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Properties

Label 12-162e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.808\times 10^{13}$
Sign $1$
Analytic cond. $4.68546$
Root an. cond. $1.13735$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 3·7-s − 8-s + 3·11-s − 12·13-s + 6·17-s + 9·19-s + 6·23-s − 15·29-s − 18·31-s + 9·35-s + 15·37-s − 3·40-s + 3·41-s − 18·43-s − 9·47-s + 15·49-s + 12·53-s + 9·55-s − 3·56-s + 6·59-s + 18·61-s − 36·65-s − 9·67-s − 12·71-s + 3·73-s + 9·77-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.13·7-s − 0.353·8-s + 0.904·11-s − 3.32·13-s + 1.45·17-s + 2.06·19-s + 1.25·23-s − 2.78·29-s − 3.23·31-s + 1.52·35-s + 2.46·37-s − 0.474·40-s + 0.468·41-s − 2.74·43-s − 1.31·47-s + 15/7·49-s + 1.64·53-s + 1.21·55-s − 0.400·56-s + 0.781·59-s + 2.30·61-s − 4.46·65-s − 1.09·67-s − 1.42·71-s + 0.351·73-s + 1.02·77-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(4.68546\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{24} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.056369742\)
\(L(\frac12)\) \(\approx\) \(2.056369742\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 3 T + 9 T^{2} - 9 T^{3} + 36 T^{4} - 12 T^{5} + 109 T^{6} - 12 p T^{7} + 36 p^{2} T^{8} - 9 p^{3} T^{9} + 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 3 T - 6 T^{2} + 50 T^{3} - 99 T^{4} - 207 T^{5} + 1401 T^{6} - 207 p T^{7} - 99 p^{2} T^{8} + 50 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T + 9 T^{2} + 9 T^{3} - 18 T^{4} + 114 T^{5} + 1225 T^{6} + 114 p T^{7} - 18 p^{2} T^{8} + 9 p^{3} T^{9} + 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 12 T + 6 p T^{2} + 386 T^{3} + 1566 T^{4} + 5886 T^{5} + 21843 T^{6} + 5886 p T^{7} + 1566 p^{2} T^{8} + 386 p^{3} T^{9} + 6 p^{5} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 6 T + 12 T^{2} - 54 T^{3} - 6 p T^{4} + 2082 T^{5} - 8345 T^{6} + 2082 p T^{7} - 6 p^{3} T^{8} - 54 p^{3} T^{9} + 12 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 9 T + 36 T^{2} - 79 T^{3} - 297 T^{4} + 4806 T^{5} - 27429 T^{6} + 4806 p T^{7} - 297 p^{2} T^{8} - 79 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T + 36 T^{2} - 180 T^{3} + 1386 T^{4} - 6954 T^{5} + 33589 T^{6} - 6954 p T^{7} + 1386 p^{2} T^{8} - 180 p^{3} T^{9} + 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 15 T + 99 T^{2} + 387 T^{3} - 162 T^{4} - 17112 T^{5} - 132695 T^{6} - 17112 p T^{7} - 162 p^{2} T^{8} + 387 p^{3} T^{9} + 99 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 18 T + 171 T^{2} + 1253 T^{3} + 7263 T^{4} + 37719 T^{5} + 206634 T^{6} + 37719 p T^{7} + 7263 p^{2} T^{8} + 1253 p^{3} T^{9} + 171 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 15 T + 60 T^{2} - 289 T^{3} + 4725 T^{4} - 17730 T^{5} - 19395 T^{6} - 17730 p T^{7} + 4725 p^{2} T^{8} - 289 p^{3} T^{9} + 60 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 3 T + 36 T^{2} + 72 T^{3} + 18 p T^{4} - 1119 T^{5} + 93799 T^{6} - 1119 p T^{7} + 18 p^{3} T^{8} + 72 p^{3} T^{9} + 36 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 18 T + 144 T^{2} + 740 T^{3} + 432 T^{4} - 23706 T^{5} - 185739 T^{6} - 23706 p T^{7} + 432 p^{2} T^{8} + 740 p^{3} T^{9} + 144 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 9 T + 9 T^{2} - 495 T^{3} - 3222 T^{4} + 3726 T^{5} + 123409 T^{6} + 3726 p T^{7} - 3222 p^{2} T^{8} - 495 p^{3} T^{9} + 9 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 6 T + 150 T^{2} - 639 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 6 T + 36 T^{2} - 261 T^{3} - 639 T^{4} + 33681 T^{5} - 161243 T^{6} + 33681 p T^{7} - 639 p^{2} T^{8} - 261 p^{3} T^{9} + 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 18 T + 153 T^{2} - 745 T^{3} - 3915 T^{4} + 107703 T^{5} - 1034862 T^{6} + 107703 p T^{7} - 3915 p^{2} T^{8} - 745 p^{3} T^{9} + 153 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 9 T + 45 T^{2} + 281 T^{3} - 1836 T^{4} - 68094 T^{5} - 564675 T^{6} - 68094 p T^{7} - 1836 p^{2} T^{8} + 281 p^{3} T^{9} + 45 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 12 T - 24 T^{2} - 738 T^{3} - 228 T^{4} + 5556 T^{5} - 117857 T^{6} + 5556 p T^{7} - 228 p^{2} T^{8} - 738 p^{3} T^{9} - 24 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 3 T - 96 T^{2} + 23 T^{3} + 2853 T^{4} + 12258 T^{5} - 46191 T^{6} + 12258 p T^{7} + 2853 p^{2} T^{8} + 23 p^{3} T^{9} - 96 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 33 T + 510 T^{2} - 4168 T^{3} + 3429 T^{4} + 380187 T^{5} - 5136507 T^{6} + 380187 p T^{7} + 3429 p^{2} T^{8} - 4168 p^{3} T^{9} + 510 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 18 T + 144 T^{2} - 720 T^{3} + 5580 T^{4} - 58968 T^{5} + 392545 T^{6} - 58968 p T^{7} + 5580 p^{2} T^{8} - 720 p^{3} T^{9} + 144 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 15 T - 78 T^{2} + 477 T^{3} + 27177 T^{4} - 70638 T^{5} - 2238167 T^{6} - 70638 p T^{7} + 27177 p^{2} T^{8} + 477 p^{3} T^{9} - 78 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 12 T + 51 T^{2} + 1277 T^{3} + 801 T^{4} - 56169 T^{5} + 617238 T^{6} - 56169 p T^{7} + 801 p^{2} T^{8} + 1277 p^{3} T^{9} + 51 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
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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

