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Properties

Label 12-126e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.002\times 10^{12}$
Sign $1$
Analytic cond. $1.03725$
Root an. cond. $1.00305$
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  − 6·2-s + 2·3-s + 21·4-s + 5-s − 12·6-s + 2·7-s − 56·8-s + 6·9-s − 6·10-s − 11-s + 42·12-s + 8·13-s − 12·14-s + 2·15-s + 126·16-s − 4·17-s − 36·18-s − 3·19-s + 21·20-s + 4·21-s + 6·22-s − 7·23-s − 112·24-s + 9·25-s − 48·26-s + 7·27-s + 42·28-s + ⋯
L(s)  = 1  − 4.24·2-s + 1.15·3-s + 21/2·4-s + 0.447·5-s − 4.89·6-s + 0.755·7-s − 19.7·8-s + 2·9-s − 1.89·10-s − 0.301·11-s + 12.1·12-s + 2.21·13-s − 3.20·14-s + 0.516·15-s + 63/2·16-s − 0.970·17-s − 8.48·18-s − 0.688·19-s + 4.69·20-s + 0.872·21-s + 1.27·22-s − 1.45·23-s − 22.8·24-s + 9/5·25-s − 9.41·26-s + 1.34·27-s + 7.93·28-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(1.03725\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3232762782\)
\(L(\frac12)\) \(\approx\) \(0.3232762782\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{6} \)
3 \( 1 - 2 T - 2 T^{2} + p^{2} T^{3} - 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 - 2 T - 4 T^{2} + 31 T^{3} - 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 - T - 8 T^{2} + 17 T^{3} + 23 T^{4} - 52 T^{5} - 11 T^{6} - 52 p T^{7} + 23 p^{2} T^{8} + 17 p^{3} T^{9} - 8 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + T - 26 T^{2} - 23 T^{3} + 37 p T^{4} + 202 T^{5} - 4853 T^{6} + 202 p T^{7} + 37 p^{3} T^{8} - 23 p^{3} T^{9} - 26 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 1408 p T^{7} - 32 p^{2} T^{8} - 42 p^{3} T^{9} + 24 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 T - 23 T^{2} - 4 p T^{3} + 410 T^{4} + 220 T^{5} - 8111 T^{6} + 220 p T^{7} + 410 p^{2} T^{8} - 4 p^{4} T^{9} - 23 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 54 p T^{7} - 153 p^{2} T^{8} - 67 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 7 T - 32 T^{2} - 83 T^{3} + 2423 T^{4} + 3946 T^{5} - 46865 T^{6} + 3946 p T^{7} + 2423 p^{2} T^{8} - 83 p^{3} T^{9} - 32 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 5 T + 4 T^{2} + 251 T^{3} + 197 T^{4} - 3418 T^{5} + 20293 T^{6} - 3418 p T^{7} + 197 p^{2} T^{8} + 251 p^{3} T^{9} + 4 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 20 T + 214 T^{2} + 1441 T^{3} + 214 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 - 11 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3} \)
41 \( 1 - 90 T^{2} + 18 T^{3} + 4410 T^{4} - 810 T^{5} - 194177 T^{6} - 810 p T^{7} + 4410 p^{2} T^{8} + 18 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 12 T - 6 T^{2} + 547 T^{3} - 6 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )( 1 + 18 T + 198 T^{2} + 1519 T^{3} + 198 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} ) \)
47 \( ( 1 - 9 T + 87 T^{2} - 657 T^{3} + 87 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 - 15 T - 33 T^{3} + 13635 T^{4} - 60360 T^{5} - 225155 T^{6} - 60360 p T^{7} + 13635 p^{2} T^{8} - 33 p^{3} T^{9} - 15 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 - 14 T + 216 T^{2} - 1589 T^{3} + 216 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 8 T + 178 T^{2} + 883 T^{3} + 178 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 + T + 89 T^{2} - 77 T^{3} + 89 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 - 7 T + 15 T^{2} + 599 T^{3} + 15 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 41986 p T^{7} - 5759 p^{2} T^{8} - 27 p^{3} T^{9} + 134 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 + 5 T + 163 T^{2} + 469 T^{3} + 163 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 2 T - 182 T^{2} - 2 T^{3} + 18788 T^{4} + 13564 T^{5} - 1721225 T^{6} + 13564 p T^{7} + 18788 p^{2} T^{8} - 2 p^{3} T^{9} - 182 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 9 T - 144 T^{2} - 1197 T^{3} + 16101 T^{4} + 73314 T^{5} - 1141967 T^{6} + 73314 p T^{7} + 16101 p^{2} T^{8} - 1197 p^{3} T^{9} - 144 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 388196 p T^{7} + 45178 p^{2} T^{8} - 2724 p^{3} T^{9} + 281 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
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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

−7.62045858941007342609328917881, −7.39327235666563156657912512035, −7.31175292918449682689485407400, −7.09042613394326803308586615805, −6.93677617308514214511416834780, −6.88601815630164515905870459924, −6.64343629647277429914394841098, −6.18356623052483934234293234208, −6.07947531546242995099900252362, −5.56369873423253165324445189779, −5.54977942797210003983703452427, −5.54941311319682606125326317940, −5.29011304663618827340426332134, −4.46608122365669305639871803248, −4.20953944430714044671990300953, −3.92107866396784158616243738204, −3.67301402463892483769100570211, −3.44125436550772866138974481360, −3.36667418652079540664379115619, −2.35854856132667627291828329605, −2.33583682642769888899118225151, −2.03750631602677229208003221670, −1.85617955315654329620368695849, −1.55177684972531063012113657708, −0.873986433272503384728038232416, 0.873986433272503384728038232416, 1.55177684972531063012113657708, 1.85617955315654329620368695849, 2.03750631602677229208003221670, 2.33583682642769888899118225151, 2.35854856132667627291828329605, 3.36667418652079540664379115619, 3.44125436550772866138974481360, 3.67301402463892483769100570211, 3.92107866396784158616243738204, 4.20953944430714044671990300953, 4.46608122365669305639871803248, 5.29011304663618827340426332134, 5.54941311319682606125326317940, 5.54977942797210003983703452427, 5.56369873423253165324445189779, 6.07947531546242995099900252362, 6.18356623052483934234293234208, 6.64343629647277429914394841098, 6.88601815630164515905870459924, 6.93677617308514214511416834780, 7.09042613394326803308586615805, 7.31175292918449682689485407400, 7.39327235666563156657912512035, 7.62045858941007342609328917881

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

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