Properties

Label 12-912e6-1.1-c1e6-0-14
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $149152.$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·5-s − 3·7-s − 3·11-s + 6·13-s + 12·17-s − 6·19-s + 15·23-s + 18·25-s + 27-s − 12·29-s + 6·31-s − 18·35-s − 12·37-s − 18·41-s + 18·43-s + 3·47-s + 15·49-s − 24·53-s − 18·55-s − 18·59-s − 9·61-s + 36·65-s + 6·67-s + 18·71-s + 21·73-s + 9·77-s − 6·79-s + ⋯
L(s)  = 1  + 2.68·5-s − 1.13·7-s − 0.904·11-s + 1.66·13-s + 2.91·17-s − 1.37·19-s + 3.12·23-s + 18/5·25-s + 0.192·27-s − 2.22·29-s + 1.07·31-s − 3.04·35-s − 1.97·37-s − 2.81·41-s + 2.74·43-s + 0.437·47-s + 15/7·49-s − 3.29·53-s − 2.42·55-s − 2.34·59-s − 1.15·61-s + 4.46·65-s + 0.733·67-s + 2.13·71-s + 2.45·73-s + 1.02·77-s − 0.675·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(149152.\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.823221737\)
\(L(\frac12)\) \(\approx\) \(8.823221737\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T^{3} + T^{6} \)
19 \( 1 + 6 T - 12 T^{2} - 169 T^{3} - 12 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 - 6 T + 18 T^{2} - 9 p T^{3} + 81 T^{4} - 87 T^{5} + 109 T^{6} - 87 p T^{7} + 81 p^{2} T^{8} - 9 p^{4} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 3 T - 6 T^{2} - 5 T^{3} + 45 T^{4} - 108 T^{5} - 705 T^{6} - 108 p T^{7} + 45 p^{2} T^{8} - 5 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 3 T - 6 T^{2} - 81 T^{3} - 129 T^{4} + 318 T^{5} + 3067 T^{6} + 318 p T^{7} - 129 p^{2} T^{8} - 81 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 6 T + 42 T^{2} - 154 T^{3} + 864 T^{4} - 3132 T^{5} + 14823 T^{6} - 3132 p T^{7} + 864 p^{2} T^{8} - 154 p^{3} T^{9} + 42 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 12 T + 54 T^{2} - 81 T^{3} + 9 p T^{4} - 3999 T^{5} + 27073 T^{6} - 3999 p T^{7} + 9 p^{3} T^{8} - 81 p^{3} T^{9} + 54 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 15 T + 144 T^{2} - 1116 T^{3} + 7191 T^{4} - 40695 T^{5} + 205741 T^{6} - 40695 p T^{7} + 7191 p^{2} T^{8} - 1116 p^{3} T^{9} + 144 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 12 T + 72 T^{2} + 234 T^{3} + 1404 T^{4} + 15492 T^{5} + 119215 T^{6} + 15492 p T^{7} + 1404 p^{2} T^{8} + 234 p^{3} T^{9} + 72 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 6 T - 42 T^{2} + 238 T^{3} + 1548 T^{4} - 4284 T^{5} - 39009 T^{6} - 4284 p T^{7} + 1548 p^{2} T^{8} + 238 p^{3} T^{9} - 42 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 6 T + 102 T^{2} + 427 T^{3} + 102 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 18 T + 189 T^{2} + 1683 T^{3} + 12609 T^{4} + 90819 T^{5} + 629218 T^{6} + 90819 p T^{7} + 12609 p^{2} T^{8} + 1683 p^{3} T^{9} + 189 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 18 T + 270 T^{2} - 3044 T^{3} + 28584 T^{4} - 235404 T^{5} + 1630749 T^{6} - 235404 p T^{7} + 28584 p^{2} T^{8} - 3044 p^{3} T^{9} + 270 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 3 T - 9 T^{2} - 513 T^{3} - 324 T^{4} + 5370 T^{5} + 215173 T^{6} + 5370 p T^{7} - 324 p^{2} T^{8} - 513 p^{3} T^{9} - 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 24 T + 243 T^{2} + 1035 T^{3} - 3843 T^{4} - 111855 T^{5} - 1075634 T^{6} - 111855 p T^{7} - 3843 p^{2} T^{8} + 1035 p^{3} T^{9} + 243 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 18 T + 144 T^{2} + 603 T^{3} - 4707 T^{4} - 98199 T^{5} - 856835 T^{6} - 98199 p