Properties

Label 32-884e16-1.1-c0e16-0-0
Degree $32$
Conductor $1.391\times 10^{47}$
Sign $1$
Analytic cond. $2.05944\times 10^{-6}$
Root an. cond. $0.664208$
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  − 16·29-s − 8·41-s − 8·53-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 + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  − 16·29-s − 8·41-s − 8·53-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 + 239-s + 241-s + 251-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 13^{16} \cdot 17^{16}\)
Sign: $1$
Analytic conductor: \(2.05944\times 10^{-6}\)
Root analytic conductor: \(0.664208\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{884} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 13^{16} \cdot 17^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007617587542\)
\(L(\frac12)\) \(\approx\) \(0.007617587542\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.01956291256799867276118492283, −2.96231531849742828239946534471, −2.95346579974800542305961535408, −2.62753160565342453313922888627, −2.56673584696770297072759354317, −2.55896967328168504622226507528, −2.53754628572748525109853490389, −2.22995995869731263848145539955, −2.19585735324789723093341616059, −2.10585389390308150948044667934, −2.04856665946721987106615956922, −1.88728834406980217021795888065, −1.88515118792304746813798763042, −1.87249548544910795489776618942, −1.79588994319590548235528916936, −1.68364917829225103476831952547, −1.65366081868462531482747031075, −1.58152813495542466743333154357, −1.52676982998994225240417367426, −1.47228763585329291814892710882, −1.39394239761761281693929727569, −1.35115960006997351740310417792, −0.821521010873783412990451230843, −0.52721632688693702275397830090, −0.095323343507969894942606698695, 0.095323343507969894942606698695, 0.52721632688693702275397830090, 0.821521010873783412990451230843, 1.35115960006997351740310417792, 1.39394239761761281693929727569, 1.47228763585329291814892710882, 1.52676982998994225240417367426, 1.58152813495542466743333154357, 1.65366081868462531482747031075, 1.68364917829225103476831952547, 1.79588994319590548235528916936, 1.87249548544910795489776618942, 1.88515118792304746813798763042, 1.88728834406980217021795888065, 2.04856665946721987106615956922, 2.10585389390308150948044667934, 2.19585735324789723093341616059, 2.22995995869731263848145539955, 2.53754628572748525109853490389, 2.55896967328168504622226507528, 2.56673584696770297072759354317, 2.62753160565342453313922888627, 2.95346579974800542305961535408, 2.96231531849742828239946534471, 3.01956291256799867276118492283

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

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