Properties

Degree 40
Conductor $ 2^{40} \cdot 3^{20} \cdot 67^{20} $
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  + 3·9-s + 16·19-s + 10·25-s − 20·37-s − 2·49-s − 16·67-s + 90·73-s + 44·79-s + 40·103-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 48·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 9-s + 3.67·19-s + 2·25-s − 3.28·37-s − 2/7·49-s − 1.95·67-s + 10.5·73-s + 4.95·79-s + 3.94·103-s + 2·121-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 + 1.69·169-s + 3.67·171-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(40\)
\( N \)  =  \(2^{40} \cdot 3^{20} \cdot 67^{20}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{804} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(40,\ 2^{40} \cdot 3^{20} \cdot 67^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )$
$L(1)$  $\approx$  $9.17346$
$L(\frac12)$  $\approx$  $9.17346$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;67\}$,\(F_p(T)\) is a polynomial of degree 40. If $p \in \{2,\;3,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 39.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} - p^{5} T^{10} + p^{6} T^{12} - p^{7} T^{14} + p^{8} T^{16} - p^{9} T^{18} + p^{10} T^{20} \)
67 \( 1 + 16 T + 189 T^{2} + 1952 T^{3} + 18569 T^{4} + 166320 T^{5} + 1416997 T^{6} + 11528512 T^{7} + 89517393 T^{8} + 659867984 T^{9} + 4560222413 T^{10} + 659867984 p T^{11} + 89517393 p^{2} T^{12} + 11528512 p^{3} T^{13} + 1416997 p^{4} T^{14} + 166320 p^{5} T^{15} + 18569 p^{6} T^{16} + 1952 p^{7} T^{17} + 189 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \)
good5 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} - p^{5} T^{10} + p^{6} T^{12} - p^{7} T^{14} + p^{8} T^{16} - p^{9} T^{18} + p^{10} T^{20} )^{2} \)
7 \( ( 1 - 4 T + 9 T^{2} - 8 T^{3} - 31 T^{4} + 180 T^{5} - 503 T^{6} + 752 T^{7} + 513 T^{8} - 7316 T^{9} + 25673 T^{10} - 7316 p T^{11} + 513 p^{2} T^{12} + 752 p^{3} T^{13} - 503 p^{4} T^{14} + 180 p^{5} T^{15} - 31 p^{6} T^{16} - 8 p^{7} T^{17} + 9 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} )( 1 + 4 T + 9 T^{2} + 8 T^{3} - 31 T^{4} - 180 T^{5} - 503 T^{6} - 752 T^{7} + 513 T^{8} + 7316 T^{9} + 25673 T^{10} + 7316 p T^{11} + 513 p^{2} T^{12} - 752 p^{3} T^{13} - 503 p^{4} T^{14} - 180 p^{5} T^{15} - 31 p^{6} T^{16} + 8 p^{7} T^{17} + 9 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} ) \)
11 \( ( 1 - p T + 5 p T^{2} - p^{2} T^{3} - p^{2} T^{4} + p^{3} T^{5} - p^{3} T^{6} - p^{4} T^{7} + 5 p^{4} T^{8} - p^{5} T^{9} + p^{5} T^{10} )^{2}( 1 + p T + 5 p T^{2} + p^{2} T^{3} - p^{2} T^{4} - p^{3} T^{5} - p^{3} T^{6} + p^{4} T^{7} + 5 p^{4} T^{8} + p^{5} T^{9} + p^{5} T^{10} )^{2} \)
13 \( ( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} - 630 T^{5} + 883 T^{6} + 6424 T^{7} - 24327 T^{8} - 34858 T^{9} + 385967 T^{10} - 34858 p T^{11} - 24327 p^{2} T^{12} + 6424 p^{3} T^{13} + 883 p^{4} T^{14} - 630 p^{5} T^{15} + 29 p^{6} T^{16} + 44 p^{7} T^{17} - 9 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} )( 1 + 2 T - 9 T^{2} - 44 T^{3} + 29 T^{4} + 630 T^{5} + 883 T^{6} - 6424 T^{7} - 24327 T^{8} + 34858 T^{9} + 385967 T^{10} + 34858 p T^{11} - 24327 p^{2} T^{12} - 6424 p^{3} T^{13} + 883 p^{4} T^{14} + 630 p^{5} T^{15} + 29 p^{6} T^{16} - 44 p^{7} T^{17} - 9 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} ) \)
17 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} + p^{5} T^{10} + p^{6} T^{12} + p^{7} T^{14} + p^{8} T^{16} + p^{9} T^{18} + p^{10} T^{20} )^{2} \)
19 \( ( 1 - 8 T + 45 T^{2} - 208 T^{3} + 809 T^{4} - 2520 T^{5} + 4789 T^{6} + 9568 T^{7} - 167535 T^{8} + 1158488 T^{9} - 6084739 T^{10} + 1158488 p T^{11} - 167535 p^{2} T^{12} + 9568 p^{3} T^{13} + 4789 p^{4} T^{14} - 2520 p^{5} T^{15} + 809 p^{6} T^{16} - 208 p^{7} T^{17} + 45 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
