) \(\approx\) \(17.23374739\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{8} + T^{16} \)
7 \( 1 \)
17 \( 1 - T^{8} + T^{16} \)
good3 \( 1 - T^{16} + T^{32} \)
5 \( ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
11 \( 1 - T^{16} + T^{32} \)
13 \( ( 1 - T )^{16}( 1 + T^{2} )^{8} \)
19 \( ( 1 - T^{8} + T^{16} )^{2} \)
23 \( 1 - T^{16} + T^{32} \)
29 \( ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2} \)
31 \( 1 - T^{16} + T^{32} \)
37 \( ( 1 - T + T^{2} )^{8}( 1 - T^{8} + T^{16} ) \)
41 \( ( 1 + T )^{16}( 1 + T^{8} )^{2} \)
43 \( ( 1 + T^{8} )^{4} \)
47 \( ( 1 - T^{4} + T^{8} )^{4} \)
53 \( ( 1 - T^{2} + T^{4} )^{4}( 1 - T^{4} + T^{8} )^{2} \)
59 \( ( 1 - T^{8} + T^{16} )^{2} \)
61 \( ( 1 + T + T^{2} )^{8}( 1 - T^{8} + T^{16} ) \)
67 \( ( 1 - T^{2} + T^{4} )^{8} \)
71 \( ( 1 + T^{16} )^{2} \)
73 \( ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
79 \( 1 - T^{16} + T^{32} \)
83 \( ( 1 + T^{8} )^{4} \)
89 \( ( 1 - T^{4} + T^{8} )^{4} \)
97 \( ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2} \)
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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

−2.20955539447899786613584365787, −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 $Z$-function along the critical line

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