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Properties

Label 12-3808e6-1.1-c1e6-0-9
Degree $12$
Conductor $3.049\times 10^{21}$
Sign $1$
Analytic cond. $7.90395\times 10^{8}$
Root an. cond. $5.51425$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s − 6·7-s − 5·9-s − 2·11-s − 4·13-s − 4·15-s + 6·17-s + 2·19-s + 12·21-s − 10·23-s − 13·25-s + 14·27-s + 2·29-s − 12·31-s + 4·33-s − 12·35-s + 2·37-s + 8·39-s − 8·43-s − 10·45-s − 22·47-s + 21·49-s − 12·51-s − 6·53-s − 4·55-s − 4·57-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 2.26·7-s − 5/3·9-s − 0.603·11-s − 1.10·13-s − 1.03·15-s + 1.45·17-s + 0.458·19-s + 2.61·21-s − 2.08·23-s − 2.59·25-s + 2.69·27-s + 0.371·29-s − 2.15·31-s + 0.696·33-s − 2.02·35-s + 0.328·37-s + 1.28·39-s − 1.21·43-s − 1.49·45-s − 3.20·47-s + 3·49-s − 1.68·51-s − 0.824·53-s − 0.539·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{30} \cdot 7^{6} \cdot 17^{6}\)
Sign: $1$
Analytic conductor: \(7.90395\times 10^{8}\)
Root analytic conductor: \(5.51425\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{30} \cdot 7^{6} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( ( 1 + T )^{6} \)
17 \( ( 1 - T )^{6} \)
good3 \( 1 + 2 T + p^{2} T^{2} + 14 T^{3} + 46 T^{4} + 56 T^{5} + 152 T^{6} + 56 p T^{7} + 46 p^{2} T^{8} + 14 p^{3} T^{9} + p^{6} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 2 T + 17 T^{2} - 34 T^{3} + 152 T^{4} - 284 T^{5} + 916 T^{6} - 284 p T^{7} + 152 p^{2} T^{8} - 34 p^{3} T^{9} + 17 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T + 40 T^{2} + 54 T^{3} + 723 T^{4} + 716 T^{5} + 8952 T^{6} + 716 p T^{7} + 723 p^{2} T^{8} + 54 p^{3} T^{9} + 40 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 4 T + 56 T^{2} + 192 T^{3} + 1563 T^{4} + 4396 T^{5} + 25608 T^{6} + 4396 p T^{7} + 1563 p^{2} T^{8} + 192 p^{3} T^{9} + 56 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 2 T + 70 T^{2} - 162 T^{3} + 2643 T^{4} - 5304 T^{5} + 62588 T^{6} - 5304 p T^{7} + 2643 p^{2} T^{8} - 162 p^{3} T^{9} + 70 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 10 T + 154 T^{2} + 1034 T^{3} + 8931 T^{4} + 44288 T^{5} + 271972 T^{6} + 44288 p T^{7} + 8931 p^{2} T^{8} + 1034 p^{3} T^{9} + 154 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 2 T + 92 T^{2} - 178 T^{3} + 175 p T^{4} - 8292 T^{5} + 176368 T^{6} - 8292 p T^{7} + 175 p^{3} T^{8} - 178 p^{3} T^{9} + 92 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 12 T + 179 T^{2} + 1590 T^{3} + 13706 T^{4} + 90978 T^{5} + 564612 T^{6} + 90978 p T^{7} + 13706 p^{2} T^{8} + 1590 p^{3} T^{9} + 179 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 2 T + 98 T^{2} - 182 T^{3} + 5435 T^{4} - 12992 T^{5} + 238228 T^{6} - 12992 p T^{7} + 5435 p^{2} T^{8} - 182 p^{3} T^{9} + 98 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 165 T^{2} - 178 T^{3} + 13040 T^{4} - 18290 T^{5} + 654988 T^{6} - 18290 p T^{7} + 13040 p^{2} T^{8} - 178 p^{3} T^{9} + 165 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 + 8 T + 169 T^{2} + 856 T^{3} + 11606 T^{4} + 41064 T^{5} + 532872 T^{6} + 41064 p T^{7} + 11606 p^{2} T^{8} + 856 p^{3} T^{9} + 169 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 22 T + 380 T^{2} + 4778 T^{3} + 50051 T^{4} + 432084 T^{5} + 3233344 T^{6} + 432084 p T^{7} + 50051 p^{2} T^{8} + 4778 p^{3} T^{9} + 380 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 6 T + 181 T^{2} + 748 T^{3} + 17288 T^{4} + 58154 T^{5} + 1097028 T^{6} + 58154 p T^{7} + 17288 p^{2} T^{8} + 748 p^{3} T^{9} + 181 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 