Properties

Degree 12
Conductor $ 43^{6} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s + 9·9-s + 38·11-s + 30·13-s − 25·16-s − 20·17-s − 80·23-s + 33·25-s − 112·31-s + 36·36-s − 172·41-s + 10·43-s + 152·44-s + 30·47-s + 144·49-s + 120·52-s − 110·53-s − 12·59-s − 150·64-s − 70·67-s − 80·68-s + 178·79-s + 55·81-s + 10·83-s − 320·92-s − 380·97-s + 342·99-s + ⋯
L(s)  = 1  + 4-s + 9-s + 3.45·11-s + 2.30·13-s − 1.56·16-s − 1.17·17-s − 3.47·23-s + 1.31·25-s − 3.61·31-s + 36-s − 4.19·41-s + 0.232·43-s + 3.45·44-s + 0.638·47-s + 2.93·49-s + 2.30·52-s − 2.07·53-s − 0.203·59-s − 2.34·64-s − 1.04·67-s − 1.17·68-s + 2.25·79-s + 0.679·81-s + 0.120·83-s − 3.47·92-s − 3.91·97-s + 3.45·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(43^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{43} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((12,\ 43^{6} ,\ ( \ : [1]^{6} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.01757\)
\(L(\frac12)\)  \(\approx\)  \(2.01757\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 12. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad43 \( 1 - 10 T + 147 T^{2} + 3140 p T^{3} + 147 p^{2} T^{4} - 10 p^{4} T^{5} + p^{6} T^{6} \)
good2 \( 1 - p^{2} T^{2} + 41 T^{4} - 57 p T^{6} + 41 p^{4} T^{8} - p^{10} T^{10} + p^{12} T^{12} \)
3 \( 1 - p^{2} T^{2} + 26 T^{4} - 254 T^{6} + 26 p^{4} T^{8} - p^{10} T^{10} + p^{12} T^{12} \)
5 \( 1 - 33 T^{2} + 1538 T^{4} - 41006 T^{6} + 1538 p^{4} T^{8} - 33 p^{8} T^{10} + p^{12} T^{12} \)
7 \( 1 - 144 T^{2} + 11511 T^{4} - 668464 T^{6} + 11511 p^{4} T^{8} - 144 p^{8} T^{10} + p^{12} T^{12} \)
11 \( ( 1 - 19 T + 411 T^{2} - 4354 T^{3} + 411 p^{2} T^{4} - 19 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
13 \( ( 1 - 15 T + 257 T^{2} - 1570 T^{3} + 257 p^{2} T^{4} - 15 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
17 \( ( 1 + 10 T + 767 T^{2} + 5655 T^{3} + 767 p^{2} T^{4} + 10 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
19 \( 1 - 691 T^{2} + 378040 T^{4} - 133433020 T^{6} + 378040 p^{4} T^{8} - 691 p^{8} T^{10} + p^{12} T^{12} \)
23 \( ( 1 + 40 T + 1387 T^{2} + 28195 T^{3} + 1387 p^{2} T^{4} + 40 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
29 \( 1 - 1621 T^{2} + 1946890 T^{4} - 2007466870 T^{6} + 1946890 p^{4} T^{8} - 1621 p^{8} T^{10} + p^{12} T^{12} \)
31 \( ( 1 + 56 T + 2591 T^{2} + 96591 T^{3} + 2591 p^{2} T^{4} + 56 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
37 \( 1 - 5259 T^{2} + 14237096 T^{4} - 24083763884 T^{6} + 14237096 p^{4} T^{8} - 5259 p^{8} T^{10} + p^{12} T^{12} \)
41 \( ( 1 + 86 T + 6451 T^{2} + 292121 T^{3} + 6451 p^{2} T^{4} + 86 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
47 \( ( 1 - 15 T + 5052 T^{2} - 79770 T^{3} + 5052 p^{2} T^{4} - 15 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
53 \( ( 1 + 55 T + 4677 T^{2} + 168490 T^{3} + 4677 p^{2} T^{4} + 55 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
59 \( ( 1 + 6 T + 4271 T^{2} + 147196 T^{3} + 4271 p^{2} T^{4} + 6 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
61 \( 1 - 6176 T^{2} + 53251015 T^{4} - 178029474320 T^{6} + 53251015 p^{4} T^{8} - 6176 p^{8} T^{10} + p^{12} T^{12} \)
67 \( ( 1 + 35 T + 9017 T^{2} + 350730 T^{3} + 9017 p^{2} T^{4} + 35 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
71 \( 1 - 22246 T^{2} + 239223215 T^{4} - 1523736510420 T^{6} + 239223215 p^{4} T^{8} - 22246 p^{8} T^{10} + p^{12} T^{12} \)
73 \( 1 - 24604 T^{2} + 277197791 T^{4} - 1859020745064 T^{6} + 277197791 p^{4} T^{8} - 24604 p^{8} T^{10} + p^{12} T^{12} \)
79 \( ( 1 - 89 T + 20896 T^{2} - 1122134 T^{3} + 20896 p^{2} T^{4} - 89 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
83 \( ( 1 - 5 T + 14767 T^{2} - 128390 T^{3} + 14767 p^{2} T^{4} - 5 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
89 \( 1 - 38876 T^{2} + 668902015 T^{4} - 6712320081320 T^{6} + 668902015 p^{4} T^{8} - 38876 p^{8} T^{10} + p^{12} T^{12} \)
97 \( ( 1 + 190 T + 26827 T^{2} + 2529295 T^{3} + 26827 p^{2} T^{4} + 190 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.942529864243737352898801206085, −8.913803150925598381039744520957, −8.682058964884733326247644908125, −8.588204109117985501052580824096, −8.018269683947631578467838493015, −7.973921695708962753195757935974, −7.42022048927744457369328758716, −7.18485048000757954750595580541, −7.05165582860087801247629754626, −6.58497577907870624809685895576, −6.56021242499329527371779198649, −6.51090801861167189275620107680, −6.32326823478957323977116468781, −5.84500601107391566529592982909, −5.45903340111289014273680447667, −5.34784569095217386448471058573, −4.47745039964027042511682861545, −4.33237981399592215235214907634, −4.05029658287028016652525092089, −3.96291013845544232158186303894, −3.45516535162130046971470401505, −3.27195805147200059555584961471, −1.99211827309154029193629155991, −1.82521485635335349723662469855, −1.65841435123056432683290311835, 1.65841435123056432683290311835, 1.82521485635335349723662469855, 1.99211827309154029193629155991, 3.27195805147200059555584961471, 3.45516535162130046971470401505, 3.96291013845544232158186303894, 4.05029658287028016652525092089, 4.33237981399592215235214907634, 4.47745039964027042511682861545, 5.34784569095217386448471058573, 5.45903340111289014273680447667, 5.84500601107391566529592982909, 6.32326823478957323977116468781, 6.51090801861167189275620107680, 6.56021242499329527371779198649, 6.58497577907870624809685895576, 7.05165582860087801247629754626, 7.18485048000757954750595580541, 7.42022048927744457369328758716, 7.973921695708962753195757935974, 8.018269683947631578467838493015, 8.588204109117985501052580824096, 8.682058964884733326247644908125, 8.913803150925598381039744520957, 8.942529864243737352898801206085

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

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