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

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·7-s + 3·9-s + 9·13-s − 8·19-s + 10·25-s − 3·31-s + 10·37-s + 19·49-s + 27·61-s + 18·63-s + 11·67-s − 180·73-s − 166·79-s + 54·91-s + 33·97-s − 20·103-s + 27·117-s − 11·121-s + 127-s + 131-s − 48·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 2.26·7-s + 9-s + 2.49·13-s − 1.83·19-s + 2·25-s − 0.538·31-s + 1.64·37-s + 19/7·49-s + 3.45·61-s + 2.26·63-s + 1.34·67-s − 21.0·73-s − 18.6·79-s + 5.66·91-s + 3.35·97-s − 1.97·103-s + 2.49·117-s − 121-s + 0.0887·127-s + 0.0873·131-s − 4.16·133-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 + ⋯

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  =  $(40,\ 2^{40} \cdot 3^{20} \cdot 67^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )$
$L(1)$  $\approx$  $0.662321$
$L(\frac12)$  $\approx$  $0.662321$
$L(\frac{3}{2})$   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 \( 1 - 11 T + 54 T^{2} + 143 T^{3} - 5191 T^{4} + 47520 T^{5} - 174923 T^{6} - 1259687 T^{7} + 25576398 T^{8} - 196941349 T^{9} + 452736173 T^{10} - 196941349 p T^{11} + 25576398 p^{2} T^{12} - 1259687 p^{3} T^{13} - 174923 p^{4} T^{14} + 47520 p^{5} T^{15} - 5191 p^{6} T^{16} + 143 p^{7} T^{17} + 54 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \)
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 \( ( 1 - 5 T + 18 T^{2} - 55 T^{3} + 149 T^{4} - 360 T^{5} + 757 T^{6} - 1265 T^{7} + 1026 T^{8} + 3725 T^{9} - 25807 T^{10} + 3725 p T^{11} + 1026 p^{2} T^{12} - 1265 p^{3} T^{13} + 757 p^{4} T^{14} - 360 p^{5} T^{15} + 149 p^{6} T^{16} - 55 p^{7} T^{17} + 18 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} )( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} - 120 T^{5} - 83 T^{6} + 923 T^{7} - 342 T^{8} - 6119 T^{9} + 8513 T^{10} - 6119 p T^{11} - 342 p^{2} T^{12} + 923 p^{3} T^{13} - 83 p^{4} T^{14} - 120 p^{5} T^{15} + 29 p^{6} T^{16} + 13 p^{7} T^{17} - 6 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} ) \)
11 \( ( 1 - p T + 6 p T^{2} - 3 p^{2} T^{3} + 15 p^{2} T^{4} - 6 p^{3} T^{5} + 25 p^{3} T^{6} - 9 p^{4} T^{7} + 34 p^{4} T^{8} - p^{6} T^{9} + 37 p^{5} T^{10} - p^{7} T^{11} + 34 p^{6} T^{12} - 9 p^{7} T^{13} + 25 p^{7} T^{14} - 6 p^{8} T^{15} + 15 p^{8} T^{16} - 3 p^{9} T^{17} + 6 p^{9} T^{18} - p^{10} T^{19} + p^{10} T^{20} )( 1 + p T + 6 p T^{2} + 3 p^{2} T^{3} + 15 p^{2} T^{4} + 6 p^{3} T^{5} + 25 p^{3} T^{6} + 9 p^{4} T^{7} + 34 p^{4} T^{8} + p^{6} T^{9} + 37 p^{5} T^{10} + p^{7} T^{11} + 34 p^{6} T^{12} + 9 p^{7} T^{13} + 25 p^{7} T^{14} + 6 p^{8} T^{15} + 15 p^{8} T^{16} + 3 p^{9} T^{17} + 6 p^{9} T^{18} + p^{10} T^{19} + p^{10} T^{20} ) \)
