Properties

Degree 12
Conductor $ 3^{18} \cdot 7^{6} $
Sign $1$
Motivic weight 1
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 + 3·4-s − 5-s + 2·7-s + 2·10-s − 7·11-s + 4·13-s − 4·14-s − 4·16-s − 5·19-s − 3·20-s + 14·22-s − 6·23-s + 9·25-s − 8·26-s + 6·28-s + 26·29-s + 8·31-s + 7·32-s − 2·35-s + 8·37-s + 10·38-s + 4·41-s − 18·43-s − 21·44-s + 12·46-s − 9·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.447·5-s + 0.755·7-s + 0.632·10-s − 2.11·11-s + 1.10·13-s − 1.06·14-s − 16-s − 1.14·19-s − 0.670·20-s + 2.98·22-s − 1.25·23-s + 9/5·25-s − 1.56·26-s + 1.13·28-s + 4.82·29-s + 1.43·31-s + 1.23·32-s − 0.338·35-s + 1.31·37-s + 1.62·38-s + 0.624·41-s − 2.74·43-s − 3.16·44-s + 1.76·46-s − 1.31·47-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(3^{18} \cdot 7^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{189} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(12,\ 3^{18} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$
$L(1)$  $\approx$  $0.989173$
$L(\frac12)$  $\approx$  $0.989173$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \(F_p\) is a polynomial of degree 12. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 11.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 - 2 T - 4 T^{2} + 31 T^{3} - 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 + p T + T^{2} - p^{2} T^{3} - 7 T^{4} - T^{5} + 7 T^{6} - p T^{7} - 7 p^{2} T^{8} - p^{5} T^{9} + p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 + T - 8 T^{2} - 17 T^{3} + 23 T^{4} + 52 T^{5} - 11 T^{6} + 52 p T^{7} + 23 p^{2} T^{8} - 17 p^{3} T^{9} - 8 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 7 T + 4 T^{2} + T^{3} + 431 T^{4} + 982 T^{5} - 893 T^{6} + 982 p T^{7} + 431 p^{2} T^{8} + p^{3} T^{9} + 4 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - 2 T + 20 T^{2} - 5 T^{3} + 20 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 18 T^{2} - 18 T^{3} + 18 T^{4} + 162 T^{5} + 4399 T^{6} + 162 p T^{7} + 18 p^{2} T^{8} - 18 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 + 5 T - 28 T^{2} - 3 p T^{3} + 997 T^{4} + 268 T^{5} - 22757 T^{6} + 268 p T^{7} + 997 p^{2} T^{8} - 3 p^{4} T^{9} - 28 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 6 T - 36 T^{2} - 102 T^{3} + 1926 T^{4} + 2526 T^{5} - 42653 T^{6} + 2526 p T^{7} + 1926 p^{2} T^{8} - 102 p^{3} T^{9} - 36 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 - 13 T + 117 T^{2} - 763 T^{3} + 117 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 8 T - 30 T^{2} + 102 T^{3} + 2506 T^{4} - 1202 T^{5} - 93509 T^{6} - 1202 p T^{7} + 2506 p^{2} T^{8} + 102 p^{3} T^{9} - 30 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 8 T - 42 T^{2} + 150 T^{3} + 3322 T^{4} - 1094 T^{5} - 153041 T^{6} - 1094 p T^{7} + 3322 p^{2} T^{8} + 150 p^{3} T^{9} - 42 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 - 2 T + 18 T^{2} + 223 T^{3} + 18 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 9 T + 117 T^{2} + 673 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 9 T - 18 T^{2} - 819 T^{3} - 2547 T^{4} + 20772 T^{5} + 299095 T^{6} + 20772 p T^{7} - 2547 p^{2} T^{8} - 819 p^{3} T^{9} - 18 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 24 T + 252 T^{2} + 2202 T^{3} + 20916 T^{4} + 146148 T^{5} + 883411 T^{6} + 146148 p T^{7} + 20916 p^{2} T^{8} + 2202 p^{3} T^{9} + 252 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 15 T - 18 T^{2} + 57 T^{3} + 16947 T^{4} - 71898 T^{5} - 430157 T^{6} - 71898 p T^{7} + 16947 p^{2} T^{8} + 57 p^{3} T^{9} - 18 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - T - 133 T^{2} - 132 T^{3} + 9781 T^{4} + 12493 T^{5} - 645074 T^{6} + 12493 p T^{7} + 9781 p^{2} T^{8} - 132 p^{3} T^{9} - 133 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 14 T - 58 T^{2} - 258 T^{3} + 20800 T^{4} + 70720 T^{5} - 964241 T^{6} + 70720 p T^{7} + 20800 p^{2} T^{8} - 258 p^{3} T^{9} - 58 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 3 T + 105 T^{2} + 669 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 7 T - 36 T^{2} + 513 T^{3} + 733 T^{4} - 46082 T^{5} - 8831 T^{6} - 46082 p T^{7} + 733 p^{2} T^{8} + 513 p^{3} T^{9} - 36 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 6 T - 132 T^{2} - 634 T^{3} + 10026 T^{4} + 16110 T^{5} - 794253 T^{6} + 16110 p T^{7} + 10026 p^{2} T^{8} - 634 p^{3} T^{9} - 132 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 3 T + 69 T^{2} - 1227 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 5 T - 149 T^{2} - 68 T^{3} + 12785 T^{4} + 45481 T^{5} - 1321850 T^{6} + 45481 p T^{7} + 12785 p^{2} T^{8} - 68 p^{3} T^{9} - 149 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 + 14 T + 307 T^{2} + 2692 T^{3} + 307 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−6.97789496896642725653303542541, −6.70481678199360586900680251303, −6.62750669789568535006674279058, −6.62151194074563675045925169339, −6.22894900122790426762571758586, −6.22829859181164130538672971329, −5.75937009812810085312094954301, −5.65753152070042961390964534084, −5.45261081915559241856239925822, −5.04616122491773614662085498031, −4.72735538671741628648836767723, −4.56370563194415387173852405404, −4.52996705262543845454074689183, −4.52901464755819055874357286326, −4.33233797969404216597051904151, −3.69153561356004538361060102442, −3.24238314136025927764919275467, −3.08503916168646420411761682599, −2.91194813142760716286759211651, −2.83725192711501270840058828502, −2.18925864776931347484174428161, −1.92959202178053118598540503962, −1.77890331276711388954673613819, −1.04657415131580579698762638850, −0.789988307387670108050688681938, 0.789988307387670108050688681938, 1.04657415131580579698762638850, 1.77890331276711388954673613819, 1.92959202178053118598540503962, 2.18925864776931347484174428161, 2.83725192711501270840058828502, 2.91194813142760716286759211651, 3.08503916168646420411761682599, 3.24238314136025927764919275467, 3.69153561356004538361060102442, 4.33233797969404216597051904151, 4.52901464755819055874357286326, 4.52996705262543845454074689183, 4.56370563194415387173852405404, 4.72735538671741628648836767723, 5.04616122491773614662085498031, 5.45261081915559241856239925822, 5.65753152070042961390964534084, 5.75937009812810085312094954301, 6.22829859181164130538672971329, 6.22894900122790426762571758586, 6.62151194074563675045925169339, 6.62750669789568535006674279058, 6.70481678199360586900680251303, 6.97789496896642725653303542541

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

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