Properties

Label 12-126e6-1.1-c1e6-0-1
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 2·3-s + 3·4-s − 2·5-s + 6·6-s − 4·7-s − 2·8-s − 6·10-s + 2·11-s + 6·12-s + 8·13-s − 12·14-s − 4·15-s − 9·16-s − 4·17-s − 3·19-s − 6·20-s − 8·21-s + 6·22-s + 14·23-s − 4·24-s − 15·25-s + 24·26-s − 5·27-s − 12·28-s − 5·29-s − 12·30-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 2.44·6-s − 1.51·7-s − 0.707·8-s − 1.89·10-s + 0.603·11-s + 1.73·12-s + 2.21·13-s − 3.20·14-s − 1.03·15-s − 9/4·16-s − 0.970·17-s − 0.688·19-s − 1.34·20-s − 1.74·21-s + 1.27·22-s + 2.91·23-s − 0.816·24-s − 3·25-s + 4.70·26-s − 0.962·27-s − 2.26·28-s − 0.928·29-s − 2.19·30-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

\(L(1)\) \(\approx\) \(2.909486503\)
\(L(\frac12)\) \(\approx\) \(2.909486503\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{3} \)
3 \( 1 - 2 T + 4 T^{2} - p T^{3} + 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 + 4 T + 2 p T^{2} + 55 T^{3} + 2 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( ( 1 + T + 9 T^{2} + 13 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - T + 27 T^{2} - 25 T^{3} + 27 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 - 7 T + 81 T^{2} - 325 T^{3} + 81 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( 1 - 20 T + 6 p T^{2} - 1398 T^{3} + 10342 T^{4} - 62234 T^{5} + 331987 T^{6} - 62234 p T^{7} + 10342 p^{2} T^{8} - 1398 p^{3} T^{9} + 6 p^{5} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 + 9 T - 6 T^{2} - 531 T^{3} - 2433 T^{4} + 3438 T^{5} + 104623 T^{6} + 3438 p T^{7} - 2433 p^{2} T^{8} - 531 p^{3} T^{9} - 6 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 + 14 T - 20 T^{2} - 154 T^{3} + 11666 T^{4} + 35126 T^{5} - 499301 T^{6} + 35126 p T^{7} + 11666 p^{2} T^{8} - 154 p^{3} T^{9} - 20 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 8 T - 114 T^{2} + 342 T^{3} + 13762 T^{4} - 13214 T^{5} - 937217 T^{6} - 13214 p T^{7} + 13762 p^{2} T^{8} + 342 p^{3} T^{9} - 114 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - T - 88 T^{2} - 243 T^{3} + 2035 T^{4} + 14290 T^{5} + 72259 T^{6} + 14290 p T^{7} + 2035 p^{2} T^{8} - 243 p^{3} T^{9} - 88 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 5 T - 138 T^{2} + 123 T^{3} + 11347 T^{4} + 21118 T^{5} - 1048937 T^{6} + 21118 p T^{7} + 11347 p^{2} T^{8} + 123 p^{3} T^{9} - 138 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
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

−7.53950574370617436301038422251, −7.17999232977680790648100722051, −7.07452238466565773544418706421, −6.66515354680586032019837820974, −6.49026569704850197614274030391, −6.40286948112792857949669210977, −6.33901154360039921833016422647, −6.04174285103951081141132427393, −5.95721394063676308002915931262, −5.56917180578508875854682927084, −5.24329848713899403295736960405, −5.17163324064884639422067900811, −4.69393714370206280513870960388, −4.54029461108714216190735535763, −4.45042070536359115715029558497, −4.03751028382164272898075285560, −3.72482347751956533205239016958, −3.65973002710619229074729051696, −3.51011320438515015745342506711, −3.22487293797384578041901895404, −3.14774410492995108083117912936, −2.48978995753422717894583952160, −2.43302439274686321422683307550, −1.91688645512099114379754115180, −0.933624983337636158523858647482, 0.933624983337636158523858647482, 1.91688645512099114379754115180, 2.43302439274686321422683307550, 2.48978995753422717894583952160, 3.14774410492995108083117912936, 3.22487293797384578041901895404, 3.51011320438515015745342506711, 3.65973002710619229074729051696, 3.72482347751956533205239016958, 4.03751028382164272898075285560, 4.45042070536359115715029558497, 4.54029461108714216190735535763, 4.69393714370206280513870960388, 5.17163324064884639422067900811, 5.24329848713899403295736960405, 5.56917180578508875854682927084, 5.95721394063676308002915931262, 6.04174285103951081141132427393, 6.33901154360039921833016422647, 6.40286948112792857949669210977, 6.49026569704850197614274030391, 6.66515354680586032019837820974, 7.07452238466565773544418706421, 7.17999232977680790648100722051, 7.53950574370617436301038422251

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

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