8 T + 186 T^{2} + 1496 T^{3} + 13399 T^{4} + 132208 T^{5} + 701004 T^{6} + 132208 p T^{7} + 13399 p^{2} T^{8} + 1496 p^{3} T^{9} + 186 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 4 T + 93 T^{2} + 30 T^{3} + 6392 T^{4} - 24154 T^{5} + 175004 T^{6} - 24154 p T^{7} + 6392 p^{2} T^{8} + 30 p^{3} T^{9} + 93 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 8 T + 311 T^{2} - 1976 T^{3} + 45014 T^{4} - 231512 T^{5} + 3832348 T^{6} - 231512 p T^{7} + 45014 p^{2} T^{8} - 1976 p^{3} T^{9} + 311 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 18 T + 302 T^{2} + 3546 T^{3} + 39563 T^{4} + 369608 T^{5} + 3434668 T^{6} + 369608 p T^{7} + 39563 p^{2} T^{8} + 3546 p^{3} T^{9} + 302 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 8 T + 5 p T^{2} - 2466 T^{3} + 59296 T^{4} - 328690 T^{5} + 5545676 T^{6} - 328690 p T^{7} + 59296 p^{2} T^{8} - 2466 p^{3} T^{9} + 5 p^{5} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 14 T + 380 T^{2} + 4610 T^{3} + 68659 T^{4} + 654836 T^{5} + 7090784 T^{6} + 654836 p T^{7} + 68659 p^{2} T^{8} + 4610 p^{3} T^{9} + 380 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 432 T^{2} + 148 T^{3} + 81587 T^{4} + 36308 T^{5} + 8735176 T^{6} + 36308 p T^{7} + 81587 p^{2} T^{8} + 148 p^{3} T^{9} + 432 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 2 T + 352 T^{2} - 58 T^{3} + 57075 T^{4} + 53468 T^{5} + 5989528 T^{6} + 53468 p T^{7} + 57075 p^{2} T^{8} - 58 p^{3} T^{9} + 352 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 2 T + 89 T^{2} + 322 T^{3} + 16432 T^{4} - 9924 T^{5} + 1614388 T^{6} - 9924 p T^{7} + 16432 p^{2} T^{8} + 322 p^{3} T^{9} + 89 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.01890694427427408822783295610, −4.92195656529938840023883681607, −4.34980540183823317500781684766, −4.33972778177312341499485912927, −4.29587508293990532033012130487, −4.13089930324050149293044638342, −3.78388545626032308706676310635, −3.66750024114580001741295949431, −3.63533281951626489136344682016, −3.50895678723596489284919733067, −3.34563087625857584241152384107, −3.33366814845751551746754572792, −3.12230710723398814600782280558, −2.77140603684487745880483767053, −2.69391575216320253958544678835, −2.62674609962138086070908280520, −2.35202316011113328905799288168, −2.34560253224922895190592095453, −2.32203404163882378370584488188, −1.95396193169749418340536233522, −1.63660433969957739827661779609, −1.44642416367983555411354163440, −1.34781187418814293042775889559, −1.20437185677027119307468636464, −1.03760336153571274653010804956, 0, 0, 0, 0, 0, 0, 1.03760336153571274653010804956, 1.20437185677027119307468636464, 1.34781187418814293042775889559, 1.44642416367983555411354163440, 1.63660433969957739827661779609, 1.95396193169749418340536233522, 2.32203404163882378370584488188, 2.34560253224922895190592095453, 2.35202316011113328905799288168, 2.62674609962138086070908280520, 2.69391575216320253958544678835, 2.77140603684487745880483767053, 3.12230710723398814600782280558, 3.33366814845751551746754572792, 3.34563087625857584241152384107, 3.50895678723596489284919733067, 3.63533281951626489136344682016, 3.66750024114580001741295949431, 3.78388545626032308706676310635, 4.13089930324050149293044638342, 4.29587508293990532033012130487, 4.33972778177312341499485912927, 4.34980540183823317500781684766, 4.92195656529938840023883681607, 5.01890694427427408822783295610

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

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