23 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} + p^{5} T^{10} + p^{6} T^{12} + p^{7} T^{14} + p^{8} T^{16} + p^{9} T^{18} + p^{10} T^{20} )^{2} \)
29 \( ( 1 - p T^{2} )^{20} \)
31 \( ( 1 - 4 T - 15 T^{2} + 184 T^{3} - 271 T^{4} - 4620 T^{5} + 26881 T^{6} + 35696 T^{7} - 976095 T^{8} + 2797804 T^{9} + 19067729 T^{10} + 2797804 p T^{11} - 976095 p^{2} T^{12} + 35696 p^{3} T^{13} + 26881 p^{4} T^{14} - 4620 p^{5} T^{15} - 271 p^{6} T^{16} + 184 p^{7} T^{17} - 15 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} )( 1 + 4 T - 15 T^{2} - 184 T^{3} - 271 T^{4} + 4620 T^{5} + 26881 T^{6} - 35696 T^{7} - 976095 T^{8} - 2797804 T^{9} + 19067729 T^{10} - 2797804 p T^{11} - 976095 p^{2} T^{12} - 35696 p^{3} T^{13} + 26881 p^{4} T^{14} + 4620 p^{5} T^{15} - 271 p^{6} T^{16} - 184 p^{7} T^{17} - 15 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} ) \)
37 \( ( 1 + 10 T + 63 T^{2} + 260 T^{3} + 269 T^{4} - 6930 T^{5} - 79253 T^{6} - 536120 T^{7} - 2428839 T^{8} - 4451950 T^{9} + 45347543 T^{10} - 4451950 p T^{11} - 2428839 p^{2} T^{12} - 536120 p^{3} T^{13} - 79253 p^{4} T^{14} - 6930 p^{5} T^{15} + 269 p^{6} T^{16} + 260 p^{7} T^{17} + 63 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
41 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} - p^{5} T^{10} + p^{6} T^{12} - p^{7} T^{14} + p^{8} T^{16} - p^{9} T^{18} + p^{10} T^{20} )^{2} \)
43 \( ( 1 - 8 T + 21 T^{2} + 176 T^{3} - 2311 T^{4} + 10920 T^{5} + 12013 T^{6} - 565664 T^{7} + 4008753 T^{8} - 7746472 T^{9} - 110404603 T^{10} - 7746472 p T^{11} + 4008753 p^{2} T^{12} - 565664 p^{3} T^{13} + 12013 p^{4} T^{14} + 10920 p^{5} T^{15} - 2311 p^{6} T^{16} + 176 p^{7} T^{17} + 21 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} )( 1 + 8 T + 21 T^{2} - 176 T^{3} - 2311 T^{4} - 10920 T^{5} + 12013 T^{6} + 565664 T^{7} + 4008753 T^{8} + 7746472 T^{9} - 110404603 T^{10} + 7746472 p T^{11} + 4008753 p^{2} T^{12} + 565664 p^{3} T^{13} + 12013 p^{4} T^{14} - 10920 p^{5} T^{15} - 2311 p^{6} T^{16} - 176 p^{7} T^{17} + 21 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} ) \)
47 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} + p^{5} T^{10} + p^{6} T^{12} + p^{7} T^{14} + p^{8} T^{16} + p^{9} T^{18} + p^{10} T^{20} )^{2} \)
53 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} - p^{5} T^{10} + p^{6} T^{12} - p^{7} T^{14} + p^{8} T^{16} - p^{9} T^{18} + p^{10} T^{20} )^{2} \)
59 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} + p^{5} T^{10} + p^{6} T^{12} + p^{7} T^{14} + p^{8} T^{16} + p^{9} T^{18} + p^{10} T^{20} )^{2} \)
61 \( ( 1 - 14 T + 135 T^{2} - 1036 T^{3} + 6269 T^{4} - 24570 T^{5} - 38429 T^{6} + 2036776 T^{7} - 26170695 T^{8} + 242146394 T^{9} - 1793637121 T^{10} + 242146394 p T^{11} - 26170695 p^{2} T^{12} + 2036776 p^{3} T^{13} - 38429 p^{4} T^{14} - 24570 p^{5} T^{15} + 6269 p^{6} T^{16} - 1036 p^{7} T^{17} + 135 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} )( 1 + 14 T + 135 T^{2} + 1036 T^{3} + 6269 T^{4} + 24570 T^{5} - 38429 T^{6} - 2036776 T^{7} - 26170695 T^{8} - 242146394 T^{9} - 1793637121 T^{10} - 242146394 p T^{11} - 26170695 p^{2} T^{12} - 2036776 p^{3} T^{13} - 38429 p^{4} T^{14} + 24570 p^{5} T^{15} + 6269 p^{6} T^{16} + 1036 p^{7} T^{17} + 135 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} ) \)
71 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} + p^{5} T^{10} + p^{6} T^{12} + p^{7} T^{14} + p^{8} T^{16} + p^{9} T^{18} + p^{10} T^{20} )^{2} \)