13 \( ( 1 - 7 T + 36 T^{2} - 161 T^{3} + 659 T^{4} - 2520 T^{5} + 9073 T^{6} - 30751 T^{7} + 97308 T^{8} - 281393 T^{9} + 704747 T^{10} - 281393 p T^{11} + 97308 p^{2} T^{12} - 30751 p^{3} T^{13} + 9073 p^{4} T^{14} - 2520 p^{5} T^{15} + 659 p^{6} T^{16} - 161 p^{7} T^{17} + 36 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} )( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} - 630 T^{5} + 883 T^{6} + 6424 T^{7} - 24327 T^{8} - 34858 T^{9} + 385967 T^{10} - 34858 p T^{11} - 24327 p^{2} T^{12} + 6424 p^{3} T^{13} + 883 p^{4} T^{14} - 630 p^{5} T^{15} + 29 p^{6} T^{16} + 44 p^{7} T^{17} - 9 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} ) \)
17 \( 1 - p T^{2} + p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} - p^{7} T^{14} + p^{9} T^{18} - p^{10} T^{20} + p^{11} T^{22} - p^{13} T^{26} + p^{14} T^{28} - p^{16} T^{32} + p^{17} T^{34} - p^{19} T^{38} + p^{20} T^{40} \)
19 \( ( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} + 1008 T^{5} - 4787 T^{6} - 23939 T^{7} + 67014 T^{8} + 521855 T^{9} - 751411 T^{10} + 521855 p T^{11} + 67014 p^{2} T^{12} - 23939 p^{3} T^{13} - 4787 p^{4} T^{14} + 1008 p^{5} T^{15} + 305 p^{6} T^{16} - 37 p^{7} T^{17} - 18 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} )( 1 + 7 T + 30 T^{2} + 77 T^{3} - 31 T^{4} - 1680 T^{5} - 11171 T^{6} - 46277 T^{7} - 111690 T^{8} + 97433 T^{9} + 2804141 T^{10} + 97433 p T^{11} - 111690 p^{2} T^{12} - 46277 p^{3} T^{13} - 11171 p^{4} T^{14} - 1680 p^{5} T^{15} - 31 p^{6} T^{16} + 77 p^{7} T^{17} + 30 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} ) \)
23 \( 1 - p T^{2} + p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} - p^{7} T^{14} + p^{9} T^{18} - p^{10} T^{20} + p^{11} T^{22} - p^{13} T^{26} + p^{14} T^{28} - p^{16} T^{32} + p^{17} T^{34} - p^{19} T^{38} + p^{20} T^{40} \)
29 \( ( 1 + p T^{2} + p^{2} T^{4} )^{10} \)
31 \( ( 1 - 4 T - 15 T^{2} + 184 T^{3} - 271 T^{4} - 4620 T^{5} + 26881 T^{6} + 35696 T^{7} - 976095 T^{8} + 2797804 T^{9} + 19067729 T^{10} + 2797804 p T^{11} - 976095 p^{2} T^{12} + 35696 p^{3} T^{13} + 26881 p^{4} T^{14} - 4620 p^{5} T^{15} - 271 p^{6} T^{16} + 184 p^{7} T^{17} - 15 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} )( 1 + 7 T + 18 T^{2} - 91 T^{3} - 1195 T^{4} - 5544 T^{5} - 1763 T^{6} + 159523 T^{7} + 1171314 T^{8} + 3253985 T^{9} - 13532839 T^{10} + 3253985 p T^{11} + 1171314 p^{2} T^{12} + 159523 p^{3} T^{13} - 1763 p^{4} T^{14} - 5544 p^{5} T^{15} - 1195 p^{6} T^{16} - 91 p^{7} T^{17} + 18 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} ) \)
37 \( ( 1 - 11 T + 84 T^{2} - 517 T^{3} + 2579 T^{4} - 9240 T^{5} + 6217 T^{6} + 273493 T^{7} - 3238452 T^{8} + 25503731 T^{9} - 160718317 T^{10} + 25503731 p T^{11} - 3238452 p^{2} T^{12} + 273493 p^{3} T^{13} + 6217 p^{4} T^{14} - 9240 p^{5} T^{15} + 2579 p^{6} T^{16} - 517 p^{7} T^{17} + 84 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} )( 1 + T - 36 T^{2} - 73 T^{3} + 1259 T^{4} + 3960 T^{5} - 42623 T^{6} - 189143 T^{7} + 1387908 T^{8} + 8386199 T^{9} - 42966397 T^{10} + 8386199 p T^{11} + 1387908 p^{2} T^{12} - 189143 p^{3} T^{13} - 42623 p^{4} T^{14} + 3960 p^{5} T^{15} + 1259 p^{6} T^{16} - 73 p^{7} T^{17} - 36 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} ) \)
