Properties

Label 12-1386e6-1.1-c1e6-0-1
Degree $12$
Conductor $7.089\times 10^{18}$
Sign $1$
Analytic cond. $1.83756\times 10^{6}$
Root an. cond. $3.32675$
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  + 3·2-s + 3·4-s − 2·8-s − 3·11-s − 9·16-s + 3·17-s + 9·19-s − 9·22-s + 3·23-s − 18·29-s + 6·31-s − 9·32-s + 9·34-s − 21·37-s + 27·38-s − 24·41-s + 6·43-s − 9·44-s + 9·46-s + 3·47-s − 6·49-s + 24·53-s − 54·58-s − 9·59-s + 6·61-s + 18·62-s + 3·64-s + ⋯
L(s)  = 1  + 2.12·2-s + 3/2·4-s − 0.707·8-s − 0.904·11-s − 9/4·16-s + 0.727·17-s + 2.06·19-s − 1.91·22-s + 0.625·23-s − 3.34·29-s + 1.07·31-s − 1.59·32-s + 1.54·34-s − 3.45·37-s + 4.37·38-s − 3.74·41-s + 0.914·43-s − 1.35·44-s + 1.32·46-s + 0.437·47-s − 6/7·49-s + 3.29·53-s − 7.09·58-s − 1.17·59-s + 0.768·61-s + 2.28·62-s + 3/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 7^{6} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(1.83756\times 10^{6}\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1386} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2584498757\)
\(L(\frac12)\) \(\approx\) \(0.2584498757\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{3} \)
3 \( 1 \)
7 \( 1 + 6 T^{2} - 20 T^{3} + 6 p T^{4} + p^{3} T^{6} \)
11 \( ( 1 + T + T^{2} )^{3} \)
good5 \( ( 1 + 4 p T^{3} + p^{3} T^{6} )^{2} \)
13 \( ( 1 + p T^{2} )^{6} \)
17 \( 1 - 3 T + 15 T^{2} - 56 T^{3} + 249 T^{4} - 1701 T^{5} + 7982 T^{6} - 1701 p T^{7} + 249 p^{2} T^{8} - 56 p^{3} T^{9} + 15 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 9 T + 12 T^{2} + 7 T^{3} + 624 T^{4} - 1317 T^{5} - 7386 T^{6} - 1317 p T^{7} + 624 p^{2} T^{8} + 7 p^{3} T^{9} + 12 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 3 T - 33 T^{2} - 28 T^{3} + 651 T^{4} + 2151 T^{5} - 18598 T^{6} + 2151 p T^{7} + 651 p^{2} T^{8} - 28 p^{3} T^{9} - 33 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 9 T + 39 T^{2} + 6 p T^{3} + 39 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 6 T - 9 T^{2} + 90 T^{3} - 450 T^{4} + 2874 T^{5} - 5389 T^{6} + 2874 p T^{7} - 450 p^{2} T^{8} + 90 p^{3} T^{9} - 9 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 21 T + 198 T^{2} + 1539 T^{3} + 12636 T^{4} + 84153 T^{5} + 488288 T^{6} + 84153 p T^{7} + 12636 p^{2} T^{8} + 1539 p^{3} T^{9} + 198 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 12 T + 96 T^{2} + 678 T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 3 T - 3 T^{2} + 146 T^{3} - 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 3 T - 105 T^{2} + 204 T^{3} + 6819 T^{4} - 5601 T^{5} - 348518 T^{6} - 5601 p T^{7} + 6819 p^{2} T^{8} + 204 p^{3} T^{9} - 105 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 8 T + 11 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{3} \)
59 \( 1 + 9 T - 48 T^{2} - 867 T^{3} - 96 T^{4} + 25137 T^{5} + 175174 T^{6} + 25137 p T^{7} - 96 p^{2} T^{8} - 867 p^{3} T^{9} - 48 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T - 84 T^{2} + 940 T^{3} + 2100 T^{4} - 34086 T^{5} + 130506 T^{6} - 34086 p T^{7} + 2100 p^{2} T^{8} + 940 p^{3} T^{9} - 84 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T - 102 T^{2} - 356 T^{3} + 6246 T^{4} - 2322 T^{5} - 485562 T^{6} - 2322 p T^{7} + 6246 p^{2} T^{8} - 356 p^{3} T^{9} - 102 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 9 T + 105 T^{2} + 1170 T^{3} + 105 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 99 T^{2} - 960 T^{3} + 2574 T^{4} + 47520 T^{5} + 292961 T^{6} + 47520 p T^{7} + 2574 p^{2} T^{8} - 960 p^{3} T^{9} - 99 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 + 12 T - 126 T^{2} - 520 T^{3} + 29010 T^{4} + 70572 T^{5} - 2137326 T^{6} + 70572 p T^{7} + 29010 p^{2} T^{8} - 520 p^{3} T^{9} - 126 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 18 T + 282 T^{2} + 2824 T^{3} + 282 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 12 T - 111 T^{2} - 1180 T^{3} + 16890 T^{4} + 90972 T^{5} - 1023511 T^{6} + 90972 p T^{7} + 16890 p^{2} T^{8} - 1180 p^{3} T^{9} - 111 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 7 T + p T^{2} )^{6} \)
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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

−4.92936932533881873615008471483, −4.92734558074786769079285904895, −4.90920284068658661807126604591, −4.58234238681438709194999843555, −4.49882386466825625984165364425, −4.20194126036395396114437804026, −4.11707555782188481031278959576, −3.76374392533741773939779140350, −3.71902958085182317952577649897, −3.63040962690912129893661876527, −3.47319416524941832648197404654, −3.43503201483177406835474677996, −2.94972481159395889462873051883, −2.90592346441474990881941689133, −2.90148668708684642037799697480, −2.88145781109617046137080430832, −2.42867313291301469815092619835, −2.05481920505225074586457652263, −1.82515632406501849460644153679, −1.74764750481368439477182597497, −1.69678511467731903539240390777, −1.22623810785327963102929021647, −0.998266759730272012902232994852, −0.53156363397190164673825902769, −0.05245301872664529235129904413, 0.05245301872664529235129904413, 0.53156363397190164673825902769, 0.998266759730272012902232994852, 1.22623810785327963102929021647, 1.69678511467731903539240390777, 1.74764750481368439477182597497, 1.82515632406501849460644153679, 2.05481920505225074586457652263, 2.42867313291301469815092619835, 2.88145781109617046137080430832, 2.90148668708684642037799697480, 2.90592346441474990881941689133, 2.94972481159395889462873051883, 3.43503201483177406835474677996, 3.47319416524941832648197404654, 3.63040962690912129893661876527, 3.71902958085182317952577649897, 3.76374392533741773939779140350, 4.11707555782188481031278959576, 4.20194126036395396114437804026, 4.49882386466825625984165364425, 4.58234238681438709194999843555, 4.90920284068658661807126604591, 4.92734558074786769079285904895, 4.92936932533881873615008471483

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

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