Properties

Degree 32
Conductor $ 2^{96} \cdot 3^{32} \cdot 7^{16} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 40·25-s − 8·49-s − 80·73-s − 112·97-s + 136·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-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 + ⋯
L(s)  = 1  − 8·25-s − 8/7·49-s − 9.36·73-s − 11.3·97-s + 12.3·121-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 + 9.84·169-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 + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{96} \cdot 3^{32} \cdot 7^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4032} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(32,\ 2^{96} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )$
$L(1)$  $\approx$  $7.578069941e-5$
$L(\frac12)$  $\approx$  $7.578069941e-5$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \(F_p\) is a polynomial of degree 32. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 31.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 + T^{2} )^{8} \)
good5 \( ( 1 + 2 p T^{2} + 54 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{4} \)
11 \( ( 1 - 34 T^{2} + 510 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
13 \( ( 1 - 32 T^{2} + 510 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
17 \( ( 1 - 22 T^{2} + 174 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
19 \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{8} \)
23 \( ( 1 + 70 T^{2} + 2262 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
29 \( ( 1 - 8 T^{2} + 354 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
31 \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{8} \)
37 \( ( 1 - 80 T^{2} + 3582 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
41 \( ( 1 + 26 T^{2} + 3342 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
43 \( ( 1 + 20 T^{2} - 1578 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
47 \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{8} \)
53 \( ( 1 + 136 T^{2} + 8898 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
59 \( ( 1 - p T^{2} )^{16} \)
61 \( ( 1 - 14 T + p T^{2} )^{8}( 1 + 14 T + p T^{2} )^{8} \)
67 \( ( 1 + 200 T^{2} + 18222 T^{4} + 200 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
71 \( ( 1 + 254 T^{2} + 26022 T^{4} + 254 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
73 \( ( 1 + 10 T + 150 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{8} \)
79 \( ( 1 + 80 T^{2} + 7278 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
83 \( ( 1 - 4 T^{2} + 5382 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
89 \( ( 1 - 166 T^{2} + 22542 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
97 \( ( 1 + 14 T + 222 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{8} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.99713703496434975495584851528, −1.98020939971037556012887490198, −1.94838550390134719203087879337, −1.85634811614904717288564751826, −1.77561053162430344545300296820, −1.56162107453976556137600948323, −1.54544819595916813189407937594, −1.52320737432118670071017131966, −1.51827853615113300400126193082, −1.46922200587654134678064935254, −1.42702576049018909315258581933, −1.30953129224655850211064709490, −1.25213980044446213158122507865, −1.11144066092977623967537942955, −1.06934599763782384526145005732, −0.990617429914980236547881348023, −0.880122482303472240044464635947, −0.816363954498810292737362315756, −0.58883657103295071608592768301, −0.42525949503236434345250898525, −0.41740603356420236361077799977, −0.24043438150772755665033707084, −0.19018907050634605734134476642, −0.15906754049462487837373373401, −0.00115759301572620047819726424, 0.00115759301572620047819726424, 0.15906754049462487837373373401, 0.19018907050634605734134476642, 0.24043438150772755665033707084, 0.41740603356420236361077799977, 0.42525949503236434345250898525, 0.58883657103295071608592768301, 0.816363954498810292737362315756, 0.880122482303472240044464635947, 0.990617429914980236547881348023, 1.06934599763782384526145005732, 1.11144066092977623967537942955, 1.25213980044446213158122507865, 1.30953129224655850211064709490, 1.42702576049018909315258581933, 1.46922200587654134678064935254, 1.51827853615113300400126193082, 1.52320737432118670071017131966, 1.54544819595916813189407937594, 1.56162107453976556137600948323, 1.77561053162430344545300296820, 1.85634811614904717288564751826, 1.94838550390134719203087879337, 1.98020939971037556012887490198, 1.99713703496434975495584851528

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

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