−7.35556027597914959278157158464, −7.06864243300658888459624370570, −6.83818532377423263228962317260, −6.66085577282787644193434851316, −6.34269798896863724004544120113, −6.06229404910769250481742251159, −5.82352390325018480979772847481, −5.72777640943653494195939986064, −5.40410070804570242472191486453, −5.34826355233230935211879246032, −5.10067758148170325622425824768, −4.99397791544914048359135871678, −4.89588655815877525823228777769, −4.57784777941659498859603044667, −4.18519048863507980954315565373, −3.66901602202065051517885507611, −3.58640312145986509602051705520, −3.43266242464513517182798164566, −3.38170396163347249519022023722, −2.45927333243607376973869572282, −2.36715758806242702386775584580, −2.36596287690234497643672623157, −1.85363419930606385773560585482, −1.52873340963037104889923350361, −0.892424071567505690977637293974, 0.892424071567505690977637293974, 1.52873340963037104889923350361, 1.85363419930606385773560585482, 2.36596287690234497643672623157, 2.36715758806242702386775584580, 2.45927333243607376973869572282, 3.38170396163347249519022023722, 3.43266242464513517182798164566, 3.58640312145986509602051705520, 3.66901602202065051517885507611, 4.18519048863507980954315565373, 4.57784777941659498859603044667, 4.89588655815877525823228777769, 4.99397791544914048359135871678, 5.10067758148170325622425824768, 5.34826355233230935211879246032, 5.40410070804570242472191486453, 5.72777640943653494195939986064, 5.82352390325018480979772847481, 6.06229404910769250481742251159, 6.34269798896863724004544120113, 6.66085577282787644193434851316, 6.83818532377423263228962317260, 7.06864243300658888459624370570, 7.35556027597914959278157158464

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

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