Properties

Degree 16
Conductor $ 2^{40} \cdot 5^{24} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·13-s − 2·17-s − 8·37-s − 8·53-s + 2·73-s + 81-s + 10·89-s + 2·97-s + 4·101-s − 2·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 2·13-s − 2·17-s − 8·37-s − 8·53-s + 2·73-s + 81-s + 10·89-s + 2·97-s + 4·101-s − 2·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{40} \cdot 5^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{4000} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{40} \cdot 5^{24} ,\ ( \ : [0]^{8} ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $0.2225169021$
$L(\frac12)$  $\approx$  $0.2225169021$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 16. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 15.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
7 \( ( 1 + T^{4} )^{4} \)
11 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
13 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
37 \( ( 1 + T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
53 \( ( 1 + T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
89 \( ( 1 - T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.64287339607775174670315998273, −3.55697033631481768419599102173, −3.42625349440374281574296387389, −3.32809014689970200764436480556, −3.29099709857617645181734339917, −3.27606640128429834635803549029, −3.09868494906398745957963937224, −3.00156338490065672218981955566, −2.66886965184114505833188087916, −2.66769380532270550784370097460, −2.46867510874005352708696823966, −2.36816929672583738871697376756, −2.07700804261814804163447586923, −2.07116336530858488929950858830, −2.00469962958422626197963829526, −1.96819925723893023740312552428, −1.93138120635499229201319970850, −1.69603992315337827643740621500, −1.48825040359220765606321554022, −1.43870518205962038865426797308, −1.40918208346695072936083268714, −0.842515910366969655842917993558, −0.66706664084150803662014498185, −0.57895258705990064752217201139, −0.14373568290828362381687220266, 0.14373568290828362381687220266, 0.57895258705990064752217201139, 0.66706664084150803662014498185, 0.842515910366969655842917993558, 1.40918208346695072936083268714, 1.43870518205962038865426797308, 1.48825040359220765606321554022, 1.69603992315337827643740621500, 1.93138120635499229201319970850, 1.96819925723893023740312552428, 2.00469962958422626197963829526, 2.07116336530858488929950858830, 2.07700804261814804163447586923, 2.36816929672583738871697376756, 2.46867510874005352708696823966, 2.66769380532270550784370097460, 2.66886965184114505833188087916, 3.00156338490065672218981955566, 3.09868494906398745957963937224, 3.27606640128429834635803549029, 3.29099709857617645181734339917, 3.32809014689970200764436480556, 3.42625349440374281574296387389, 3.55697033631481768419599102173, 3.64287339607775174670315998273

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

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