Properties

Degree 32
Conductor $ 2^{80} \cdot 3^{48} \cdot 7^{16} $
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  + 8·13-s + 64·25-s + 40·37-s − 8·49-s − 56·73-s − 16·97-s + 56·109-s − 64·121-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 + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 2.21·13-s + 64/5·25-s + 6.57·37-s − 8/7·49-s − 6.55·73-s − 1.62·97-s + 5.36·109-s − 5.81·121-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 + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{80} \cdot 3^{48} \cdot 7^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(32,\ 2^{80} \cdot 3^{48} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )$
$L(1)$  $\approx$  $73.35505464$
$L(\frac12)$  $\approx$  $73.35505464$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 + T^{2} )^{8} \)
good5 \( ( 1 - 16 T^{2} + 111 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
11 \( ( 1 + 32 T^{2} + 622 T^{4} + 8576 T^{6} + 99379 T^{8} + 8576 p^{2} T^{10} + 622 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 - 2 T + 2 p T^{2} - 6 p T^{3} + 434 T^{4} - 6 p^{2} T^{5} + 2 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
17 \( ( 1 - 24 T^{2} + 788 T^{4} - 17736 T^{6} + 292518 T^{8} - 17736 p^{2} T^{10} + 788 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 - 36 T^{2} + 746 T^{4} - 8304 T^{6} + 72147 T^{8} - 8304 p^{2} T^{10} + 746 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 + 64 T^{2} + 2055 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
29 \( ( 1 - 180 T^{2} + 15368 T^{4} - 805116 T^{6} + 28222446 T^{8} - 805116 p^{2} T^{10} + 15368 p^{4} T^{12} - 180 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 112 T^{2} + 4690 T^{4} - 84064 T^{6} + 1025299 T^{8} - 84064 p^{2} T^{10} + 4690 p^{4} T^{12} - 112 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 - 10 T + 146 T^{2} - 1056 T^{3} + 8039 T^{4} - 1056 p T^{5} + 146 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
41 \( ( 1 - 168 T^{2} + 14246 T^{4} - 874752 T^{6} + 41345307 T^{8} - 874752 p^{2} T^{10} + 14246 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 - 264 T^{2} + 32912 T^{4} - 2530824 T^{6} + 131089566 T^{8} - 2530824 p^{2} T^{10} + 32912 p^{4} T^{12} - 264 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( ( 1 + 140 T^{2} + 12088 T^{4} + 786212 T^{6} + 39897838 T^{8} + 786212 p^{2} T^{10} + 12088 p^{4} T^{12} + 140 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 264 T^{2} + 32204 T^{4} - 2520504 T^{6} + 149086662 T^{8} - 2520504 p^{2} T^{10} + 32204 p^{4} T^{12} - 264 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( ( 1 + 260 T^{2} + 30952 T^{4} + 2376716 T^{6} + 147668398 T^{8} + 2376716 p^{2} T^{10} + 30952 p^{4} T^{12} + 260 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 + 80 T^{2} - 528 T^{3} + 2286 T^{4} - 528 p T^{5} + 80 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
67 \( ( 1 - 352 T^{2} + 62320 T^{4} - 7098256 T^{6} + 563157790 T^{8} - 7098256 p^{2} T^{10} + 62320 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
71 \( ( 1 + 216 T^{2} + 28598 T^{4} + 2870208 T^{6} + 226923531 T^{8} + 2870208 p^{2} T^{10} + 28598 p^{4} T^{12} + 216 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 + 14 T + 278 T^{2} + 2994 T^{3} + 29954 T^{4} + 2994 p T^{5} + 278 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
79 \( ( 1 + 56 T^{2} + 14896 T^{4} + 945080 T^{6} + 108009022 T^{8} + 945080 p^{2} T^{10} + 14896 p^{4} T^{12} + 56 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( ( 1 + 516 T^{2} + 125336 T^{4} + 18717900 T^{6} + 1875461838 T^{8} + 18717900 p^{2} T^{10} + 125336 p^{4} T^{12} + 516 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 - 336 T^{2} + 65102 T^{4} - 8829312 T^{6} + 891379059 T^{8} - 8829312 p^{2} T^{10} + 65102 p^{4} T^{12} - 336 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 + 4 T + 344 T^{2} + 1068 T^{3} + 48302 T^{4} + 1068 p T^{5} + 344 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
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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

−1.81552149453283991106919097199, −1.69220839994654039657814650829, −1.63084880231314225902820459323, −1.54441485892614483064867657993, −1.52626658171478029599031979934, −1.50767556363360682612811344730, −1.42435812566157622128542707853, −1.31546352526388467386483074985, −1.31094906334666250221423650975, −1.29021626087301612774485678751, −1.28717850208480018127886828470, −1.14196646124642502667827024022, −1.07775470928758121181908565081, −1.05682590600848104032037128752, −0.991611024287603634920958340091, −0.961048464826185538801467684908, −0.77303807237320767883596877030, −0.77063613472039240075296921307, −0.75798516337044596031206769850, −0.62142533459574712590356305878, −0.53554948187952728603564943825, −0.36430875051775939586685352683, −0.25175146107464040511601135566, −0.22187799429114881856570736105, −0.11950987000577543486535294567, 0.11950987000577543486535294567, 0.22187799429114881856570736105, 0.25175146107464040511601135566, 0.36430875051775939586685352683, 0.53554948187952728603564943825, 0.62142533459574712590356305878, 0.75798516337044596031206769850, 0.77063613472039240075296921307, 0.77303807237320767883596877030, 0.961048464826185538801467684908, 0.991611024287603634920958340091, 1.05682590600848104032037128752, 1.07775470928758121181908565081, 1.14196646124642502667827024022, 1.28717850208480018127886828470, 1.29021626087301612774485678751, 1.31094906334666250221423650975, 1.31546352526388467386483074985, 1.42435812566157622128542707853, 1.50767556363360682612811344730, 1.52626658171478029599031979934, 1.54441485892614483064867657993, 1.63084880231314225902820459323, 1.69220839994654039657814650829, 1.81552149453283991106919097199

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

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