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Properties

Label 32-2420e16-1.1-c0e16-0-0
Degree $32$
Conductor $1.384\times 10^{54}$
Sign $1$
Analytic cond. $20.4908$
Root an. cond. $1.09897$
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  + 2·4-s + 2·5-s + 4·9-s + 16-s + 4·20-s + 3·25-s + 8·36-s + 8·45-s + 4·49-s + 2·80-s + 6·81-s − 16·89-s + 6·100-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2·4-s + 2·5-s + 4·9-s + 16-s + 4·20-s + 3·25-s + 8·36-s + 8·45-s + 4·49-s + 2·80-s + 6·81-s − 16·89-s + 6·100-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 5^{16} \cdot 11^{32}\)
Sign: $1$
Analytic conductor: \(20.4908\)
Root analytic conductor: \(1.09897\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 5^{16} \cdot 11^{32} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5123477757\)
\(L(\frac12)\) \(\approx\) \(0.5123477757\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
5 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
11 \( 1 \)
good3 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
7 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
13 \( ( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} )^{2} \)
17 \( ( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} )^{2} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
23 \( ( 1 + T^{2} )^{16} \)
29 \( ( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
41 \( ( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} )^{2} \)
43 \( ( 1 + T^{2} )^{16} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
53 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
67 \( ( 1 + T^{2} )^{16} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
89 \( ( 1 + T + T^{2} )^{16} \)
97 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
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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

−2.43599153130760708981189747424, −2.41330699717075573141660091499, −2.25791152243750879319911747687, −2.25758585989804738324011751180, −2.20745606477093292026214539586, −2.09785836373458753988251062682, −1.96626193972691274927910780021, −1.95414533936478115995726726457, −1.89435142462664199256272273851, −1.76042315003817961140560929883, −1.73327079136630249809094168009, −1.63785994357045433053676727460, −1.60447717131878820656306270994, −1.55007861733360227312543729199, −1.51811711462649571761672869725, −1.41153479287785952230433658114, −1.31493821684464034790271773667, −1.17161406114652531257774566200, −1.15496679170433662353411358342, −0.959469600595375861125479650487, −0.939900672500997440260709684574, −0.927313695032740414478264223804, −0.908928970927098720786272196198, −0.872971390990113599407966490477, −0.06366206240582024986722537067, 0.06366206240582024986722537067, 0.872971390990113599407966490477, 0.908928970927098720786272196198, 0.927313695032740414478264223804, 0.939900672500997440260709684574, 0.959469600595375861125479650487, 1.15496679170433662353411358342, 1.17161406114652531257774566200, 1.31493821684464034790271773667, 1.41153479287785952230433658114, 1.51811711462649571761672869725, 1.55007861733360227312543729199, 1.60447717131878820656306270994, 1.63785994357045433053676727460, 1.73327079136630249809094168009, 1.76042315003817961140560929883, 1.89435142462664199256272273851, 1.95414533936478115995726726457, 1.96626193972691274927910780021, 2.09785836373458753988251062682, 2.20745606477093292026214539586, 2.25758585989804738324011751180, 2.25791152243750879319911747687, 2.41330699717075573141660091499, 2.43599153130760708981189747424

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

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