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Properties

Label 32-1020e16-1.1-c0e16-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.4909862440\)
\(L(\frac12)\) \(\approx\) \(0.4909862440\)
\(L(1)\) 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

−2.88330304702176192527217125431, −2.70410768000419378180703777729, −2.69327570477689414155649533067, −2.56471305480360113165921321398, −2.48397349363565068338920974242, −2.44062101059661745477876156856, −2.42504723112797056058625906956, −2.34288528506834513253007846185, −2.23652275594591082697147598909, −2.21523844245786414118111678529, −2.09147345166363936232908207961, −2.04148980092772743228856824280, −1.95597603373145905146990454571, −1.85370632273633537598927648189, −1.74720524987380151883600201197, −1.62471948237383795379034278580, −1.50372496417796516090769864050, −1.30866386034064000342885965218, −1.30253379521264541731535652725, −1.22664354114791953041737218556, −1.20271570339249303862981533680, −1.17893749332507079889893939180, −0.791382404187776367628833591577, −0.66753310552010901134122348317, −0.63732969388957838996982461109, 0.63732969388957838996982461109, 0.66753310552010901134122348317, 0.791382404187776367628833591577, 1.17893749332507079889893939180, 1.20271570339249303862981533680, 1.22664354114791953041737218556, 1.30253379521264541731535652725, 1.30866386034064000342885965218, 1.50372496417796516090769864050, 1.62471948237383795379034278580, 1.74720524987380151883600201197, 1.85370632273633537598927648189, 1.95597603373145905146990454571, 2.04148980092772743228856824280, 2.09147345166363936232908207961, 2.21523844245786414118111678529, 2.23652275594591082697147598909, 2.34288528506834513253007846185, 2.42504723112797056058625906956, 2.44062101059661745477876156856, 2.48397349363565068338920974242, 2.56471305480360113165921321398, 2.69327570477689414155649533067, 2.70410768000419378180703777729, 2.88330304702176192527217125431

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

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