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Properties

Label 12-861e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.074\times 10^{17}$
Sign $1$
Analytic cond. $105604.$
Root an. cond. $2.62204$
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  + 3·3-s + 6·4-s + 3·7-s + 3·9-s + 18·12-s − 6·13-s + 12·16-s + 6·17-s + 9·19-s + 9·21-s − 6·23-s + 3·25-s − 2·27-s + 18·28-s − 3·31-s + 18·36-s + 3·37-s − 18·39-s − 6·41-s − 42·43-s + 36·48-s + 15·49-s + 18·51-s − 36·52-s + 6·53-s + 27·57-s − 18·59-s + ⋯
L(s)  = 1  + 1.73·3-s + 3·4-s + 1.13·7-s + 9-s + 5.19·12-s − 1.66·13-s + 3·16-s + 1.45·17-s + 2.06·19-s + 1.96·21-s − 1.25·23-s + 3/5·25-s − 0.384·27-s + 3.40·28-s − 0.538·31-s + 3·36-s + 0.493·37-s − 2.88·39-s − 0.937·41-s − 6.40·43-s + 5.19·48-s + 15/7·49-s + 2.52·51-s − 4.99·52-s + 0.824·53-s + 3.57·57-s − 2.34·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 7^{6} \cdot 41^{6}\)
Sign: $1$
Analytic conductor: \(105604.\)
Root analytic conductor: \(2.62204\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 7^{6} \cdot 41^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.674499851\)
\(L(\frac12)\) \(\approx\) \(8.674499851\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T + T^{2} )^{3} \)
7 \( 1 - 3 T - 6 T^{2} + 37 T^{3} - 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
41 \( ( 1 + T )^{6} \)
good2 \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \)
5 \( 1 - 3 T^{2} + 28 T^{3} - 6 T^{4} - 42 T^{5} + 401 T^{6} - 42 p T^{7} - 6 p^{2} T^{8} + 28 p^{3} T^{9} - 3 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 3 T^{2} - 100 T^{3} - 24 T^{4} + 150 T^{5} + 5063 T^{6} + 150 p T^{7} - 24 p^{2} T^{8} - 100 p^{3} T^{9} - 3 p^{4} T^{10} + p^{6} T^{12} \)
13 \( ( 1 + 3 T + 24 T^{2} + 43 T^{3} + 24 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{3} \)
19 \( 1 - 9 T + 9 T^{2} + 46 T^{3} + 693 T^{4} - 3609 T^{5} + 6390 T^{6} - 3609 p T^{7} + 693 p^{2} T^{8} + 46 p^{3} T^{9} + 9 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 6 T - 33 T^{2} - 134 T^{3} + 1530 T^{4} + 132 p T^{5} - 29137 T^{6} + 132 p^{2} T^{7} + 1530 p^{2} T^{8} - 134 p^{3} T^{9} - 33 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 39 T^{2} - 112 T^{3} + 39 p T^{4} + p^{3} T^{6} )^{2} \)
31 \( 1 + 3 T - 15 T^{2} - 86 T^{3} - 405 T^{4} + 315 T^{5} + 39318 T^{6} + 315 p T^{7} - 405 p^{2} T^{8} - 86 p^{3} T^{9} - 15 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 11 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3} \)
43 \( ( 1 + 21 T + 264 T^{2} + 2069 T^{3} + 264 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 39 T^{2} - 548 T^{3} - 312 T^{4} + 10686 T^{5} + 211235 T^{6} + 10686 p T^{7} - 312 p^{2} T^{8} - 548 p^{3} T^{9} - 39 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 6 T - 63 T^{2} + 118 T^{3} + 2418 T^{4} + 10866 T^{5} - 196087 T^{6} + 10866 p T^{7} + 2418 p^{2} T^{8} + 118 p^{3} T^{9} - 63 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 18 T + 111 T^{2} + 306 T^{3} + 1050 T^{4} - 14022 T^{5} - 282161 T^{6} - 14022 p T^{7} + 1050 p^{2} T^{8} + 306 p^{3} T^{9} + 111 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T - 39 T^{2} + 450 T^{3} + 1050 T^{4} - 534 p T^{5} - 379 p T^{6} - 534 p^{2} T^{7} + 1050 p^{2} T^{8} + 450 p^{3} T^{9} - 39 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 3 T - 177 T^{2} - 248 T^{3} + 20925 T^{4} + 13653 T^{5} - 1594398 T^{6} + 13653 p T^{7} + 20925 p^{2} T^{8} - 248 p^{3} T^{9} - 177 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 141 T^{2} + 144 T^{3} + 141 p T^{4} + p^{3} T^{6} )^{2} \)
73 \( 1 - 21 T + 87 T^{2} - 776 T^{3} + 405 p T^{4} - 205011 T^{5} + 362118 T^{6} - 205011 p T^{7} + 405 p^{3} T^{8} - 776 p^{3} T^{9} + 87 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 21 T + 129 T^{2} - 230 T^{3} + 1485 T^{4} + 153837 T^{5} + 2042502 T^{6} + 153837 p T^{7} + 1485 p^{2} T^{8} - 230 p^{3} T^{9} + 129 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 6 T + 81 T^{2} + 194 T^{3} + 81 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 12 T - 69 T^{2} - 480 T^{3} + 8196 T^{4} - 36582 T^{5} - 1464203 T^{6} - 36582 p T^{7} + 8196 p^{2} T^{8} - 480 p^{3} T^{9} - 69 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 18 T + 207 T^{2} - 1660 T^{3} + 207 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.44849364329614145303879181172, −5.24301923428664390608309679583, −5.06155515202373218430355130207, −4.77824616131442474236448606472, −4.77817107128800868385190486823, −4.77624045988388026060815612794, −4.46423796734062632391566853913, −4.17201424183751846445132792178, −3.87171521844227780051834535057, −3.80127611751353686203705361559, −3.33564115006375693129322165444, −3.26748960388508659568591033854, −3.25360662017491586150078869522, −3.22530981891750119989669103766, −2.86702189231908063230868945898, −2.69867613248757695729624363587, −2.48981720973145589065988477464, −2.37048349498805544536662527765, −2.04252302771863366079069511299, −1.84968371553826701174054429451, −1.75910863596172308839068701778, −1.61971077762429832480937917051, −1.40295552170556915920382186925, −1.01029758725268068588374011878, −0.25928849703570603498450021626, 0.25928849703570603498450021626, 1.01029758725268068588374011878, 1.40295552170556915920382186925, 1.61971077762429832480937917051, 1.75910863596172308839068701778, 1.84968371553826701174054429451, 2.04252302771863366079069511299, 2.37048349498805544536662527765, 2.48981720973145589065988477464, 2.69867613248757695729624363587, 2.86702189231908063230868945898, 3.22530981891750119989669103766, 3.25360662017491586150078869522, 3.26748960388508659568591033854, 3.33564115006375693129322165444, 3.80127611751353686203705361559, 3.87171521844227780051834535057, 4.17201424183751846445132792178, 4.46423796734062632391566853913, 4.77624045988388026060815612794, 4.77817107128800868385190486823, 4.77824616131442474236448606472, 5.06155515202373218430355130207, 5.24301923428664390608309679583, 5.44849364329614145303879181172

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

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