↑

Properties

Label 12-5586e6-1.1-c1e6-0-0
Degree $12$
Conductor $3.038\times 10^{22}$
Sign $1$
Analytic cond. $7.87532\times 10^{9}$
Root an. cond. $6.67865$
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  − 6·2-s − 6·3-s + 21·4-s + 36·6-s − 56·8-s + 21·9-s + 4·11-s − 126·12-s + 126·16-s − 4·17-s − 126·18-s + 6·19-s − 24·22-s + 16·23-s + 336·24-s − 10·25-s − 56·27-s + 8·29-s − 252·32-s − 24·33-s + 24·34-s + 441·36-s − 36·38-s − 8·41-s + 8·43-s + 84·44-s − 96·46-s + ⋯
L(s)  = 1  − 4.24·2-s − 3.46·3-s + 21/2·4-s + 14.6·6-s − 19.7·8-s + 7·9-s + 1.20·11-s − 36.3·12-s + 63/2·16-s − 0.970·17-s − 29.6·18-s + 1.37·19-s − 5.11·22-s + 3.33·23-s + 68.5·24-s − 2·25-s − 10.7·27-s + 1.48·29-s − 44.5·32-s − 4.17·33-s + 4.11·34-s + 73.5·36-s − 5.83·38-s − 1.24·41-s + 1.21·43-s + 12.6·44-s − 14.1·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 7^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(7.87532\times 10^{9}\)
Root analytic conductor: \(6.67865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 7^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3221352810\)
\(L(\frac12)\) \(\approx\) \(0.3221352810\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{6} \)
3 \( ( 1 + T )^{6} \)
7 \( 1 \)
19 \( ( 1 - T )^{6} \)
good5 \( 1 + 2 p T^{2} - 8 T^{3} + 57 T^{4} - 112 T^{5} + 264 T^{6} - 112 p T^{7} + 57 p^{2} T^{8} - 8 p^{3} T^{9} + 2 p^{5} T^{10} + p^{6} T^{12} \)
11 \( 1 - 4 T + 12 T^{2} + 20 T^{3} - 17 T^{4} - 632 T^{5} + 3768 T^{6} - 632 p T^{7} - 17 p^{2} T^{8} + 20 p^{3} T^{9} + 12 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 42 T^{2} + 64 T^{3} + 745 T^{4} + 2312 T^{5} + 9624 T^{6} + 2312 p T^{7} + 745 p^{2} T^{8} + 64 p^{3} T^{9} + 42 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 4 T + 50 T^{2} + 116 T^{3} + 959 T^{4} + 520 T^{5} + 13660 T^{6} + 520 p T^{7} + 959 p^{2} T^{8} + 116 p^{3} T^{9} + 50 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 16 T + 172 T^{2} - 1224 T^{3} + 7129 T^{4} - 34096 T^{5} + 165400 T^{6} - 34096 p T^{7} + 7129 p^{2} T^{8} - 1224 p^{3} T^{9} + 172 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 8 T + 134 T^{2} - 776 T^{3} + 7959 T^{4} - 36048 T^{5} + 283220 T^{6} - 36048 p T^{7} + 7959 p^{2} T^{8} - 776 p^{3} T^{9} + 134 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 142 T^{2} + 40 T^{3} + 9377 T^{4} + 3152 T^{5} + 368088 T^{6} + 3152 p T^{7} + 9377 p^{2} T^{8} + 40 p^{3} T^{9} + 142 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 + 2 p T^{2} + 328 T^{3} + 4041 T^{4} + 15024 T^{5} + 205800 T^{6} + 15024 p T^{7} + 4041 p^{2} T^{8} + 328 p^{3} T^{9} + 2 p^{5} T^{10} + p^{6} T^{12} \)
41 \( 1 + 8 T + 180 T^{2} + 1192 T^{3} + 14839 T^{4} + 83728 T^{5} + 753832 T^{6} + 83728 p T^{7} + 14839 p^{2} T^{8} + 1192 p^{3} T^{9} + 180 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 8 T + 206 T^{2} - 1400 T^{3} + 18955 T^{4} - 107952 T^{5} + 1027900 T^{6} - 107952 p T^{7} + 18955 p^{2} T^{8} - 1400 p^{3} T^{9} + 206 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 16 T + 198 T^{2} + 912 T^{3} + 1601 T^{4} - 56216 T^{5} - 425816 T^{6} - 56216 p T^{7} + 1601 p^{2} T^{8} + 912 p^{3} T^{9} + 198 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 4 T + 42 T^{2} + 452 T^{3} + 7191 T^{4} + 28936 T^{5} + 257196 T^{6} + 28936 p T^{7} + 7191 p^{2} T^{8} + 452 p^{3} T^{9} + 42 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 24 T + 454 T^{2} + 5336 T^{3} + 56075 T^{4} + 455328 T^{5} + 3780716 T^{6} + 455328 p T^{7} + 56075 p^{2} T^{8} + 5336 p^{3} T^{9} + 454 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 164 T^{2} + 96 T^{3} + 16191 T^{4} + 11680 T^{5} + 1145832 T^{6} + 11680 p T^{7} + 16191 p^{2} T^{8} + 96 p^{3} T^{9} + 164 p^{4} T^{10} + p^{6} T^{12} \)
67 \( 1 + 20 T + 492 T^{2} + 6396 T^{3} + 89631 T^{4} + 838616 T^{5} + 8196472 T^{6} + 838616 p T^{7} + 89631 p^{2} T^{8} + 6396 p^{3} T^{9} + 492 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 24 T + 562 T^{2} - 8392 T^{3} + 111519 T^{4} - 1176944 T^{5} + 10883644 T^{6} - 1176944 p T^{7} + 111519 p^{2} T^{8} - 8392 p^{3} T^{9} + 562 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 16 T + 468 T^{2} - 5232 T^{3} + 85623 T^{4} - 716672 T^{5} + 8292840 T^{6} - 716672 p T^{7} + 85623 p^{2} T^{8} - 5232 p^{3} T^{9} + 468 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 40 T + 1068 T^{2} - 20064 T^{3} + 296953 T^{4} - 3534544 T^{5} + 34592120 T^{6} - 3534544 p T^{7} + 296953 p^{2} T^{8} - 20064 p^{3} T^{9} + 1068 p^{4} T^{10} - 40 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 24 T + 482 T^{2} + 5992 T^{3} + 68711 T^{4} + 604912 T^{5} + 5873788 T^{6} + 604912 p T^{7} + 68711 p^{2} T^{8} + 5992 p^{3} T^{9} + 482 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 24 T + 560 T^{2} + 8728 T^{3} + 122195 T^{4} + 1402096 T^{5} + 14348896 T^{6} + 1402096 p T^{7} + 122195 p^{2} T^{8} + 8728 p^{3} T^{9} + 560 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 4 T + 274 T^{2} - 1524 T^{3} + 47135 T^{4} - 216008 T^{5} + 5557532 T^{6} - 216008 p T^{7} + 47135 p^{2} T^{8} - 1524 p^{3} T^{9} + 274 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
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

