↑

Properties

Label 12-675e6-1.1-c1e6-0-0
Degree $12$
Conductor $9.459\times 10^{16}$
Sign $1$
Analytic cond. $24518.0$
Root an. cond. $2.32161$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5·7-s − 2·11-s + 4·13-s − 5·14-s + 16-s − 4·17-s + 8·19-s + 2·22-s − 3·23-s − 4·26-s + 5·28-s − 7·29-s − 8·31-s − 3·32-s + 4·34-s − 12·37-s − 8·38-s − 13·41-s + 10·43-s − 2·44-s + 3·46-s − 13·47-s + 24·49-s + 4·52-s − 4·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.88·7-s − 0.603·11-s + 1.10·13-s − 1.33·14-s + 1/4·16-s − 0.970·17-s + 1.83·19-s + 0.426·22-s − 0.625·23-s − 0.784·26-s + 0.944·28-s − 1.29·29-s − 1.43·31-s − 0.530·32-s + 0.685·34-s − 1.97·37-s − 1.29·38-s − 2.03·41-s + 1.52·43-s − 0.301·44-s + 0.442·46-s − 1.89·47-s + 24/7·49-s + 0.554·52-s − 0.549·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 5^{12}\)
Sign: $1$
Analytic conductor: \(24518.0\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 5^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3899089104\)
\(L(\frac12)\) \(\approx\) \(0.3899089104\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + T - T^{3} - p T^{4} + T^{5} + 11 T^{6} + p T^{7} - p^{3} T^{8} - p^{3} T^{9} + p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 5 T + T^{2} + 2 p T^{3} + 73 T^{4} - 173 T^{5} - 82 T^{6} - 173 p T^{7} + 73 p^{2} T^{8} + 2 p^{4} T^{9} + p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T - 21 T^{2} - 14 T^{3} + 26 p T^{4} - 58 T^{5} - 3673 T^{6} - 58 p T^{7} + 26 p^{3} T^{8} - 14 p^{3} T^{9} - 21 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 4 T - 19 T^{2} + 60 T^{3} + 370 T^{4} - 536 T^{5} - 4235 T^{6} - 536 p T^{7} + 370 p^{2} T^{8} + 60 p^{3} T^{9} - 19 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 + 2 T + 43 T^{2} + 56 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 - 4 T + 53 T^{2} - 148 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 + 3 T - 27 T^{2} + 66 T^{3} + 405 T^{4} - 2625 T^{5} - 9794 T^{6} - 2625 p T^{7} + 405 p^{2} T^{8} + 66 p^{3} T^{9} - 27 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 7 T - 9 T^{2} - 304 T^{3} - 803 T^{4} + 101 p T^{5} + 36038 T^{6} + 101 p^{2} T^{7} - 803 p^{2} T^{8} - 304 p^{3} T^{9} - 9 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 8 T + p T^{2} + 208 T^{3} - 158 T^{4} - 7756 T^{5} - 34897 T^{6} - 7756 p T^{7} - 158 p^{2} T^{8} + 208 p^{3} T^{9} + p^{5} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 6 T + 99 T^{2} + 440 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 13 T + 27 T^{2} - 292 T^{3} + 445 T^{4} + 22279 T^{5} + 169790 T^{6} + 22279 p T^{7} + 445 p^{2} T^{8} - 292 p^{3} T^{9} + 27 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 10 T - 25 T^{2} + 462 T^{3} + 1690 T^{4} - 17990 T^{5} + 34975 T^{6} - 17990 p T^{7} + 1690 p^{2} T^{8} + 462 p^{3} T^{9} - 25 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 13 T + 39 T^{2} - 16 T^{3} - 299 T^{4} - 19253 T^{5} - 232366 T^{6} - 19253 p T^{7} - 299 p^{2} T^{8} - 16 p^{3} T^{9} + 39 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 2 T + 139 T^{2} + 188 T^{3} + 139 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 2 T - 153 T^{2} - 110 T^{3} + 14962 T^{4} + 3194 T^{5} - 1012513 T^{6} + 3194 p T^{7} + 14962 p^{2} T^{8} - 110 p^{3} T^{9} - 153 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + T - 145 T^{2} - 240 T^{3} + 12217 T^{4} + 14087 T^{5} - 812786 T^{6} + 14087 p T^{7} + 12217 p^{2} T^{8} - 240 p^{3} T^{9} - 145 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 11 T - 41 T^{2} + 152 T^{3} + 4753 T^{4} + 32755 T^{5} - 764902 T^{6} + 32755 p T^{7} + 4753 p^{2} T^{8} + 152 p^{3} T^{9} - 41 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 10 T + 121 T^{2} - 712 T^{3} + 121 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 8 T + 155 T^{2} - 1040 T^{3} + 155 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 + 2 T - 149 T^{2} - 374 T^{3} + 10642 T^{4} + 16154 T^{5} - 772597 T^{6} + 16154 p T^{7} + 10642 p^{2} T^{8} - 374 p^{3} T^{9} - 149 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 15 T - 51 T^{2} + 678 T^{3} + 17133 T^{4} - 80979 T^{5} - 925778 T^{6} - 80979 p T^{7} + 17133 p^{2} T^{8} + 678 p^{3} T^{9} - 51 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 - 3 T + p T^{2} )^{6} \)
97 \( 1 + 18 T + 69 T^{2} + 214 T^{3} + 906 T^{4} - 163158 T^{5} - 2743251 T^{6} - 163158 p T^{7} + 906 p^{2} T^{8} + 214 p^{3} T^{9} + 69 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.45527944220360320357748034937, −5.39227794385416358831091265240, −5.27707201120011134649945293660, −5.06038852451021414638018842775, −5.04417334236876156555514805270, −4.85272215278122223556681619845, −4.78177851836079994314696131305, −4.44044203061591774518737120080, −4.22945125352899184850075259714, −3.91279426596856913573648175749, −3.64500975778967013080768340106, −3.58336647593880266171882191332, −3.55530562150653588454359337568, −3.49258447037491906021984847057, −3.22122422569522911619692401394, −2.73715037349610513840168530819, −2.32630054808145339630130504395, −2.28573164272010022827509085849, −2.19740894355872632037123980973, −2.03114645312125324038751132377, −1.50283543159909682813111091049, −1.49364915273591953502380306065, −1.09400445785701086496586887870, −1.03262825331258798862399361285, −0.12071564774306242017003622109, 0.12071564774306242017003622109, 1.03262825331258798862399361285, 1.09400445785701086496586887870, 1.49364915273591953502380306065, 1.50283543159909682813111091049, 2.03114645312125324038751132377, 2.19740894355872632037123980973, 2.28573164272010022827509085849, 2.32630054808145339630130504395, 2.73715037349610513840168530819, 3.22122422569522911619692401394, 3.49258447037491906021984847057, 3.55530562150653588454359337568, 3.58336647593880266171882191332, 3.64500975778967013080768340106, 3.91279426596856913573648175749, 4.22945125352899184850075259714, 4.44044203061591774518737120080, 4.78177851836079994314696131305, 4.85272215278122223556681619845, 5.04417334236876156555514805270, 5.06038852451021414638018842775, 5.27707201120011134649945293660, 5.39227794385416358831091265240, 5.45527944220360320357748034937

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

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