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Properties

Label 32-984e16-1.1-c0e16-0-0
Degree $32$
Conductor $7.725\times 10^{47}$
Sign $1$
Analytic cond. $1.14402\times 10^{-5}$
Root an. cond. $0.700770$
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-s − 4·17-s − 16·67-s + 81-s − 4·89-s + 2·121-s + 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 + ⋯
L(s)  = 1  + 16-s − 4·17-s − 16·67-s + 81-s − 4·89-s + 2·121-s + 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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{48} \cdot 3^{16} \cdot 41^{16}\)
Sign: $1$
Analytic conductor: \(1.14402\times 10^{-5}\)
Root analytic conductor: \(0.700770\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{48} \cdot 3^{16} \cdot 41^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005940510025\)
\(L(\frac12)\) \(\approx\) \(0.005940510025\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
3 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
41 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
good5 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
7 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
11 \( ( 1 + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
13 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
19 \( ( 1 + T^{2} )^{8}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
29 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
43 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
47 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
53 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
59 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
61 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
67 \( ( 1 + T )^{16}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
71 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
73 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
79 \( ( 1 + T^{8} )^{4} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
97 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
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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

−2.89949768914676687500891291021, −2.83850252366302666433846365852, −2.79447339986914763966568913865, −2.53534537537866374716475013900, −2.52579859214845822385762480440, −2.49636601451567028148191541970, −2.37100184950858989258057002767, −2.35375078997196806621935713207, −2.35073133612271114269729541947, −2.16438609730424965040166821238, −2.15915758136982775393210527720, −2.07857854555378119213110535092, −1.88738505720139685521230975679, −1.83743190297713270156427852779, −1.64585198046592635186583080897, −1.60382354151829398447131311954, −1.53213415896797882060669003342, −1.47358229936521453418016369015, −1.37405933941348794764124086923, −1.25885570119708666740550869740, −1.17885750635250723298046698876, −1.13537652305015730341117688843, −1.09758725210947538873308542636, −0.68031397855853713896612268016, −0.05482509595201981468866092995, 0.05482509595201981468866092995, 0.68031397855853713896612268016, 1.09758725210947538873308542636, 1.13537652305015730341117688843, 1.17885750635250723298046698876, 1.25885570119708666740550869740, 1.37405933941348794764124086923, 1.47358229936521453418016369015, 1.53213415896797882060669003342, 1.60382354151829398447131311954, 1.64585198046592635186583080897, 1.83743190297713270156427852779, 1.88738505720139685521230975679, 2.07857854555378119213110535092, 2.15915758136982775393210527720, 2.16438609730424965040166821238, 2.35073133612271114269729541947, 2.35375078997196806621935713207, 2.37100184950858989258057002767, 2.49636601451567028148191541970, 2.52579859214845822385762480440, 2.53534537537866374716475013900, 2.79447339986914763966568913865, 2.83850252366302666433846365852, 2.89949768914676687500891291021

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

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