Properties

Degree 32
Conductor $ 17^{16} \cdot 43^{16} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 24·13-s + 56·23-s − 64·59-s + 96·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 288·169-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 + ⋯
L(s)  = 1  − 6.65·13-s + 11.6·23-s − 8.33·59-s + 10.8·79-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 22.1·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16} \cdot 43^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16} \cdot 43^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(17^{16} \cdot 43^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{731} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(32,\ 17^{16} \cdot 43^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )$
$L(1)$  $\approx$  $2.09188$
$L(\frac12)$  $\approx$  $2.09188$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;43\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad17 \( 1 + 79967 T^{8} + p^{8} T^{16} \)
43 \( ( 1 + p^{4} T^{8} )^{2} \)
good2 \( ( 1 + p^{4} T^{8} )^{4} \)
3 \( ( 1 + p^{8} T^{16} )^{2} \)
5 \( ( 1 + p^{8} T^{16} )^{2} \)
7 \( ( 1 + p^{8} T^{16} )^{2} \)
11 \( ( 1 + 199 T^{4} + p^{4} T^{8} )^{2}( 1 + 10319 T^{8} + p^{8} T^{16} ) \)
13 \( ( 1 + 3 T + p T^{2} )^{8}( 1 - 17 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 + p^{4} T^{8} )^{4} \)
23 \( ( 1 - 7 T + p T^{2} )^{8}( 1 + 540719 T^{8} + p^{8} T^{16} ) \)
29 \( ( 1 + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 1561 T^{4} + p^{4} T^{8} )^{2}( 1 + 589679 T^{8} + p^{8} T^{16} ) \)
37 \( ( 1 + p^{8} T^{16} )^{2} \)
41 \( ( 1 - 1841 T^{4} + p^{4} T^{8} )^{2}( 1 - 2262241 T^{8} + p^{8} T^{16} ) \)
47 \( ( 1 - 6983806 T^{8} + p^{8} T^{16} )^{2} \)
53 \( ( 1 + 63 T^{2} + p^{2} T^{4} )^{4}( 1 - 1649 T^{4} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 8 T + p T^{2} )^{8}( 1 - 4046 T^{4} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 39816433 T^{8} + p^{8} T^{16} )^{2} \)
71 \( ( 1 + p^{8} T^{16} )^{2} \)
73 \( ( 1 + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 12 T + p T^{2} )^{8}( 1 + 73045634 T^{8} + p^{8} T^{16} ) \)
83 \( ( 1 - 93091441 T^{8} + p^{8} T^{16} )^{2} \)
89 \( ( 1 + p^{2} T^{4} )^{8} \)
97 \( ( 1 + 18431 T^{4} + p^{4} T^{8} )^{2}( 1 + 162643199 T^{8} + p^{8} T^{16} ) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.80021182392715715590433274685, −2.74155726713514193213384104394, −2.49484570194487245473473578513, −2.46791567754583328729401503250, −2.43579154902726021476492979937, −2.42621406795068358130534023001, −2.39879415646494346275757477088, −2.33414469757957925729261403225, −2.27237243837536742143792983319, −2.00124796174905465826421713184, −1.91139666797616460774983117229, −1.80988955943073015625644373954, −1.64158720805134176462891535082, −1.63879845114971410305473469140, −1.50637238504644656825039950127, −1.34846778454871446358978195154, −1.22028323507647595613900731734, −1.15092702876879522838142360742, −0.977055366957075231561650685697, −0.975998848378082664600197881796, −0.926348790280399283107982104649, −0.68473308982348851222084806348, −0.44033068918331144713624285261, −0.32101999828414267623015145419, −0.12738536170039738841728891413, 0.12738536170039738841728891413, 0.32101999828414267623015145419, 0.44033068918331144713624285261, 0.68473308982348851222084806348, 0.926348790280399283107982104649, 0.975998848378082664600197881796, 0.977055366957075231561650685697, 1.15092702876879522838142360742, 1.22028323507647595613900731734, 1.34846778454871446358978195154, 1.50637238504644656825039950127, 1.63879845114971410305473469140, 1.64158720805134176462891535082, 1.80988955943073015625644373954, 1.91139666797616460774983117229, 2.00124796174905465826421713184, 2.27237243837536742143792983319, 2.33414469757957925729261403225, 2.39879415646494346275757477088, 2.42621406795068358130534023001, 2.43579154902726021476492979937, 2.46791567754583328729401503250, 2.49484570194487245473473578513, 2.74155726713514193213384104394, 2.80021182392715715590433274685

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

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