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Properties

Label 12-666e6-1.1-c1e6-0-5
Degree $12$
Conductor $8.727\times 10^{16}$
Sign $1$
Analytic cond. $22620.8$
Root an. cond. $2.30608$
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  − 3·5-s + 6·7-s − 8-s + 9·11-s + 9·19-s + 15·23-s + 15·25-s − 18·31-s − 18·35-s + 9·37-s + 3·40-s − 6·41-s + 12·43-s + 3·47-s + 18·49-s + 18·53-s − 27·55-s − 6·56-s + 6·59-s − 12·61-s − 3·67-s + 6·71-s − 36·73-s + 54·77-s + 30·79-s − 6·83-s − 9·88-s + ⋯
L(s)  = 1  − 1.34·5-s + 2.26·7-s − 0.353·8-s + 2.71·11-s + 2.06·19-s + 3.12·23-s + 3·25-s − 3.23·31-s − 3.04·35-s + 1.47·37-s + 0.474·40-s − 0.937·41-s + 1.82·43-s + 0.437·47-s + 18/7·49-s + 2.47·53-s − 3.64·55-s − 0.801·56-s + 0.781·59-s − 1.53·61-s − 0.366·67-s + 0.712·71-s − 4.21·73-s + 6.15·77-s + 3.37·79-s − 0.658·83-s − 0.959·88-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(22620.8\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 37^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.755453713\)
\(L(\frac12)\) \(\approx\) \(8.755453713\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{3} + T^{6} \)
3 \( 1 \)
37 \( 1 - 9 T + 54 T^{2} - 305 T^{3} + 54 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 + 3 T - 6 T^{2} - 38 T^{3} - 51 T^{4} + 117 T^{5} + 581 T^{6} + 117 p T^{7} - 51 p^{2} T^{8} - 38 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 6 T + 18 T^{2} - 51 T^{3} + 99 T^{4} - 69 T^{5} - 19 T^{6} - 69 p T^{7} + 99 p^{2} T^{8} - 51 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 9 T + 24 T^{2} - 83 T^{3} + 687 T^{4} - 2058 T^{5} + 3347 T^{6} - 2058 p T^{7} + 687 p^{2} T^{8} - 83 p^{3} T^{9} + 24 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 36 T^{2} + 27 T^{3} + 63 p T^{4} + 549 T^{5} + 12977 T^{6} + 549 p T^{7} + 63 p^{3} T^{8} + 27 p^{3} T^{9} + 36 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 126 T^{3} + 10963 T^{6} - 126 p^{3} T^{9} + p^{6} T^{12} \)
19 \( 1 - 9 T + 63 T^{2} - 295 T^{3} + 1188 T^{4} - 3510 T^{5} + 11397 T^{6} - 3510 p T^{7} + 1188 p^{2} T^{8} - 295 p^{3} T^{9} + 63 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 15 T + 102 T^{2} - 459 T^{3} + 1905 T^{4} - 8304 T^{5} + 37591 T^{6} - 8304 p T^{7} + 1905 p^{2} T^{8} - 459 p^{3} T^{9} + 102 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 24 T^{2} + 342 T^{3} - 120 T^{4} - 4104 T^{5} + 61315 T^{6} - 4104 p T^{7} - 120 p^{2} T^{8} + 342 p^{3} T^{9} - 24 p^{4} T^{10} + p^{6} T^{12} \)
31 \( ( 1 + 9 T + 117 T^{2} + 575 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 6 T + 36 T^{2} - 54 T^{3} - 288 T^{4} - 5232 T^{5} + 44551 T^{6} - 5232 p T^{7} - 288 p^{2} T^{8} - 54 p^{3} T^{9} + 36 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( ( 1 - 6 T + 48 T^{2} - 49 T^{3} + 48 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 3 T - 87 T^{2} + 310 T^{3} + 81 p T^{4} - 8259 T^{5} - 155986 T^{6} - 8259 p T^{7} + 81 p^{3} T^{8} + 310 p^{3} T^{9} - 87 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 18 T + 144 T^{2} - 549 T^{3} - 4113 T^{4} + 79569 T^{5} - 665387 T^{6} + 79569 p T^{7} - 4113 p^{2} T^{8} - 549 p^{3} T^{9} + 144 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 6 T + 120 T^{2} - 83 T^{3} + 3477 T^{4} + 46503 T^{5} + 35645 T^{6} + 46503 p T^{7} + 3477 