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Properties

Label 32-3267e16-1.1-c0e16-0-1
Degree $32$
Conductor $1.684\times 10^{56}$
Sign $1$
Analytic cond. $2494.02$
Root an. cond. $1.27688$
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  + 2·5-s − 16-s + 25-s + 2·31-s + 4·37-s + 2·47-s + 4·53-s − 2·59-s + 8·67-s − 4·71-s − 2·80-s + 2·97-s − 2·103-s + 2·113-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·155-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  + 2·5-s − 16-s + 25-s + 2·31-s + 4·37-s + 2·47-s + 4·53-s − 2·59-s + 8·67-s − 4·71-s − 2·80-s + 2·97-s − 2·103-s + 2·113-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·155-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{48} \cdot 11^{32}\)
Sign: $1$
Analytic conductor: \(2494.02\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{48} \cdot 11^{32} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.219516972\)
\(L(\frac12)\) \(\approx\) \(4.219516972\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
7 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
23 \( ( 1 - T^{2} + T^{4} )^{8} \)
29 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{4} \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
43 \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
53 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{4} \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
61 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
67 \( ( 1 - T )^{16}( 1 + T + T^{2} )^{8} \)
71 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{4} \)
73 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
79 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
83 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
89 \( ( 1 + T^{2} )^{16} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
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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

−2.31374810703187447804778837341, −2.19209606933399459689158884847, −2.13326198831872801570686208993, −2.09398522342391422671063326131, −2.04039511351220061194192252184, −2.01891408509430050084110518319, −1.99477141188730011714524947587, −1.98695291646014666979973795395, −1.94081623471609119191244817187, −1.90015704873942300471707223656, −1.82524649508512336969972772767, −1.44162861834733743164888102913, −1.41781804331677058417518292933, −1.39428330215029798847470279326, −1.37607424651176604397270519754, −1.23867089066437498108644143062, −1.04670719947890724920250971628, −1.02230226057611068899110721824, −0.986024687277895092403832581013, −0.943320439875923627742846021233, −0.918877734647918231234413956972, −0.821829644646311025597775298674, −0.66363393707226686707493003493, −0.65466692722784065060143111047, −0.22561377763466394589823670952, 0.22561377763466394589823670952, 0.65466692722784065060143111047, 0.66363393707226686707493003493, 0.821829644646311025597775298674, 0.918877734647918231234413956972, 0.943320439875923627742846021233, 0.986024687277895092403832581013, 1.02230226057611068899110721824, 1.04670719947890724920250971628, 1.23867089066437498108644143062, 1.37607424651176604397270519754, 1.39428330215029798847470279326, 1.41781804331677058417518292933, 1.44162861834733743164888102913, 1.82524649508512336969972772767, 1.90015704873942300471707223656, 1.94081623471609119191244817187, 1.98695291646014666979973795395, 1.99477141188730011714524947587, 2.01891408509430050084110518319, 2.04039511351220061194192252184, 2.09398522342391422671063326131, 2.13326198831872801570686208993, 2.19209606933399459689158884847, 2.31374810703187447804778837341

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

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