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Properties

Label 12-8037e6-1.1-c1e6-0-0
Degree $12$
Conductor $2.695\times 10^{23}$
Sign $1$
Analytic cond. $6.98596\times 10^{10}$
Root an. cond. $8.01097$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 2·4-s + 4·5-s + 2·7-s + 14·8-s − 16·10-s + 10·11-s − 8·13-s − 8·14-s − 21·16-s + 3·17-s − 6·19-s + 8·20-s − 40·22-s − 7·25-s + 32·26-s + 4·28-s − 2·31-s − 14·32-s − 12·34-s + 8·35-s − 9·37-s + 24·38-s + 56·40-s − 18·41-s + 9·43-s + 20·44-s + ⋯
L(s)  = 1  − 2.82·2-s + 4-s + 1.78·5-s + 0.755·7-s + 4.94·8-s − 5.05·10-s + 3.01·11-s − 2.21·13-s − 2.13·14-s − 5.25·16-s + 0.727·17-s − 1.37·19-s + 1.78·20-s − 8.52·22-s − 7/5·25-s + 6.27·26-s + 0.755·28-s − 0.359·31-s − 2.47·32-s − 2.05·34-s + 1.35·35-s − 1.47·37-s + 3.89·38-s + 8.85·40-s − 2.81·41-s + 1.37·43-s + 3.01·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 19^{6} \cdot 47^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 19^{6} \cdot 47^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 19^{6} \cdot 47^{6}\)
Sign: $1$
Analytic conductor: \(6.98596\times 10^{10}\)
Root analytic conductor: \(8.01097\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{12} \cdot 19^{6} \cdot 47^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( ( 1 + T )^{6} \)
47 \( ( 1 - T )^{6} \)
good2 \( ( 1 + p T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + p^{3} T^{5} + p^{3} T^{6} )^{2} \)
5 \( 1 - 4 T + 23 T^{2} - 14 p T^{3} + 249 T^{4} - 596 T^{5} + 1589 T^{6} - 596 p T^{7} + 249 p^{2} T^{8} - 14 p^{4} T^{9} + 23 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 2 T + 31 T^{2} - 51 T^{3} + 459 T^{4} - 625 T^{5} + 4066 T^{6} - 625 p T^{7} + 459 p^{2} T^{8} - 51 p^{3} T^{9} + 31 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( ( 1 - 5 T + 39 T^{2} - 111 T^{3} + 39 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 + 8 T + 77 T^{2} + 368 T^{3} + 2037 T^{4} + 7120 T^{5} + 2397 p T^{6} + 7120 p T^{7} + 2037 p^{2} T^{8} + 368 p^{3} T^{9} + 77 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 3 T + 75 T^{2} - 196 T^{3} + 2552 T^{4} - 5744 T^{5} + 53231 T^{6} - 5744 p T^{7} + 2552 p^{2} T^{8} - 196 p^{3} T^{9} + 75 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 116 T^{2} - 7 T^{3} + 6021 T^{4} - 420 T^{5} + 178531 T^{6} - 420 p T^{7} + 6021 p^{2} T^{8} - 7 p^{3} T^{9} + 116 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 + 159 T^{2} - 14 T^{3} + 10901 T^{4} - 1204 T^{5} + 413585 T^{6} - 1204 p T^{7} + 10901 p^{2} T^{8} - 14 p^{3} T^{9} + 159 p^{4} T^{10} + p^{6} T^{12} \)
31 \( 1 + 2 T + 144 T^{2} + 259 T^{3} + 9697 T^{4} + 14702 T^{5} + 382403 T^{6} + 14702 p T^{7} + 9697 p^{2} T^{8} + 259 p^{3} T^{9} + 144 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 9 T + 168 T^{2} + 1122 T^{3} + 13049 T^{4} + 68721 T^{5} + 599564 T^{6} + 68721 p T^{7} + 13049 p^{2} T^{8} + 1122 p^{3} T^{9} + 168 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 18 T + 282 T^{2} + 2678 T^{3} + 24284 T^{4} + 165316 T^{5} + 1178303 T^{6} + 165316 p T^{7} + 24284 p^{2} T^{8} + 2678 p^{3} T^{9} + 282 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 9 T + 199 T^{2} - 1516 T^{3} + 18842 T^{4} - 114176 T^{5} + 1039451 T^{6} - 114176 p T^{7} + 18842 p^{2} T^{8} - 1516 p^{3} T^{9} + 199 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 20 T + 331 T^{2} + 3296 T^{3} + 32643 T^{4} + 245470 T^{5} + 2014567 T^{6} + 245470 p T^{7} + 32643 p^{2} T^{8} + 3296 p^{3} T^{9} + 331 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 19 T + 321 T^{2} + 4342 T^{3} + 46963 T^{4} + 439027 T^{5} + 3715246 T^{6} + 439027 p T^{7} + 46963 p^{2} T^{8} + 4342 p^{3} T^{9} + 321 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 254 T^{2} - 287 T^{3} + 30937 T^{4} - 42378 T^{5} + 2346877 T^{6} - 42378 p T^{7} + 30937 p^{2} T^{8} - 287 p^{3} T^{9} + 254 p^{4} T^{10} + p^{6} T^{12} \)
