Properties

Label 12-1323e6-1.1-c1e6-0-8
Degree $12$
Conductor $5.362\times 10^{18}$
Sign $1$
Analytic cond. $1.39002\times 10^{6}$
Root an. cond. $3.25026$
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  + 2·2-s − 4-s + 5·5-s − 2·8-s + 10·10-s − 2·11-s + 3·13-s + 3·16-s + 12·17-s − 3·19-s − 5·20-s − 4·22-s + 17·25-s + 6·26-s + 29-s + 6·31-s − 2·32-s + 24·34-s + 3·37-s − 6·38-s − 10·40-s + 22·41-s + 3·43-s + 2·44-s − 18·47-s + 34·50-s − 3·52-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s + 2.23·5-s − 0.707·8-s + 3.16·10-s − 0.603·11-s + 0.832·13-s + 3/4·16-s + 2.91·17-s − 0.688·19-s − 1.11·20-s − 0.852·22-s + 17/5·25-s + 1.17·26-s + 0.185·29-s + 1.07·31-s − 0.353·32-s + 4.11·34-s + 0.493·37-s − 0.973·38-s − 1.58·40-s + 3.43·41-s + 0.457·43-s + 0.301·44-s − 2.62·47-s + 4.80·50-s − 0.416·52-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1.39002\times 10^{6}\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(24.46479097\)
\(L(\frac12)\) \(\approx\) \(24.46479097\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( ( 1 - T + p T^{2} - 3 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
5 \( 1 - p T + 8 T^{2} - 7 T^{3} + 9 T^{4} + 62 T^{5} - 299 T^{6} + 62 p T^{7} + 9 p^{2} T^{8} - 7 p^{3} T^{9} + 8 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T - 10 T^{2} + 34 T^{3} + 48 T^{4} - 416 T^{5} + 31 T^{6} - 416 p T^{7} + 48 p^{2} T^{8} + 34 p^{3} T^{9} - 10 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \)
17 \( 1 - 12 T + 54 T^{2} - 210 T^{3} + 1350 T^{4} - 5898 T^{5} + 19735 T^{6} - 5898 p T^{7} + 1350 p^{2} T^{8} - 210 p^{3} T^{9} + 54 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 36 T^{2} + 18 T^{3} + 468 T^{4} - 324 T^{5} - 5393 T^{6} - 324 p T^{7} + 468 p^{2} T^{8} + 18 p^{3} T^{9} - 36 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 - T - 82 T^{2} + 31 T^{3} + 4425 T^{4} - 758 T^{5} - 148595 T^{6} - 758 p T^{7} + 4425 p^{2} T^{8} + 31 p^{3} T^{9} - 82 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 3 T + 69 T^{2} - 213 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 3 T - 48 T^{2} + 435 T^{3} + 231 T^{4} - 8724 T^{5} + 60581 T^{6} - 8724 p T^{7} + 231 p^{2} T^{8} + 435 p^{3} T^{9} - 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 22 T + 206 T^{2} - 1802 T^{3} + 18432 T^{4} - 135116 T^{5} + 808243 T^{6} - 135116 p T^{7} + 18432 p^{2} T^{8} - 1802 p^{3} T^{9} + 206 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 13170 p T^{7} + 123 p^{2} T^{8} + 569 p^{3} T^{9} - 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 + 18 T + 90 T^{2} + 378 T^{3} + 7848 T^{4} + 52668 T^{5} + 160459 T^{6} + 52668 p T^{7} + 7848 p^{2} T^{8} + 378 p^{3} T^{9} + 90 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 + 9 T + 171 T^{2} + 999 T^{3} + 171 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 6 T + 162 T^{2} - 665 T^{3} + 162 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 - 6 T^{2} + 683 T^{3} - 6 p T^{4} + p^{3} T^{6} )^{2} \)
71 \( ( 1 + 9 T + 207 T^{2} + 1197 T^{3} + 207 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 3 T - 42 T^{2} + 1209 T^{3} - 3165 T^{4} - 28380 T^{5} + 1003961 T^{6} - 28380 p T^{7} - 3165 p^{2} T^{8} + 1209 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 - 15 T + 189 T^{2} - 1601 T^{3} + 189 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 12 T - 144 T^{2} + 582 T^{3} + 34812 T^{4} - 90444 T^{5} - 2656433 T^{6} - 90444 p T^{7} + 34812 p^{2} T^{8} + 582 p^{3} T^{9} - 144 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 2 T - 112 T^{2} + 1238 T^{3} + 1662 T^{4} - 59806 T^{5} + 720895 T^{6} - 59806 p T^{7} + 1662 p^{2} T^{8} + 1238 p^{3} T^{9} - 112 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 3 T - 168 T^{2} - 573 T^{3} + 14223 T^{4} + 78504 T^{5} - 1297807 T^{6} + 78504 p T^{7} + 14223 p^{2} T^{8} - 573 p^{3} T^{9} - 168 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.08144259704426812142944922539, −4.98877602546986830548731473105, −4.69179542635325791528519206505, −4.62269970865391802463004384528, −4.50297923704865235988465907203, −4.47438997810788147694569975501, −4.34456310924181443946902815707, −3.80310330032698322195558099698, −3.71779209579323638829133581267, −3.61063158211119605973804945418, −3.52184256725330665546435667056, −3.48946403948986409787338978783, −2.97858776886615032297766455506, −2.94143375880927461776752814589, −2.76513015153184071339578959636, −2.49096823132822699803004694876, −2.47579428703840871778787577310, −2.24953764418134427057046053530, −1.76837755677781027413895749337, −1.54136423197958814037728501605, −1.50135411808745831132655887918, −1.45180098131268483888199772694, −0.975115404953303915485986360416, −0.62733038170574024586408531600, −0.55770895285854709590045493652, 0.55770895285854709590045493652, 0.62733038170574024586408531600, 0.975115404953303915485986360416, 1.45180098131268483888199772694, 1.50135411808745831132655887918, 1.54136423197958814037728501605, 1.76837755677781027413895749337, 2.24953764418134427057046053530, 2.47579428703840871778787577310, 2.49096823132822699803004694876, 2.76513015153184071339578959636, 2.94143375880927461776752814589, 2.97858776886615032297766455506, 3.48946403948986409787338978783, 3.52184256725330665546435667056, 3.61063158211119605973804945418, 3.71779209579323638829133581267, 3.80310330032698322195558099698, 4.34456310924181443946902815707, 4.47438997810788147694569975501, 4.50297923704865235988465907203, 4.62269970865391802463004384528, 4.69179542635325791528519206505, 4.98877602546986830548731473105, 5.08144259704426812142944922539

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

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