73 \( ( 1 - 10 T + p T^{2} )^{10}( 1 + 10 T + 27 T^{2} - 460 T^{3} - 6571 T^{4} - 32130 T^{5} + 158383 T^{6} + 3929320 T^{7} + 27731241 T^{8} - 9527950 T^{9} - 2119660093 T^{10} - 9527950 p T^{11} + 27731241 p^{2} T^{12} + 3929320 p^{3} T^{13} + 158383 p^{4} T^{14} - 32130 p^{5} T^{15} - 6571 p^{6} T^{16} - 460 p^{7} T^{17} + 27 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} ) \)
79 \( ( 1 - 4 T + p T^{2} )^{10}( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} - 55692 T^{5} + 9073 T^{6} + 4363376 T^{7} - 18170271 T^{8} - 272025620 T^{9} + 2523553889 T^{10} - 272025620 p T^{11} - 18170271 p^{2} T^{12} + 4363376 p^{3} T^{13} + 9073 p^{4} T^{14} - 55692 p^{5} T^{15} + 2705 p^{6} T^{16} + 568 p^{7} T^{17} - 63 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} ) \)
83 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} + p^{5} T^{10} + p^{6} T^{12} + p^{7} T^{14} + p^{8} T^{16} + p^{9} T^{18} + p^{10} T^{20} )^{2} \)
89 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} + p^{5} T^{10} + p^{6} T^{12} + p^{7} T^{14} + p^{8} T^{16} + p^{9} T^{18} + p^{10} T^{20} )^{2} \)
97 \( ( 1 - 14 T + 99 T^{2} - 28 T^{3} - 9211 T^{4} + 131670 T^{5} - 949913 T^{6} + 526792 T^{7} + 84766473 T^{8} - 1237829446 T^{9} + 9107264363 T^{10} - 1237829446 p T^{11} + 84766473 p^{2} T^{12} + 526792 p^{3} T^{13} - 949913 p^{4} T^{14} + 131670 p^{5} T^{15} - 9211 p^{6} T^{16} - 28 p^{7} T^{17} + 99 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} )( 1 + 14 T + 99 T^{2} + 28 T^{3} - 9211 T^{4} - 131670 T^{5} - 949913 T^{6} - 526792 T^{7} + 84766473 T^{8} + 1237829446 T^{9} + 9107264363 T^{10} + 1237829446 p T^{11} + 84766473 p^{2} T^{12} - 526792 p^{3} T^{13} - 949913 p^{4} T^{14} - 131670 p^{5} T^{15} - 9211 p^{6} T^{16} + 28 p^{7} T^{17} + 99 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} ) \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.23224949719835760957542690367, −2.15758067506660991869531148657, −2.13722620084439560032136645816, −2.09156115006110979873075139247, −2.08565717034992582689485617635, −2.05344342786902451312619556157, −2.03983713517344529464595399512, −1.99007463274771737683916455701, −1.84287196062897168595852298711, −1.64031235169965337597291560820, −1.49541170821458975316981055400, −1.44046715901648137671174425880, −1.31401557785857073433834958468, −1.24276202379162434812750458360, −1.15803288825886686491589806569, −1.14443104812082336923551578578, −1.07511815008898047306910656026, −1.07110217817557382051227124621, −1.03873672709058412323974492871, −0.74726171647869634602790386046, −0.68795928121257683908574283989, −0.52449979567729259925374496248, −0.51882023754894552378988458545, −0.40515711029627455510024832309, −0.086910742605287533092897733626, 0.086910742605287533092897733626, 0.40515711029627455510024832309, 0.51882023754894552378988458545, 0.52449979567729259925374496248, 0.68795928121257683908574283989, 0.74726171647869634602790386046, 1.03873672709058412323974492871, 1.07110217817557382051227124621, 1.07511815008898047306910656026, 1.14443104812082336923551578578, 1.15803288825886686491589806569, 1.24276202379162434812750458360, 1.31401557785857073433834958468, 1.44046715901648137671174425880, 1.49541170821458975316981055400, 1.64031235169965337597291560820, 1.84287196062897168595852298711, 1.99007463274771737683916455701, 2.03983713517344529464595399512, 2.05344342786902451312619556157, 2.08565717034992582689485617635, 2.09156115006110979873075139247, 2.13722620084439560032136645816, 2.15758067506660991869531148657, 2.23224949719835760957542690367

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

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