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Properties

Label 12-507e6-1.1-c1e6-0-1
Degree $12$
Conductor $1.698\times 10^{16}$
Sign $1$
Analytic cond. $4402.61$
Root an. cond. $2.01206$
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  + 2-s − 3·3-s + 4·4-s − 8·5-s − 3·6-s + 10·7-s + 5·8-s + 3·9-s − 8·10-s − 11-s − 12·12-s + 10·14-s + 24·15-s + 10·16-s + 7·17-s + 3·18-s + 11·19-s − 32·20-s − 30·21-s − 22-s − 2·23-s − 15·24-s + 12·25-s + 2·27-s + 40·28-s + 8·29-s + 24·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 2·4-s − 3.57·5-s − 1.22·6-s + 3.77·7-s + 1.76·8-s + 9-s − 2.52·10-s − 0.301·11-s − 3.46·12-s + 2.67·14-s + 6.19·15-s + 5/2·16-s + 1.69·17-s + 0.707·18-s + 2.52·19-s − 7.15·20-s − 6.54·21-s − 0.213·22-s − 0.417·23-s − 3.06·24-s + 12/5·25-s + 0.384·27-s + 7.55·28-s + 1.48·29-s + 4.38·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(4402.61\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.678650042\)
\(L(\frac12)\) \(\approx\) \(2.678650042\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T + T^{2} )^{3} \)
13 \( 1 \)
good2 \( 1 - T - 3 T^{2} + p T^{3} + 5 T^{4} - 11 T^{6} + 5 p^{2} T^{8} + p^{4} T^{9} - 3 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
5 \( ( 1 + 4 T + 18 T^{2} + 39 T^{3} + 18 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
7 \( 1 - 10 T + 48 T^{2} - 26 p T^{3} + 650 T^{4} - 1998 T^{5} + 5419 T^{6} - 1998 p T^{7} + 650 p^{2} T^{8} - 26 p^{4} T^{9} + 48 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + T - 2 T^{2} + 45 T^{3} - 3 T^{4} - 8 T^{5} + 2883 T^{6} - 8 p T^{7} - 3 p^{2} T^{8} + 45 p^{3} T^{9} - 2 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 7 T - 16 T^{2} + 35 T^{3} + 1405 T^{4} - 2478 T^{5} - 16135 T^{6} - 2478 p T^{7} + 1405 p^{2} T^{8} + 35 p^{3} T^{9} - 16 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 11 T + 54 T^{2} - 127 T^{3} - 139 T^{4} + 3600 T^{5} - 22229 T^{6} + 3600 p T^{7} - 139 p^{2} T^{8} - 127 p^{3} T^{9} + 54 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 2 T - 22 T^{2} - 298 T^{3} - 272 T^{4} + 3216 T^{5} + 37295 T^{6} + 3216 p T^{7} - 272 p^{2} T^{8} - 298 p^{3} T^{9} - 22 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 8 T - 28 T^{2} + 106 T^{3} + 2428 T^{4} - 2772 T^{5} - 73261 T^{6} - 2772 p T^{7} + 2428 p^{2} T^{8} + 106 p^{3} T^{9} - 28 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 8 T + 70 T^{2} + 299 T^{3} + 70 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 14 T + 22 T^{2} - 182 T^{3} + 8060 T^{4} - 35000 T^{5} - 17101 T^{6} - 35000 p T^{7} + 8060 p^{2} T^{8} - 182 p^{3} T^{9} + 22 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - T - 120 T^{2} + p T^{3} + 9599 T^{4} - 1560 T^{5} - 457559 T^{6} - 1560 p T^{7} + 9599 p^{2} T^{8} + p^{4} T^{9} - 120 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T - 95 T^{2} + 262 T^{3} + 5483 T^{4} - 8563 T^{5} - 240346 T^{6} - 8563 p T^{7} + 5483 p^{2} T^{8} + 262 p^{3} T^{9} - 95 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 - 9 T + 21 T^{2} + 65 T^{3} + 21 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 + 13 T + 199 T^{2} + 1407 T^{3} + 199 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 14 T + 19 T^{2} - 714 T^{3} - 1458 T^{4} + 658 p T^{5} + 427287 T^{6} + 658 p^{2} T^{7} - 1458 p^{2} T^{8} - 714 p^{3} T^{9} + 19 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 13 T - 26 T^{2} + 191 T^{3} + 8411 T^{4} - 4888 T^{5} - 636643 T^{6} - 4888 p T^{7} + 8411 p^{2} T^{8} + 191 p^{3} T^{9} - 26 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 5 T - 154 T^{2} + 251 T^{3} + 17183 T^{4} - 5082 T^{5} - 1329117 T^{6} - 5082 p T^{7} + 17183 p^{2} T^{8} + 251 p^{3} T^{9} - 154 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 6 T - 98 T^{2} + 22 T^{3} + 6096 T^{4} - 31528 T^{5} - 587649 T^{6} - 31528 p T^{7} + 6096 p^{2} T^{8} + 22 p^{3} T^{9} - 98 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 + 18 T + 320 T^{2} + 2795 T^{3} + 320 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( ( 1 + 9 T + 215 T^{2} + 1253 T^{3} + 215 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 - 16 T + 304 T^{2} - 2613 T^{3} + 304 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 5 T - 234 T^{2} - 487 T^{3} + 39575 T^{4} + 39864 T^{5} - 3964415 T^{6} + 39864 p T^{7} + 39575 p^{2} T^{8} - 487 p^{3} T^{9} - 234 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 5 T - p T^{2} - 1048 T^{3} + 4145 T^{4} + 98013 T^{5} + 194334 T^{6} + 98013 p T^{7} + 4145 p^{2} T^{8} - 1048 p^{3} T^{9} - p^{5} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.86001939548272916020245456512, −5.63644624154163384996238472288, −5.60509209746762363061836574989, −5.25277623105440961769563329001, −5.13866452189399809355512809703, −5.06781234843590150447712330282, −4.89654716539588817058792672746, −4.55679021945064720091158865421, −4.34546275082258980487243656190, −4.32972556674601684195301237049, −4.22023033205328564997282158615, −3.93191154327233013365840452052, −3.87260986627731040037512834077, −3.42991587260895836032164602708, −3.30355837569467803352091487550, −3.21826261881645412198445317402, −2.97916692576758770219796862989, −2.52194600600372058720521402482, −2.35576055467604213561843172810, −1.86656844632471711841863355583, −1.79008867974181324963761932205, −1.54846859320585799697169659112, −1.14445736488256366674524030737, −0.941650694877956833792088186429, −0.40335543681034526766362535944, 0.40335543681034526766362535944, 0.941650694877956833792088186429, 1.14445736488256366674524030737, 1.54846859320585799697169659112, 1.79008867974181324963761932205, 1.86656844632471711841863355583, 2.35576055467604213561843172810, 2.52194600600372058720521402482, 2.97916692576758770219796862989, 3.21826261881645412198445317402, 3.30355837569467803352091487550, 3.42991587260895836032164602708, 3.87260986627731040037512834077, 3.93191154327233013365840452052, 4.22023033205328564997282158615, 4.32972556674601684195301237049, 4.34546275082258980487243656190, 4.55679021945064720091158865421, 4.89654716539588817058792672746, 5.06781234843590150447712330282, 5.13866452189399809355512809703, 5.25277623105440961769563329001, 5.60509209746762363061836574989, 5.63644624154163384996238472288, 5.86001939548272916020245456512

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

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