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Properties

Label 12-3381e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.494\times 10^{21}$
Sign $1$
Analytic cond. $3.87198\times 10^{8}$
Root an. cond. $5.19590$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 6·3-s + 2·4-s + 2·5-s + 18·6-s + 2·8-s + 21·9-s − 6·10-s − 3·11-s − 12·12-s − 12·15-s − 4·16-s + 8·17-s − 63·18-s + 19·19-s + 4·20-s + 9·22-s − 6·23-s − 12·24-s − 13·25-s − 56·27-s − 12·29-s + 36·30-s − 2·31-s + 18·33-s − 24·34-s + 42·36-s + ⋯
L(s)  = 1  − 2.12·2-s − 3.46·3-s + 4-s + 0.894·5-s + 7.34·6-s + 0.707·8-s + 7·9-s − 1.89·10-s − 0.904·11-s − 3.46·12-s − 3.09·15-s − 16-s + 1.94·17-s − 14.8·18-s + 4.35·19-s + 0.894·20-s + 1.91·22-s − 1.25·23-s − 2.44·24-s − 2.59·25-s − 10.7·27-s − 2.22·29-s + 6.57·30-s − 0.359·31-s + 3.13·33-s − 4.11·34-s + 7·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 7^{12} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(3.87198\times 10^{8}\)
Root analytic conductor: \(5.19590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 7^{12} \cdot 23^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5075486328\)
\(L(\frac12)\) \(\approx\) \(0.5075486328\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T )^{6} \)
7 \( 1 \)
23 \( ( 1 + T )^{6} \)
good2 \( 1 + 3 T + 7 T^{2} + 13 T^{3} + 23 T^{4} + 41 T^{5} + 15 p^{2} T^{6} + 41 p T^{7} + 23 p^{2} T^{8} + 13 p^{3} T^{9} + 7 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 2 T + 17 T^{2} - 34 T^{3} + 156 T^{4} - 292 T^{5} + 944 T^{6} - 292 p T^{7} + 156 p^{2} T^{8} - 34 p^{3} T^{9} + 17 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 3 T + 28 T^{2} + 43 T^{3} + 324 T^{4} + 431 T^{5} + 3610 T^{6} + 431 p T^{7} + 324 p^{2} T^{8} + 43 p^{3} T^{9} + 28 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 37 T^{2} - 36 T^{3} + 750 T^{4} - 924 T^{5} + 11452 T^{6} - 924 p T^{7} + 750 p^{2} T^{8} - 36 p^{3} T^{9} + 37 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 8 T + 72 T^{2} - 354 T^{3} + 2100 T^{4} - 8360 T^{5} + 41090 T^{6} - 8360 p T^{7} + 2100 p^{2} T^{8} - 354 p^{3} T^{9} + 72 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - p T + 184 T^{2} - 1263 T^{3} + 7512 T^{4} - 40731 T^{5} + 193142 T^{6} - 40731 p T^{7} + 7512 p^{2} T^{8} - 1263 p^{3} T^{9} + 184 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
29 \( 1 + 12 T + 108 T^{2} + 514 T^{3} + 1608 T^{4} - 2940 T^{5} - 34870 T^{6} - 2940 p T^{7} + 1608 p^{2} T^{8} + 514 p^{3} T^{9} + 108 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + 4446 T^{4} - 3142 T^{5} + 138366 T^{6} - 3142 p T^{7} + 4446 p^{2} T^{8} + 2 p^{4} T^{9} + 102 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 10 T + 198 T^{2} + 1566 T^{3} + 17098 T^{4} + 105634 T^{5} + 824306 T^{6} + 105634 p T^{7} + 17098 p^{2} T^{8} + 1566 p^{3} T^{9} + 198 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 3 T + 84 T^{2} - 197 T^{3} + 5372 T^{4} - 9739 T^{5} + 214354 T^{6} - 9739 p T^{7} + 5372 p^{2} T^{8} - 197 p^{3} T^{9} + 84 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 2 T + 133 T^{2} + 412 T^{3} + 8850 T^{4} + 37052 T^{5} + 428672 T^{6} + 37052 p T^{7} + 8850 p^{2} T^{8} + 412 p^{3} T^{9} + 133 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 7 T + 114 T^{2} - 933 T^{3} + 9391 T^{4} - 61470 T^{5} + 587004 T^{6} - 61470 p T^{7} + 9391 p^{2} T^{8} - 933 p^{3} T^{9} + 114 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 5 T + 233 T^{2} + 892 T^{3} + 24754 T^{4} + 74098 T^{5} + 1613358 T^{6} + 74098 p T^{7} + 24754 p^{2} T^{8} + 892 p^{3} T^{9} + 233 