Properties

Degree 40
Conductor $ 2^{100} \cdot 3^{60} \cdot 7^{20} $
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  + 20·7-s + 36·25-s − 36·31-s + 210·49-s − 64·79-s + 56·97-s + 12·103-s + 68·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 166·169-s + 173-s + 720·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 7.55·7-s + 36/5·25-s − 6.46·31-s + 30·49-s − 7.20·79-s + 5.68·97-s + 1.18·103-s + 6.18·121-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 + 12.7·169-s + 0.0760·173-s + 54.4·175-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100} \cdot 3^{60} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100} \cdot 3^{60} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(40\)
\( N \)  =  \(2^{100} \cdot 3^{60} \cdot 7^{20}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(40,\ 2^{100} \cdot 3^{60} \cdot 7^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )$
$L(1)$  $\approx$  $335.6693434$
$L(\frac12)$  $\approx$  $335.6693434$
$L(\frac{3}{2})$   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 40. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 39.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 - T )^{20} \)
good5 \( ( 1 - 18 T^{2} + 187 T^{4} - 1608 T^{6} + 10781 T^{8} - 57906 T^{10} + 10781 p^{2} T^{12} - 1608 p^{4} T^{14} + 187 p^{6} T^{16} - 18 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
11 \( ( 1 - 34 T^{2} + 589 T^{4} - 8472 T^{6} + 122234 T^{8} - 1528172 T^{10} + 122234 p^{2} T^{12} - 8472 p^{4} T^{14} + 589 p^{6} T^{16} - 34 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
13 \( ( 1 - 83 T^{2} + 3533 T^{4} - 98556 T^{6} + 1975738 T^{8} - 29558418 T^{10} + 1975738 p^{2} T^{12} - 98556 p^{4} T^{14} + 3533 p^{6} T^{16} - 83 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
17 \( ( 1 + 39 T^{2} + 455 T^{4} - 2382 T^{6} - 47987 T^{8} + 145305 T^{10} - 47987 p^{2} T^{12} - 2382 p^{4} T^{14} + 455 p^{6} T^{16} + 39 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
19 \( ( 1 - 70 T^{2} + 175 p T^{4} - 107448 T^{6} + 2823098 T^{8} - 58351460 T^{10} + 2823098 p^{2} T^{12} - 107448 p^{4} T^{14} + 175 p^{7} T^{16} - 70 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
23 \( ( 1 + 139 T^{2} + 10081 T^{4} + 21108 p T^{6} + 16965806 T^{8} + 447075730 T^{10} + 16965806 p^{2} T^{12} + 21108 p^{5} T^{14} + 10081 p^{6} T^{16} + 139 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
29 \( ( 1 - 189 T^{2} + 17145 T^{4} - 999612 T^{6} + 42327774 T^{8} - 1383568894 T^{10} + 42327774 p^{2} T^{12} - 999612 p^{4} T^{14} + 17145 p^{6} T^{16} - 189 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
31 \( ( 1 + 9 T + 171 T^{2} + 1092 T^{3} + 11058 T^{4} + 50402 T^{5} + 11058 p T^{6} + 1092 p^{2} T^{7} + 171 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
37 \( ( 1 - 22 T^{2} + 1771 T^{4} + 13920 T^{6} + 526445 T^{8} + 112600450 T^{10} + 526445 p^{2} T^{12} + 13920 p^{4} T^{14} + 1771 p^{6} T^{16} - 22 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
41 \( ( 1 + 190 T^{2} + 16651 T^{4} + 899952 T^{6} + 36306749 T^{8} + 1400716342 T^{10} + 36306749 p^{2} T^{12} + 899952 p^{4} T^{14} + 16651 p^{6} T^{16} + 190 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
43 \( ( 1 - 131 T^{2} + 5615 T^{4} - 93162 T^{6} + 5954173 T^{8} - 455929005 T^{10} + 5954173 p^{2} T^{12} - 93162 p^{4} T^{14} + 5615 p^{6} T^{16} - 131 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
47 \( ( 1 + 110 T^{2} + 11187 T^{4} + 742128 T^{6} + 47768349 T^{8} + 2326511142 T^{10} + 47768349 p^{2} T^{12} + 742128 p^{4} T^{14} + 11187 p^{6} T^{16} + 110 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