41 \( 1 + p T^{2} - p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} + p^{7} T^{14} - p^{9} T^{18} - p^{10} T^{20} - p^{11} T^{22} + p^{13} T^{26} + p^{14} T^{28} - p^{16} T^{32} - p^{17} T^{34} + p^{19} T^{38} + p^{20} T^{40} \)
43 \( ( 1 - 13 T + 126 T^{2} - 1079 T^{3} + 8609 T^{4} - 65520 T^{5} + 481573 T^{6} - 3443089 T^{7} + 24052518 T^{8} - 164629907 T^{9} + 1105930517 T^{10} - 164629907 p T^{11} + 24052518 p^{2} T^{12} - 3443089 p^{3} T^{13} + 481573 p^{4} T^{14} - 65520 p^{5} T^{15} + 8609 p^{6} T^{16} - 1079 p^{7} T^{17} + 126 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} )( 1 + 13 T + 126 T^{2} + 1079 T^{3} + 8609 T^{4} + 65520 T^{5} + 481573 T^{6} + 3443089 T^{7} + 24052518 T^{8} + 164629907 T^{9} + 1105930517 T^{10} + 164629907 p T^{11} + 24052518 p^{2} T^{12} + 3443089 p^{3} T^{13} + 481573 p^{4} T^{14} + 65520 p^{5} T^{15} + 8609 p^{6} T^{16} + 1079 p^{7} T^{17} + 126 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} ) \)
47 \( 1 - p T^{2} + p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} - p^{7} T^{14} + p^{9} T^{18} - p^{10} T^{20} + p^{11} T^{22} - p^{13} T^{26} + p^{14} T^{28} - p^{16} T^{32} + p^{17} T^{34} - p^{19} T^{38} + p^{20} T^{40} \)
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 \( ( 1 - 14 T + 135 T^{2} - 1036 T^{3} + 6269 T^{4} - 24570 T^{5} - 38429 T^{6} + 2036776 T^{7} - 26170695 T^{8} + 242146394 T^{9} - 1793637121 T^{10} + 242146394 p T^{11} - 26170695 p^{2} T^{12} + 2036776 p^{3} T^{13} - 38429 p^{4} T^{14} - 24570 p^{5} T^{15} + 6269 p^{6} T^{16} - 1036 p^{7} T^{17} + 135 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} )( 1 - 13 T + 108 T^{2} - 611 T^{3} + 1355 T^{4} + 19656 T^{5} - 338183 T^{6} + 3197363 T^{7} - 20936556 T^{8} + 77136085 T^{9} + 274360811 T^{10} + 77136085 p T^{11} - 20936556 p^{2} T^{12} + 3197363 p^{3} T^{13} - 338183 p^{4} T^{14} + 19656 p^{5} T^{15} + 1355 p^{6} T^{16} - 611 p^{7} T^{17} + 108 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} ) \)
71 \( 1 - p T^{2} + p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} - p^{7} T^{14} + p^{9} T^{18} - p^{10} T^{20} + p^{11} T^{22} - p^{13} T^{26} + p^{14} T^{28} - p^{16} T^{32} + p^{17} T^{34} - p^{19} T^{38} + p^{20} T^{40} \)
73 \( ( 1 + 17 T + p T^{2} )^{10}( 1 + 10 T + 27 T^{2} - 460 T^{3} - 6571 T^{4} - 32130 T^{5} + 158383 T^{6} + 3929320 T^{7} + 27731241 T^{8} - 9527950 T^{9} - 2119660093 T^{10} - 9527950 p T^{11} + 27731241 p^{2} T^{12} + 3929320 p^{3} T^{13} + 158383 p^{4} T^{14} - 32130 p^{5} T^{15} - 6571 p^{6} T^{16} - 460 p^{7} T^{17} + 27 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} ) \)
79 \( ( 1 + 17 T + p T^{2} )^{10}( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} - 55692 T^{5} + 9073 T^{6} + 4363376 T^{7} - 18170271 T^{8} - 272025620 T^{9} + 2523553889 T^{10} - 272025620 p T^{11} - 18170271 p^{2} T^{12} + 4363376 p^{3} T^{13} + 9073 p^{4} T^{14} - 55692 p^{5} T^{15} + 2705 p^{6} T^{16} + 568 p^{7} T^{17} - 63 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} ) \)
