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Properties

Label 12-2646e6-1.1-c1e6-0-2
Degree $12$
Conductor $3.432\times 10^{20}$
Sign $1$
Analytic cond. $8.89614\times 10^{7}$
Root an. cond. $4.59656$
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 + 3·4-s + 10·5-s − 2·8-s + 30·10-s − 2·11-s + 2·13-s − 9·16-s − 4·17-s + 3·19-s + 30·20-s − 6·22-s − 14·23-s + 37·25-s + 6·26-s + 5·29-s + 14·31-s − 9·32-s − 12·34-s − 9·37-s + 9·38-s − 20·40-s − 12·41-s + 18·43-s − 6·44-s − 42·46-s + 3·47-s + ⋯
L(s)  = 1  + 2.12·2-s + 3/2·4-s + 4.47·5-s − 0.707·8-s + 9.48·10-s − 0.603·11-s + 0.554·13-s − 9/4·16-s − 0.970·17-s + 0.688·19-s + 6.70·20-s − 1.27·22-s − 2.91·23-s + 37/5·25-s + 1.17·26-s + 0.928·29-s + 2.51·31-s − 1.59·32-s − 2.05·34-s − 1.47·37-s + 1.45·38-s − 3.16·40-s − 1.87·41-s + 2.74·43-s − 0.904·44-s − 6.19·46-s + 0.437·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{18} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(8.89614\times 10^{7}\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(19.28038805\)
\(L(\frac12)\) \(\approx\) \(19.28038805\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( ( 1 - p T + 19 T^{2} - 47 T^{3} + 19 p T^{4} - p^{3} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 + T + 7 T^{2} + 5 p T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 2 T - 32 T^{2} + 2 p T^{3} + 730 T^{4} - 230 T^{5} - 10729 T^{6} - 230 p T^{7} + 730 p^{2} T^{8} + 2 p^{4} T^{9} - 32 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 T + 9 T^{2} + 92 T^{3} + 58 T^{4} - 20 T^{5} + 5393 T^{6} - 20 p T^{7} + 58 p^{2} T^{8} + 92 p^{3} T^{9} + 9 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T - 42 T^{2} + 61 T^{3} + 69 p T^{4} - 726 T^{5} - 27501 T^{6} - 726 p T^{7} + 69 p^{3} T^{8} + 61 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 + 7 T + 73 T^{2} + 319 T^{3} + 73 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 - 5 T - 30 T^{2} + 371 T^{3} - 185 T^{4} - 6020 T^{5} + 44357 T^{6} - 6020 p T^{7} - 185 p^{2} T^{8} + 371 p^{3} T^{9} - 30 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 14 T + 58 T^{2} - 250 T^{3} + 2992 T^{4} - 9728 T^{5} - 11857 T^{6} - 9728 p T^{7} + 2992 p^{2} T^{8} - 250 p^{3} T^{9} + 58 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 4875 p T^{7} + 1293 p^{2} T^{8} - 268 p^{3} T^{9} - 21 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 12 T - 18 T^{2} - 78 T^{3} + 7470 T^{4} + 24546 T^{5} - 158105 T^{6} + 24546 p T^{7} + 7470 p^{2} T^{8} - 78 p^{3} T^{9} - 18 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 18 T + 114 T^{2} - 682 T^{3} + 7188 T^{4} - 33492 T^{5} + 63039 T^{6} - 33492 p T^{7} + 7188 p^{2} T^{8} - 682 p^{3} T^{9} + 114 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 3 T - 108 T^{2} + 267 T^{3} + 7263 T^{4} - 9786 T^{5} - 360137 T^{6} - 9786 p T^{7} + 7263 p^{2} T^{8} + 267 p^{3} T^{9} - 108 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 9 T - 36 T^{2} - 873 T^{3} - 1179 T^{4} + 26334 T^{5} + 272077 T^{6} + 26334 p T^{7} - 1179 p^{2} T^{8} - 873 p^{3} T^{9} - 36 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 4 T - 60 T^{2} + 994 T^{3} - 1304 T^{4} - 464 p T^{5} + 7381 p T^{6} - 464 p^{2} T^{7} - 1304 p^{2} T^{8} + 994 p^{3} T^{9} - 60 