Properties

Label 32-776e16-1.1-c0e16-0-0
Degree $32$
Conductor $1.729\times 10^{46}$
Sign $1$
Analytic cond. $2.56036\times 10^{-7}$
Root an. cond. $0.622313$
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  + 8·3-s + 28·9-s + 48·27-s + 8·73-s + 2·81-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 + 64·219-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 8·3-s + 28·9-s + 48·27-s + 8·73-s + 2·81-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 + 64·219-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 97^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 97^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{48} \cdot 97^{16}\)
Sign: $1$
Analytic conductor: \(2.56036\times 10^{-7}\)
Root analytic conductor: \(0.622313\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{776} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{48} \cdot 97^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.298164756\)
\(L(\frac12)\) \(\approx\) \(3.298164756\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.05198291454220799478992853258, −3.00024912088580320235515188920, −2.98091723723471126131280703385, −2.83609896731445773924294241840, −2.61493525083454288700410068608, −2.59016389294132225993947804094, −2.54786791482735560708900481277, −2.54121745258923744840732474087, −2.49248839389802738513101195040, −2.34982065203221303377489236118, −2.24363234608343360947470994504, −2.18840753687828050195450009974, −2.17731098216373787259311447202, −2.10276769839194210628504296801, −2.08121939136187110948610532022, −1.89836678669043984228881747410, −1.76532367359657227337553504339, −1.69374815170021849135477260065, −1.65146817030295823442688008730, −1.53226258952428566210194819814, −1.42097680109733257138346677014, −1.37961915618935007594675964211, −1.02491095737652947525179142036, −0.919627321279888668156759251419, −0.870998555614100719394191661839, 0.870998555614100719394191661839, 0.919627321279888668156759251419, 1.02491095737652947525179142036, 1.37961915618935007594675964211, 1.42097680109733257138346677014, 1.53226258952428566210194819814, 1.65146817030295823442688008730, 1.69374815170021849135477260065, 1.76532367359657227337553504339, 1.89836678669043984228881747410, 2.08121939136187110948610532022, 2.10276769839194210628504296801, 2.17731098216373787259311447202, 2.18840753687828050195450009974, 2.24363234608343360947470994504, 2.34982065203221303377489236118, 2.49248839389802738513101195040, 2.54121745258923744840732474087, 2.54786791482735560708900481277, 2.59016389294132225993947804094, 2.61493525083454288700410068608, 2.83609896731445773924294241840, 2.98091723723471126131280703385, 3.00024912088580320235515188920, 3.05198291454220799478992853258

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

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