83 \( 1 - p T^{2} + p^{3} T^{6} - p^{4} T^{8} + p^{6} T^{12} - p^{7} T^{14} + p^{9} T^{18} - p^{10} T^{20} + p^{11} T^{22} - p^{13} T^{26} + p^{14} T^{28} - p^{16} T^{32} + p^{17} T^{34} - p^{19} T^{38} + p^{20} T^{40} \)
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 \( ( 1 - 19 T + 264 T^{2} - 3173 T^{3} + 34679 T^{4} - 351120 T^{5} + 3307417 T^{6} - 28782283 T^{7} + 226043928 T^{8} - 1502953181 T^{9} + 6629849423 T^{10} - 1502953181 p T^{11} + 226043928 p^{2} T^{12} - 28782283 p^{3} T^{13} + 3307417 p^{4} T^{14} - 351120 p^{5} T^{15} + 34679 p^{6} T^{16} - 3173 p^{7} T^{17} + 264 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} )( 1 - 14 T + 99 T^{2} - 28 T^{3} - 9211 T^{4} + 131670 T^{5} - 949913 T^{6} + 526792 T^{7} + 84766473 T^{8} - 1237829446 T^{9} + 9107264363 T^{10} - 1237829446 p T^{11} + 84766473 p^{2} T^{12} + 526792 p^{3} T^{13} - 949913 p^{4} T^{14} + 131670 p^{5} T^{15} - 9211 p^{6} T^{16} - 28 p^{7} T^{17} + 99 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} ) \)
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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

−2.39771650576121480330341376894, −2.36386591405024418678036366116, −2.24362751798995147354041188690, −1.91794909297684928661364841755, −1.83475340032958111751164529891, −1.75381957194294533166424924937, −1.73630593151091948153853871440, −1.69798548666627572894048408576, −1.69654422135277408042758808645, −1.68144095028887655586289540288, −1.65614640634787770831844324184, −1.52849314858197882712409510626, −1.44222247659244239236892694565, −1.29699180805480834515896753256, −1.25927592708598596762209839966, −1.21301817204718841709713289958, −1.14791170346109205893118592776, −1.14482444600786693838432202829, −1.11918731349471667907563301902, −1.02981508869601206801733379309, −0.61259460629678066482681671046, −0.45396738612953854568685746704, −0.31867768979834763096175439973, −0.21247503377628491252959026537, −0.05486594066442224109001702813, 0.05486594066442224109001702813, 0.21247503377628491252959026537, 0.31867768979834763096175439973, 0.45396738612953854568685746704, 0.61259460629678066482681671046, 1.02981508869601206801733379309, 1.11918731349471667907563301902, 1.14482444600786693838432202829, 1.14791170346109205893118592776, 1.21301817204718841709713289958, 1.25927592708598596762209839966, 1.29699180805480834515896753256, 1.44222247659244239236892694565, 1.52849314858197882712409510626, 1.65614640634787770831844324184, 1.68144095028887655586289540288, 1.69654422135277408042758808645, 1.69798548666627572894048408576, 1.73630593151091948153853871440, 1.75381957194294533166424924937, 1.83475340032958111751164529891, 1.91794909297684928661364841755, 2.24362751798995147354041188690, 2.36386591405024418678036366116, 2.39771650576121480330341376894

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

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