Properties

Label 32-1028e16-1.1-c0e16-0-0
Degree $32$
Conductor $1.556\times 10^{48}$
Sign $1$
Analytic cond. $2.30358\times 10^{-5}$
Root an. cond. $0.716267$
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  − 8·4-s + 36·16-s − 120·64-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  − 8·4-s + 36·16-s − 120·64-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 257^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 257^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 257^{16}\)
Sign: $1$
Analytic conductor: \(2.30358\times 10^{-5}\)
Root analytic conductor: \(0.716267\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1028} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 257^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005620386069\)
\(L(\frac12)\) \(\approx\) \(0.005620386069\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{8} \)
257 \( ( 1 + T )^{16} \)
good3 \( 1 + T^{32} \)
5 \( ( 1 + T^{8} )^{2}( 1 + T^{16} ) \)
7 \( 1 + T^{32} \)
11 \( ( 1 + T^{8} )^{4} \)
13 \( ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2} \)
17 \( ( 1 + T^{16} )^{2} \)
19 \( 1 + T^{32} \)
23 \( ( 1 + T^{8} )^{4} \)
29 \( ( 1 + T^{16} )^{2} \)
31 \( ( 1 + T^{16} )^{2} \)
37 \( ( 1 + T^{4} )^{4}( 1 + T^{16} ) \)
41 \( ( 1 + T^{8} )^{2}( 1 + T^{16} ) \)
43 \( 1 + T^{32} \)
47 \( 1 + T^{32} \)
53 \( ( 1 + T^{8} )^{2}( 1 + T^{16} ) \)
59 \( ( 1 + T^{16} )^{2} \)
61 \( ( 1 + T^{16} )^{2} \)
67 \( ( 1 + T^{8} )^{4} \)
71 \( 1 + T^{32} \)
73 \( ( 1 + T^{16} )^{2} \)
79 \( ( 1 + T^{16} )^{2} \)
83 \( 1 + T^{32} \)
89 \( ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2} \)
97 \( ( 1 + T^{4} )^{4}( 1 + T^{16} ) \)
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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

−2.88216865134159214637786775699, −2.83317436680373887543639993497, −2.80729428691038351153446275896, −2.69829654176970545030798434124, −2.68462711100802008411557846431, −2.55716486024389098907590815810, −2.50085070408490939972414146035, −2.37532307034652720143064142655, −2.17639511347170166686621380374, −2.08731970455126486790200488525, −2.02202125426084213664154871796, −1.83226508500775960564427770243, −1.82562943313818704567200651219, −1.78372075867876704961122071100, −1.77820330494842889831256155229, −1.37828472835755954226645719106, −1.33718969264319444772434139319, −1.21694136403485106242364710213, −1.21394696471838532946309467986, −1.07657613422300720616541096577, −1.06743200876436915685386767151, −0.951166093145613542744074044393, −0.869211434005572108358086717545, −0.64902038155353741455681052179, −0.12597442389847513736200472585, 0.12597442389847513736200472585, 0.64902038155353741455681052179, 0.869211434005572108358086717545, 0.951166093145613542744074044393, 1.06743200876436915685386767151, 1.07657613422300720616541096577, 1.21394696471838532946309467986, 1.21694136403485106242364710213, 1.33718969264319444772434139319, 1.37828472835755954226645719106, 1.77820330494842889831256155229, 1.78372075867876704961122071100, 1.82562943313818704567200651219, 1.83226508500775960564427770243, 2.02202125426084213664154871796, 2.08731970455126486790200488525, 2.17639511347170166686621380374, 2.37532307034652720143064142655, 2.50085070408490939972414146035, 2.55716486024389098907590815810, 2.68462711100802008411557846431, 2.69829654176970545030798434124, 2.80729428691038351153446275896, 2.83317436680373887543639993497, 2.88216865134159214637786775699

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

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