T^{7} - 4707 p^{2} T^{8} + 603 p^{3} T^{9} + 144 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 9 T + 36 T^{2} - 628 T^{3} - 2682 T^{4} - 1089 T^{5} + 320655 T^{6} - 1089 p T^{7} - 2682 p^{2} T^{8} - 628 p^{3} T^{9} + 36 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 6 T + 12 T^{2} - 338 T^{3} - 1512 T^{4} + 49140 T^{5} - 303915 T^{6} + 49140 p T^{7} - 1512 p^{2} T^{8} - 338 p^{3} T^{9} + 12 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 18 T + 270 T^{2} - 2736 T^{3} + 25812 T^{4} - 198576 T^{5} + 1613305 T^{6} - 198576 p T^{7} + 25812 p^{2} T^{8} - 2736 p^{3} T^{9} + 270 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 21 T + 156 T^{2} - 4 T^{3} - 6993 T^{4} + 46629 T^{5} - 275871 T^{6} + 46629 p T^{7} - 6993 p^{2} T^{8} - 4 p^{3} T^{9} + 156 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 6 T + 78 T^{2} - 800 T^{3} + 3348 T^{4} + 23760 T^{5} + 1487457 T^{6} + 23760 p T^{7} + 3348 p^{2} T^{8} - 800 p^{3} T^{9} + 78 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 15 T + 12 T^{2} - 1791 T^{3} - 9705 T^{4} + 103974 T^{5} + 2042971 T^{6} + 103974 p T^{7} - 9705 p^{2} T^{8} - 1791 p^{3} T^{9} + 12 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 15 T + 108 T^{2} - 396 T^{3} - 6489 T^{4} + 85359 T^{5} - 603863 T^{6} + 85359 p T^{7} - 6489 p^{2} T^{8} - 396 p^{3} T^{9} + 108 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 9 T + 90 T^{2} - 826 T^{3} - 243 T^{4} + 27189 T^{5} - 54627 T^{6} + 27189 p T^{7} - 243 p^{2} T^{8} - 826 p^{3} T^{9} + 90 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.46715402335344388622429710712, −5.29755353569939446277988069589, −5.13538314088787886548295390264, −4.99982938116745042693977079073, −4.71225642709999125180684564740, −4.64452785294446608155357250658, −4.52456741605647699539850553082, −4.14915968120378601622293514389, −3.86992115488630051825484937009, −3.78700849310709285713004863350, −3.42773618394178229042746561023, −3.38753954003257173597165807640, −3.33291841155477083199561106358, −3.08223815905244983724436294723, −2.86994894853912012362345238755, −2.83614885236449872481432629461, −2.45606402740254242538130868334, −2.28762985587180160207075257781, −1.96275492818626141448336595777, −1.73579350098789440633448833991, −1.56767296172337965774789387403, −1.50532600032258618109653884492, −1.05264543327172294822139512840, −0.792670850154611033897023385964, −0.45127061799494670711253750165, 0.45127061799494670711253750165, 0.792670850154611033897023385964, 1.05264543327172294822139512840, 1.50532600032258618109653884492, 1.56767296172337965774789387403, 1.73579350098789440633448833991, 1.96275492818626141448336595777, 2.28762985587180160207075257781, 2.45606402740254242538130868334, 2.83614885236449872481432629461, 2.86994894853912012362345238755, 3.08223815905244983724436294723, 3.33291841155477083199561106358, 3.38753954003257173597165807640, 3.42773618394178229042746561023, 3.78700849310709285713004863350, 3.86992115488630051825484937009, 4.14915968120378601622293514389, 4.52456741605647699539850553082, 4.64452785294446608155357250658, 4.71225642709999125180684564740, 4.99982938116745042693977079073, 5.13538314088787886548295390264, 5.29755353569939446277988069589, 5.46715402335344388622429710712

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

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