p^{2} T^{8} - 83 p^{3} T^{9} + 120 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 12 T + 48 T^{2} + 220 T^{3} - 4320 T^{4} - 57240 T^{5} - 269733 T^{6} - 57240 p T^{7} - 4320 p^{2} T^{8} + 220 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 3 T - 36 T^{2} - 1140 T^{3} - 4194 T^{4} + 29397 T^{5} + 912401 T^{6} + 29397 p T^{7} - 4194 p^{2} T^{8} - 1140 p^{3} T^{9} - 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 6 T - 144 T^{2} + 1080 T^{3} + 2700 T^{4} - 46464 T^{5} + 353557 T^{6} - 46464 p T^{7} + 2700 p^{2} T^{8} + 1080 p^{3} T^{9} - 144 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 + 18 T + 306 T^{2} + 2681 T^{3} + 306 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 30 T + 360 T^{2} - 2028 T^{3} + 5490 T^{4} - 36318 T^{5} + 477341 T^{6} - 36318 p T^{7} + 5490 p^{2} T^{8} - 2028 p^{3} T^{9} + 360 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 6 T - 12 T^{2} - 719 T^{3} - 5007 T^{4} + 73899 T^{5} + 904205 T^{6} + 73899 p T^{7} - 5007 p^{2} T^{8} - 719 p^{3} T^{9} - 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 33 T + 516 T^{2} - 4394 T^{3} + 6204 T^{4} + 396549 T^{5} - 5838061 T^{6} + 396549 p T^{7} + 6204 p^{2} T^{8} - 4394 p^{3} T^{9} + 516 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 42 T + 897 T^{2} - 14982 T^{3} + 217518 T^{4} - 2621634 T^{5} + 27146693 T^{6} - 2621634 p T^{7} + 217518 p^{2} T^{8} - 14982 p^{3} T^{9} + 897 p^{4} T^{10} - 42 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.45440563984686329177594010423, −5.45208684878546398190310548809, −5.42818348129997615917702026707, −5.07907031061977699291731928341, −4.85144363619797312680508948650, −4.71018018271171822501779513174, −4.61159273714614770343798943314, −4.58403895264579036840563196224, −4.23270763117236086800550356660, −3.91363980016965994458943995121, −3.82508861720299846939518830841, −3.76036816228542401235675350557, −3.66123128764100168503709583198, −3.16138782679095722754651403756, −3.11000203027628009556026200891, −3.10760652144914542197810080933, −2.67825700355668946773564181268, −2.46629524802840144290019969435, −2.13816211043781266132257226173, −1.82046391257734116991192025389, −1.66975853339689439329240680885, −1.11177034891272356292159556253, −1.06352130561113219681091042686, −1.06286865811390778990520866352, −0.65217048266542788411252693929, 0.65217048266542788411252693929, 1.06286865811390778990520866352, 1.06352130561113219681091042686, 1.11177034891272356292159556253, 1.66975853339689439329240680885, 1.82046391257734116991192025389, 2.13816211043781266132257226173, 2.46629524802840144290019969435, 2.67825700355668946773564181268, 3.10760652144914542197810080933, 3.11000203027628009556026200891, 3.16138782679095722754651403756, 3.66123128764100168503709583198, 3.76036816228542401235675350557, 3.82508861720299846939518830841, 3.91363980016965994458943995121, 4.23270763117236086800550356660, 4.58403895264579036840563196224, 4.61159273714614770343798943314, 4.71018018271171822501779513174, 4.85144363619797312680508948650, 5.07907031061977699291731928341, 5.42818348129997615917702026707, 5.45208684878546398190310548809, 5.45440563984686329177594010423

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

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