53 \( ( 1 - 193 T^{2} + 20557 T^{4} - 1818204 T^{6} + 128583722 T^{8} - 7326778022 T^{10} + 128583722 p^{2} T^{12} - 1818204 p^{4} T^{14} + 20557 p^{6} T^{16} - 193 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
59 \( ( 1 - 449 T^{2} + 96439 T^{4} - 13053078 T^{6} + 1232082461 T^{8} - 84671750375 T^{10} + 1232082461 p^{2} T^{12} - 13053078 p^{4} T^{14} + 96439 p^{6} T^{16} - 449 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
61 \( ( 1 - 186 T^{2} + 23517 T^{4} - 2152152 T^{6} + 175128378 T^{8} - 188535020 p T^{10} + 175128378 p^{2} T^{12} - 2152152 p^{4} T^{14} + 23517 p^{6} T^{16} - 186 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
67 \( ( 1 - 495 T^{2} + 120093 T^{4} - 18487716 T^{6} + 1986970986 T^{8} - 155229602234 T^{10} + 1986970986 p^{2} T^{12} - 18487716 p^{4} T^{14} + 120093 p^{6} T^{16} - 495 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
71 \( ( 1 + 343 T^{2} + 52613 T^{4} + 4802116 T^{6} + 309151594 T^{8} + 19321038474 T^{10} + 309151594 p^{2} T^{12} + 4802116 p^{4} T^{14} + 52613 p^{6} T^{16} + 343 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
73 \( ( 1 + 149 T^{2} + 432 T^{3} + 14410 T^{4} + 51408 T^{5} + 14410 p T^{6} + 432 p^{2} T^{7} + 149 p^{3} T^{8} + p^{5} T^{10} )^{4} \)
79 \( ( 1 + 16 T + 335 T^{2} + 3202 T^{3} + 40117 T^{4} + 296322 T^{5} + 40117 p T^{6} + 3202 p^{2} T^{7} + 335 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
83 \( ( 1 - 138 T^{2} + 13627 T^{4} - 1173792 T^{6} + 994015 p T^{8} - 3517947618 T^{10} + 994015 p^{3} T^{12} - 1173792 p^{4} T^{14} + 13627 p^{6} T^{16} - 138 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
89 \( ( 1 + 547 T^{2} + 153205 T^{4} + 28413396 T^{6} + 3828024122 T^{8} + 389779473874 T^{10} + 3828024122 p^{2} T^{12} + 28413396 p^{4} T^{14} + 153205 p^{6} T^{16} + 547 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
97 \( ( 1 - 14 T + 341 T^{2} - 3480 T^{3} + 53038 T^{4} - 434964 T^{5} + 53038 p T^{6} - 3480 p^{2} T^{7} + 341 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.61309620630767262458384837422, −1.53219022979678868185813734659, −1.52838110028575547339440084388, −1.52058632398388125349210657854, −1.35416911155772787478846040610, −1.24004388674825188779154683446, −1.21538550115896424608599754056, −1.20690863888324417125441423233, −1.19260855309888481872266146145, −1.17701477855712497661007101952, −1.16199786682216951785504856091, −1.00119368716194071473003383782, −0.871504363753324296784721997554, −0.857668939277944377439496754356, −0.840687105754759906877950431285, −0.78295354212760430029894827203, −0.64371242796815150789677727192, −0.59920809127733180469888515757, −0.57909269510595634065377613383, −0.55030907290514497197411751403, −0.40227707903944276014061178720, −0.38181355290503168494738758838, −0.38096039076953683893972454969, −0.11083299151493257602250471233, −0.10234184678814731813795591011, 0.10234184678814731813795591011, 0.11083299151493257602250471233, 0.38096039076953683893972454969, 0.38181355290503168494738758838, 0.40227707903944276014061178720, 0.55030907290514497197411751403, 0.57909269510595634065377613383, 0.59920809127733180469888515757, 0.64371242796815150789677727192, 0.78295354212760430029894827203, 0.840687105754759906877950431285, 0.857668939277944377439496754356, 0.871504363753324296784721997554, 1.00119368716194071473003383782, 1.16199786682216951785504856091, 1.17701477855712497661007101952, 1.19260855309888481872266146145, 1.20690863888324417125441423233, 1.21538550115896424608599754056, 1.24004388674825188779154683446, 1.35416911155772787478846040610, 1.52058632398388125349210657854, 1.52838110028575547339440084388, 1.53219022979678868185813734659, 1.61309620630767262458384837422

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

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