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 4 T - 32 T^{2} + 650 T^{3} + 292 T^{4} - 19532 T^{5} + 306323 T^{6} - 19532 p T^{7} + 292 p^{2} T^{8} + 650 p^{3} T^{9} - 32 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 5 T - 118 T^{2} + 327 T^{3} + 8263 T^{4} - 1138 T^{5} - 609341 T^{6} - 1138 p T^{7} + 8263 p^{2} T^{8} + 327 p^{3} T^{9} - 118 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 7 T + 163 T^{2} + 895 T^{3} + 163 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 25 T + 254 T^{2} - 2073 T^{3} + 20533 T^{4} - 115046 T^{5} + 366817 T^{6} - 115046 p T^{7} + 20533 p^{2} T^{8} - 2073 p^{3} T^{9} + 254 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 7 T - 44 T^{2} + 19 T^{3} - 1043 T^{4} + 28016 T^{5} + 109223 T^{6} + 28016 p T^{7} - 1043 p^{2} T^{8} + 19 p^{3} T^{9} - 44 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 8 T - 180 T^{2} + 518 T^{3} + 29404 T^{4} - 32420 T^{5} - 2713585 T^{6} - 32420 p T^{7} + 29404 p^{2} T^{8} + 518 p^{3} T^{9} - 180 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 9 T - 180 T^{2} - 729 T^{3} + 31041 T^{4} + 54846 T^{5} - 2925911 T^{6} + 54846 p T^{7} + 31041 p^{2} T^{8} - 729 p^{3} T^{9} - 180 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 28 T + 257 T^{2} - 2820 T^{3} + 59506 T^{4} - 545924 T^{5} + 3126001 T^{6} - 545924 p T^{7} + 59506 p^{2} T^{8} - 2820 p^{3} T^{9} + 257 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.77284085007911669632767663643, −4.39895763694553797256286418995, −4.38182416356089857901573434537, −4.14079198770771820368563678717, −4.05540813532629377504447923372, −3.98938763013626370649639004378, −3.69392261832816217191935521782, −3.53026728226374695063446698909, −3.38307198855084322250640505783, −3.30698184618755927877814865014, −3.13207312624467507636956722996, −2.79860340199975724241502689734, −2.77838004949234959925705862453, −2.44347804178324953485902333107, −2.26389189478395841312066142178, −2.18807736357311127855259370649, −2.16673592917480656095787711524, −2.11067133706490015377796716405, −1.94587507594735887258855661482, −1.50141574611203974581791229245, −1.42709510953193061549702397041, −1.21988274899989367409107153294, −0.956690062478187076041672127339, −0.56034531404794795838742398385, −0.21010193264107049735090376633, 0.21010193264107049735090376633, 0.56034531404794795838742398385, 0.956690062478187076041672127339, 1.21988274899989367409107153294, 1.42709510953193061549702397041, 1.50141574611203974581791229245, 1.94587507594735887258855661482, 2.11067133706490015377796716405, 2.16673592917480656095787711524, 2.18807736357311127855259370649, 2.26389189478395841312066142178, 2.44347804178324953485902333107, 2.77838004949234959925705862453, 2.79860340199975724241502689734, 3.13207312624467507636956722996, 3.30698184618755927877814865014, 3.38307198855084322250640505783, 3.53026728226374695063446698909, 3.69392261832816217191935521782, 3.98938763013626370649639004378, 4.05540813532629377504447923372, 4.14079198770771820368563678717, 4.38182416356089857901573434537, 4.39895763694553797256286418995, 4.77284085007911669632767663643

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

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