−4.15645067710845043136511639695, −4.02005423251484065188681172973, −3.70953001275283046813336243955, −3.59406364247476067027897674029, −3.53351044073395402681432801977, −3.52617595960885969539205657715, −3.47216425122398568668328771235, −2.95805295519383752787733985617, −2.76918094261223001464796394918, −2.73747737176859734305625025744, −2.56678718091091408075609275963, −2.52445754953868129828039903543, −2.49887449169470038949810748798, −1.82276255839367897202432926990, −1.70076887283437863356134126295, −1.61150140686520767782156114051, −1.58395592708180058650905280863, −1.57824287297520251639690212969, −1.33725127417469044667545215767, −1.04853234748252422548936104631, −1.00008722512009399997588461148, −0.53098894971628434764038253085, −0.51701596106289184196998569475, −0.41679327508158865646351975146, −0.33764433985117655155917930613, 0.33764433985117655155917930613, 0.41679327508158865646351975146, 0.51701596106289184196998569475, 0.53098894971628434764038253085, 1.00008722512009399997588461148, 1.04853234748252422548936104631, 1.33725127417469044667545215767, 1.57824287297520251639690212969, 1.58395592708180058650905280863, 1.61150140686520767782156114051, 1.70076887283437863356134126295, 1.82276255839367897202432926990, 2.49887449169470038949810748798, 2.52445754953868129828039903543, 2.56678718091091408075609275963, 2.73747737176859734305625025744, 2.76918094261223001464796394918, 2.95805295519383752787733985617, 3.47216425122398568668328771235, 3.52617595960885969539205657715, 3.53351044073395402681432801977, 3.59406364247476067027897674029, 3.70953001275283046813336243955, 4.02005423251484065188681172973, 4.15645067710845043136511639695

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

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