67 \( 1 + 11 T + 325 T^{2} + 2724 T^{3} + 45390 T^{4} + 302404 T^{5} + 3767923 T^{6} + 302404 p T^{7} + 45390 p^{2} T^{8} + 2724 p^{3} T^{9} + 325 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 4 T + 49 T^{2} + 51 T^{3} + 6323 T^{4} + 26503 T^{5} + 637182 T^{6} + 26503 p T^{7} + 6323 p^{2} T^{8} + 51 p^{3} T^{9} + 49 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 7 T + 185 T^{2} + 672 T^{3} + 12168 T^{4} - 8428 T^{5} + 529409 T^{6} - 8428 p T^{7} + 12168 p^{2} T^{8} + 672 p^{3} T^{9} + 185 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 16 T + 306 T^{2} + 2608 T^{3} + 28767 T^{4} + 165024 T^{5} + 1883740 T^{6} + 165024 p T^{7} + 28767 p^{2} T^{8} + 2608 p^{3} T^{9} + 306 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 7 T + 396 T^{2} + 2254 T^{3} + 71371 T^{4} + 331807 T^{5} + 90888 p T^{6} + 331807 p T^{7} + 71371 p^{2} T^{8} + 2254 p^{3} T^{9} + 396 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 33 T + 817 T^{2} - 13486 T^{3} + 190860 T^{4} - 2159444 T^{5} + 22244565 T^{6} - 2159444 p T^{7} + 190860 p^{2} T^{8} - 13486 p^{3} T^{9} + 817 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 34 T + 880 T^{2} + 15486 T^{3} + 234990 T^{4} + 2851746 T^{5} + 30897163 T^{6} + 2851746 p T^{7} + 234990 p^{2} T^{8} + 15486 p^{3} T^{9} + 880 p^{4} T^{10} + 34 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.50632809344538285544646527902, −4.23890199504061087232027683899, −3.98617148576325527202546963073, −3.92908797048522494014495680593, −3.82784996315588437482520399851, −3.81641452398758424886083381794, −3.70498894841501328283536023385, −3.54104696273800961928017059182, −3.14529472260722086216303217776, −3.12561612158013953714965373771, −3.08804804629659407991473909764, −2.83876264915164775484686019842, −2.68721466694536653821809784331, −2.44508278444112968966350496328, −2.38070096384981686871208973875, −2.02720848257640575671940296118, −1.96367986539576600764677174208, −1.80622730817413078744235387250, −1.78930304800512111978655653184, −1.50707655294898519253113916912, −1.31161664958098143608497715862, −1.29362272557770776439860526633, −1.28008406870026559651820749566, −1.09706176254690194968412883005, −0.879201654812118362696909576014, 0, 0, 0, 0, 0, 0, 0.879201654812118362696909576014, 1.09706176254690194968412883005, 1.28008406870026559651820749566, 1.29362272557770776439860526633, 1.31161664958098143608497715862, 1.50707655294898519253113916912, 1.78930304800512111978655653184, 1.80622730817413078744235387250, 1.96367986539576600764677174208, 2.02720848257640575671940296118, 2.38070096384981686871208973875, 2.44508278444112968966350496328, 2.68721466694536653821809784331, 2.83876264915164775484686019842, 3.08804804629659407991473909764, 3.12561612158013953714965373771, 3.14529472260722086216303217776, 3.54104696273800961928017059182, 3.70498894841501328283536023385, 3.81641452398758424886083381794, 3.82784996315588437482520399851, 3.92908797048522494014495680593, 3.98617148576325527202546963073, 4.23890199504061087232027683899, 4.50632809344538285544646527902

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

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