Properties

Degree 16
Conductor $ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 10·16-s + 8·49-s − 20·64-s + 32·79-s + 80·109-s + 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 32·196-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 2·4-s + 5/2·16-s + 8/7·49-s − 5/2·64-s + 3.60·79-s + 7.66·109-s + 7.27·121-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 + 4.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 2.28·196-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $8.726549331$
$L(\frac12)$  $\approx$  $8.726549331$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{4} \)
3 \( 1 \)
5 \( 1 \)
7 \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
good11 \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 - p T^{2} )^{8} \)
23 \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 - p T^{2} )^{8} \)
37 \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \)
59 \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \)
67 \( ( 1 + p T^{2} )^{8} \)
71 \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 4 T + p T^{2} )^{8} \)
83 \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \)
89 \( ( 1 + 88 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \)
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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

−3.65387939743807060257785817763, −3.28973003205975117040239598282, −3.28885593833010005628792800811, −3.26427332861122874806959297135, −3.18965029217898984918144506402, −3.02870912591315724614564595060, −3.01818952441766136096994608524, −2.95728327215938398313625027279, −2.88894076939788356151535083180, −2.43559521041802132733773474054, −2.27121932542645040762886019685, −2.04795996349006498180300292354, −2.02416664421675885721315756439, −2.00926931752304343086770830201, −1.90466968651535076307675757003, −1.81882865060693184619663308319, −1.79810788007788625237553595835, −1.41810944225660202738530486471, −1.08854058031871183066468123363, −0.842415004010476236910189537230, −0.799780946583711719781980224577, −0.70167316635430877823994150649, −0.67798866511102393119037935785, −0.48610428837674666607064709768, −0.27390117608502847315366615123, 0.27390117608502847315366615123, 0.48610428837674666607064709768, 0.67798866511102393119037935785, 0.70167316635430877823994150649, 0.799780946583711719781980224577, 0.842415004010476236910189537230, 1.08854058031871183066468123363, 1.41810944225660202738530486471, 1.79810788007788625237553595835, 1.81882865060693184619663308319, 1.90466968651535076307675757003, 2.00926931752304343086770830201, 2.02416664421675885721315756439, 2.04795996349006498180300292354, 2.27121932542645040762886019685, 2.43559521041802132733773474054, 2.88894076939788356151535083180, 2.95728327215938398313625027279, 3.01818952441766136096994608524, 3.02870912591315724614564595060, 3.18965029217898984918144506402, 3.26427332861122874806959297135, 3.28885593833010005628792800811, 3.28973003205975117040239598282, 3.65387939743807060257785817763

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

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