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Properties

Label 12-126e6-1.1-c1e6-0-3
Degree $12$
Conductor $4.002\times 10^{12}$
Sign $1$
Analytic cond. $1.03725$
Root an. cond. $1.00305$
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  + 6·2-s − 2·3-s + 21·4-s − 5·5-s − 12·6-s + 4·7-s + 56·8-s − 30·10-s − 11-s − 42·12-s − 2·13-s + 24·14-s + 10·15-s + 126·16-s − 4·17-s − 3·19-s − 105·20-s − 8·21-s − 6·22-s − 7·23-s − 112·24-s + 19·25-s − 12·26-s + 5·27-s + 84·28-s − 5·29-s + 60·30-s + ⋯
L(s)  = 1  + 4.24·2-s − 1.15·3-s + 21/2·4-s − 2.23·5-s − 4.89·6-s + 1.51·7-s + 19.7·8-s − 9.48·10-s − 0.301·11-s − 12.1·12-s − 0.554·13-s + 6.41·14-s + 2.58·15-s + 63/2·16-s − 0.970·17-s − 0.688·19-s − 23.4·20-s − 1.74·21-s − 1.27·22-s − 1.45·23-s − 22.8·24-s + 19/5·25-s − 2.35·26-s + 0.962·27-s + 15.8·28-s − 0.928·29-s + 10.9·30-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(1.03725\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.807760453\)
\(L(\frac12)\) \(\approx\) \(7.807760453\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + 4 T^{2} + p T^{3} + 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 - 4 T + 2 p T^{2} - 55 T^{3} + 2 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 + p T + 6 T^{2} + T^{3} + 31 T^{4} + 68 T^{5} + 29 T^{6} + 68 p T^{7} + 31 p^{2} T^{8} + p^{3} T^{9} + 6 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 + T - 6 T^{2} - 103 T^{3} - 83 T^{4} + 32 p T^{5} + 457 p T^{6} + 32 p^{2} T^{7} - 83 p^{2} T^{8} - 103 p^{3} T^{9} - 6 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 2 T - 32 T^{2} - 2 p T^{3} + 730 T^{4} + 230 T^{5} - 10729 T^{6} + 230 p T^{7} + 730 p^{2} T^{8} - 2 p^{4} T^{9} - 32 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 T + 9 T^{2} + 92 T^{3} + 58 T^{4} - 20 T^{5} + 5393 T^{6} - 20 p T^{7} + 58 p^{2} T^{8} + 92 p^{3} T^{9} + 9 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 42 T^{2} - 61 T^{3} + 69 p T^{4} + 726 T^{5} - 27501 T^{6} + 726 p T^{7} + 69 p^{3} T^{8} - 61 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 7 T - 24 T^{2} - 127 T^{3} + 1417 T^{4} + 3484 T^{5} - 22393 T^{6} + 3484 p T^{7} + 1417 p^{2} T^{8} - 127 p^{3} T^{9} - 24 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 5 T - 30 T^{2} - 371 T^{3} - 185 T^{4} + 6020 T^{5} + 44357 T^{6} + 6020 p T^{7} - 185 p^{2} T^{8} - 371 p^{3} T^{9} - 30 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 14 T + 138 T^{2} - 841 T^{3} + 138 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 4875 p T^{7} + 1293 p^{2} T^{8} - 268 p^{3} T^{9} - 21 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 12 T - 18 T^{2} - 78 T^{3} + 7470 T^{4} + 24546 T^{5} - 158105 T^{6} + 24546 p T^{7} + 7470 p^{2} T^{8} - 78 p^{3} T^{9} - 18 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 18 T + 114 T^{2} - 682 T^{3} + 7188 T^{4} - 33492 T^{5} + 63039 T^{6} - 33492 p T^{7} + 7188 p^{2} T^{8} - 682 p^{3} T^{9} + 114 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 3 T + 117 T^{2} + 309 T^{3} + 117 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 - 9 T - 36 T^{2} + 873 T^{3} - 1179 T^{4} - 26334 T^{5} + 272077 T^{6} - 26334 p T^{7} - 1179 p^{2} T^{8} + 873 p^{3} T^{9} - 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 + 4 T + 76 T^{2} + 11 p T^{3} + 76 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 4 T + 48 T^{2} - 229 T^{3} + 48 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 + 5 T + 143 T^{2} + 521 T^{3} + 143 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 - 7 T + 163 T^{2} - 895 T^{3} + 163 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 25 T + 254 T^{2} + 2073 T^{3} + 20533 T^{4} + 115046 T^{5} + 366817 T^{6} + 115046 p T^{7} + 20533 p^{2} T^{8} + 2073 p^{3} T^{9} + 254 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 + 7 T + 93 T^{2} + 335 T^{3} + 93 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 8 T - 180 T^{2} + 518 T^{3} + 29404 T^{4} - 32420 T^{5} - 2713585 T^{6} - 32420 p T^{7} + 29404 p^{2} T^{8} + 518 p^{3} T^{9} - 180 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 9 T - 180 T^{2} - 729 T^{3} + 31041 T^{4} + 54846 T^{5} - 2925911 T^{6} + 54846 p T^{7} + 31041 p^{2} T^{8} - 729 p^{3} T^{9} - 180 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 28 T + 257 T^{2} + 2820 T^{3} + 59506 T^{4} + 545924 T^{5} + 3126001 T^{6} + 545924 p T^{7} + 59506 p^{2} T^{8} + 2820 p^{3} T^{9} + 257 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
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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

−7.27126321677831000274552548427, −7.00872472023068219161933604050, −6.93747758323668509053848696241, −6.78591588393007215246654466618, −6.56328901889907596491356894861, −6.35402688273999265460144535290, −6.32830505874455246230444707695, −5.71166129204748888227721673535, −5.68096524436360357168280617324, −5.53604112642445751447978710296, −5.31840217814920452606396244660, −5.05737023458522022568593748150, −4.78123518836355121981460972426, −4.48598043478844694603052868366, −4.36834588794797656110852059494, −4.33425611917981257235581876737, −4.26703273620212417774205280082, −3.99408284332197900928906549926, −3.52685007260981365350541557936, −3.07300917069112844660040710051, −2.92163865274782315884676932020, −2.72792928975173492036662640354, −2.54431991180380188749178154192, −1.77243743364827373841162369696, −1.46141373801436768922324694872, 1.46141373801436768922324694872, 1.77243743364827373841162369696, 2.54431991180380188749178154192, 2.72792928975173492036662640354, 2.92163865274782315884676932020, 3.07300917069112844660040710051, 3.52685007260981365350541557936, 3.99408284332197900928906549926, 4.26703273620212417774205280082, 4.33425611917981257235581876737, 4.36834588794797656110852059494, 4.48598043478844694603052868366, 4.78123518836355121981460972426, 5.05737023458522022568593748150, 5.31840217814920452606396244660, 5.53604112642445751447978710296, 5.68096524436360357168280617324, 5.71166129204748888227721673535, 6.32830505874455246230444707695, 6.35402688273999265460144535290, 6.56328901889907596491356894861, 6.78591588393007215246654466618, 6.93747758323668509053848696241, 7.00872472023068219161933604050, 7.27126321677831000274552548427

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

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