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 21 T + 383 T^{2} - 4874 T^{3} + 55520 T^{4} - 512716 T^{5} + 4340148 T^{6} - 512716 p T^{7} + 55520 p^{2} T^{8} - 4874 p^{3} T^{9} + 383 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 29 T + 519 T^{2} - 6864 T^{3} + 76610 T^{4} - 725530 T^{5} + 6068108 T^{6} - 725530 p T^{7} + 76610 p^{2} T^{8} - 6864 p^{3} T^{9} + 519 p^{4} T^{10} - 29 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 16 T + 331 T^{2} - 3372 T^{3} + 42142 T^{4} - 325644 T^{5} + 3289140 T^{6} - 325644 p T^{7} + 42142 p^{2} T^{8} - 3372 p^{3} T^{9} + 331 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 6 T + 81 T^{2} + 954 T^{3} + 16088 T^{4} + 109494 T^{5} + 789148 T^{6} + 109494 p T^{7} + 16088 p^{2} T^{8} + 954 p^{3} T^{9} + 81 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 8 T + 338 T^{2} + 2864 T^{3} + 51790 T^{4} + 413816 T^{5} + 4736906 T^{6} + 413816 p T^{7} + 51790 p^{2} T^{8} + 2864 p^{3} T^{9} + 338 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 12 T + 286 T^{2} + 2932 T^{3} + 46918 T^{4} + 386436 T^{5} + 4532214 T^{6} + 386436 p T^{7} + 46918 p^{2} T^{8} + 2932 p^{3} T^{9} + 286 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 24 T + 464 T^{2} - 5982 T^{3} + 73184 T^{4} - 760172 T^{5} + 7511590 T^{6} - 760172 p T^{7} + 73184 p^{2} T^{8} - 5982 p^{3} T^{9} + 464 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 18 T + 585 T^{2} - 7670 T^{3} + 135752 T^{4} - 1335970 T^{5} + 16308088 T^{6} - 1335970 p T^{7} + 135752 p^{2} T^{8} - 7670 p^{3} T^{9} + 585 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 22 T + 676 T^{2} - 9762 T^{3} + 170704 T^{4} - 1808334 T^{5} + 22292226 T^{6} - 1808334 p T^{7} + 170704 p^{2} T^{8} - 9762 p^{3} T^{9} + 676 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.75880784867699697073310797571, −4.12736472782488055895854608515, −4.03519347376598280038601652166, −4.02761997090267600960412804491, −3.97267259333569621843299216345, −3.86264354727196350534428339958, −3.50360382820155254678503599022, −3.46510746407034967097011532556, −3.38335139177811212016693406609, −3.30067444654186427024768113055, −2.88018319260492007037376280829, −2.78284804831848698792816865340, −2.35541148312317034428221145546, −2.24985829218949329905430065741, −2.10146625295548882971377706597, −2.01344281338171160521297748216, −1.69329063108792627054312654941, −1.54014590685614539395762650509, −1.51919451818873788452295873720, −1.20563037594870880208542780995, −0.815830964029018728311806663114, −0.75302430383553227464502494061, −0.44512638882851918837032058326, −0.44287314956391619596839376745, −0.43630361739468189891374772311, 0.43630361739468189891374772311, 0.44287314956391619596839376745, 0.44512638882851918837032058326, 0.75302430383553227464502494061, 0.815830964029018728311806663114, 1.20563037594870880208542780995, 1.51919451818873788452295873720, 1.54014590685614539395762650509, 1.69329063108792627054312654941, 2.01344281338171160521297748216, 2.10146625295548882971377706597, 2.24985829218949329905430065741, 2.35541148312317034428221145546, 2.78284804831848698792816865340, 2.88018319260492007037376280829, 3.30067444654186427024768113055, 3.38335139177811212016693406609, 3.46510746407034967097011532556, 3.50360382820155254678503599022, 3.86264354727196350534428339958, 3.97267259333569621843299216345, 4.02761997090267600960412804491, 4.03519347376598280038601652166, 4.12736472782488055895854608515, 4.75880784867699697073310797571

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

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