Properties

Label 12-5070e6-1.1-c1e6-0-7
Degree $12$
Conductor $1.698\times 10^{22}$
Sign $1$
Analytic cond. $4.40261\times 10^{9}$
Root an. cond. $6.36271$
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  + 6·3-s − 3·4-s + 21·9-s − 18·12-s + 6·16-s − 2·17-s − 6·23-s − 3·25-s + 56·27-s + 12·29-s − 63·36-s − 2·43-s + 36·48-s + 36·49-s − 12·51-s − 10·53-s + 10·61-s − 10·64-s + 6·68-s − 36·69-s − 18·75-s − 34·79-s + 126·81-s + 72·87-s + 18·92-s + 9·100-s − 4·101-s + ⋯
L(s)  = 1  + 3.46·3-s − 3/2·4-s + 7·9-s − 5.19·12-s + 3/2·16-s − 0.485·17-s − 1.25·23-s − 3/5·25-s + 10.7·27-s + 2.22·29-s − 10.5·36-s − 0.304·43-s + 5.19·48-s + 36/7·49-s − 1.68·51-s − 1.37·53-s + 1.28·61-s − 5/4·64-s + 0.727·68-s − 4.33·69-s − 2.07·75-s − 3.82·79-s + 14·81-s + 7.71·87-s + 1.87·92-s + 9/10·100-s − 0.398·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(4.40261\times 10^{9}\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(34.28871445\)
\(L(\frac12)\) \(\approx\) \(34.28871445\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{3} \)
3 \( ( 1 - T )^{6} \)
5 \( ( 1 + T^{2} )^{3} \)
13 \( 1 \)
good7 \( 1 - 36 T^{2} + 572 T^{4} - 5165 T^{6} + 572 p^{2} T^{8} - 36 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 52 T^{2} + 1248 T^{4} - 17485 T^{6} + 1248 p^{2} T^{8} - 52 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 + T + 49 T^{2} + 33 T^{3} + 49 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 - 45 T^{2} + 981 T^{4} - 16657 T^{6} + 981 p^{2} T^{8} - 45 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 + 3 T + 65 T^{2} + 139 T^{3} + 65 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 - 6 T + 92 T^{2} - 349 T^{3} + 92 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 148 T^{2} + 10084 T^{4} - 399493 T^{6} + 10084 p^{2} T^{8} - 148 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 - 168 T^{2} + 12584 T^{4} - 572537 T^{6} + 12584 p^{2} T^{8} - 168 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 201 T^{2} + 18237 T^{4} - 957345 T^{6} + 18237 p^{2} T^{8} - 201 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 + T + 99 T^{2} + 127 T^{3} + 99 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 247 T^{2} + 26702 T^{4} - 1626339 T^{6} + 26702 p^{2} T^{8} - 247 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 + 5 T + 151 T^{2} + 531 T^{3} + 151 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 109 T^{2} + 8801 T^{4} - 572769 T^{6} + 8801 p^{2} T^{8} - 109 p^{4} T^{10} + p^{6} T^{12} \)
61 \( ( 1 - 5 T + 119 T^{2} - 639 T^{3} + 119 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 333 T^{2} + 50297 T^{4} - 4342241 T^{6} + 50297 p^{2} T^{8} - 333 p^{4} T^{10} + p^{6} T^{12} \)
71 \( 1 - 197 T^{2} + 22737 T^{4} - 1782761 T^{6} + 22737 p^{2} T^{8} - 197 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 - 197 T^{2} + 349 p T^{4} - 2222489 T^{6} + 349 p^{3} T^{8} - 197 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 17 T + 261 T^{2} + 2393 T^{3} + 261 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 464 T^{2} + 92388 T^{4} - 10086149 T^{6} + 92388 p^{2} T^{8} - 464 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 317 T^{2} + 52833 T^{4} - 5701697 T^{6} + 52833 p^{2} T^{8} - 317 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 - 180 T^{2} + 28548 T^{4} - 2623849 T^{6} + 28548 p^{2} T^{8} - 180 p^{4} T^{10} + p^{6} T^{12} \)
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

−4.07523242207742324768878946079, −4.01469109404993626550775701898, −3.98943821474265216107456935233, −3.86299703256792629377356151257, −3.67932824773429239189129645546, −3.45201005321863784008249913921, −3.33965599897169529337572240737, −3.21897176794877205058099768335, −3.18242949256690682902741266082, −2.94527732800480714358800559275, −2.72964872658782798221365078219, −2.53308543290099184876610628448, −2.52611180823501859915081578130, −2.39953637414684366189137695183, −2.31496352406142050048716002246, −2.18228403429936001327572603428, −1.77914692444168124076394746784, −1.74763689616276904540935217587, −1.46239234145134094408035837368, −1.37861187783885375232121307100, −1.20304211082827295795677161140, −1.02400202235327133038177864624, −0.60062776880591691273033623984, −0.43512809398834130974023422540, −0.40890069576058870502851439435, 0.40890069576058870502851439435, 0.43512809398834130974023422540, 0.60062776880591691273033623984, 1.02400202235327133038177864624, 1.20304211082827295795677161140, 1.37861187783885375232121307100, 1.46239234145134094408035837368, 1.74763689616276904540935217587, 1.77914692444168124076394746784, 2.18228403429936001327572603428, 2.31496352406142050048716002246, 2.39953637414684366189137695183, 2.52611180823501859915081578130, 2.53308543290099184876610628448, 2.72964872658782798221365078219, 2.94527732800480714358800559275, 3.18242949256690682902741266082, 3.21897176794877205058099768335, 3.33965599897169529337572240737, 3.45201005321863784008249913921, 3.67932824773429239189129645546, 3.86299703256792629377356151257, 3.98943821474265216107456935233, 4.01469109404993626550775701898, 4.07523242207742324768878946079

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

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