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Properties

Label 12-77e6-1.1-c1e6-0-0
Degree $12$
Conductor $208422380089$
Sign $1$
Analytic cond. $0.0540265$
Root an. cond. $0.784122$
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  − 3·3-s + 3·4-s − 6·5-s + 2·8-s + 9·9-s − 3·11-s − 9·12-s + 6·13-s + 18·15-s + 6·16-s + 3·17-s − 9·19-s − 18·20-s − 6·24-s + 24·25-s − 20·27-s − 6·29-s − 9·31-s + 9·32-s + 9·33-s + 27·36-s − 18·39-s − 12·40-s + 18·41-s − 9·44-s − 54·45-s + 3·47-s + ⋯
L(s)  = 1  − 1.73·3-s + 3/2·4-s − 2.68·5-s + 0.707·8-s + 3·9-s − 0.904·11-s − 2.59·12-s + 1.66·13-s + 4.64·15-s + 3/2·16-s + 0.727·17-s − 2.06·19-s − 4.02·20-s − 1.22·24-s + 24/5·25-s − 3.84·27-s − 1.11·29-s − 1.61·31-s + 1.59·32-s + 1.56·33-s + 9/2·36-s − 2.88·39-s − 1.89·40-s + 2.81·41-s − 1.35·44-s − 8.04·45-s + 0.437·47-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(7^{6} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(0.0540265\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{6} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3953341351\)
\(L(\frac12)\) \(\approx\) \(0.3953341351\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + 17 T^{3} + p^{3} T^{6} \)
11 \( ( 1 + T + T^{2} )^{3} \)
good2 \( 1 - 3 T^{2} - p T^{3} + 3 T^{4} + 3 T^{5} - T^{6} + 3 p T^{7} + 3 p^{2} T^{8} - p^{4} T^{9} - 3 p^{4} T^{10} + p^{6} T^{12} \)
3 \( 1 + p T - 7 T^{3} + p T^{4} + 2 p^{2} T^{5} + 19 T^{6} + 2 p^{3} T^{7} + p^{3} T^{8} - 7 p^{3} T^{9} + p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 + 6 T + 12 T^{2} + 22 T^{3} + 18 p T^{4} + 174 T^{5} + 191 T^{6} + 174 p T^{7} + 18 p^{3} T^{8} + 22 p^{3} T^{9} + 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - 3 T + 33 T^{2} - 79 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 3 T - 6 T^{2} - 95 T^{3} + 45 T^{4} + 1038 T^{5} + 4025 T^{6} + 1038 p T^{7} + 45 p^{2} T^{8} - 95 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 9 T + 6 T^{2} - 27 T^{3} + 1041 T^{4} + 3924 T^{5} + 803 T^{6} + 3924 p T^{7} + 1041 p^{2} T^{8} - 27 p^{3} T^{9} + 6 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 12 T^{2} - 214 T^{3} - 132 T^{4} + 1284 T^{5} + 32471 T^{6} + 1284 p T^{7} - 132 p^{2} T^{8} - 214 p^{3} T^{9} - 12 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 + 3 T + 51 T^{2} + 225 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 9 T - 36 T^{2} - 101 T^{3} + 4869 T^{4} + 10872 T^{5} - 109689 T^{6} + 10872 p T^{7} + 4869 p^{2} T^{8} - 101 p^{3} T^{9} - 36 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 75 T^{2} + 144 T^{3} + 2850 T^{4} - 5400 T^{5} - 101635 T^{6} - 5400 p T^{7} + 2850 p^{2} T^{8} + 144 p^{3} T^{9} - 75 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 - 9 T + 129 T^{2} - 739 T^{3} + 129 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 120 T^{2} + 9 T^{3} + 120 p T^{4} + p^{3} T^{6} )^{2} \)
47 \( 1 - 3 T - 54 T^{2} - 271 T^{3} + 1131 T^{4} + 13722 T^{5} + 2903 T^{6} + 13722 p T^{7} + 1131 p^{2} T^{8} - 271 p^{3} T^{9} - 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 9 T - 24 T^{2} - 45 T^{3} + 1005 T^{4} - 22914 T^{5} - 269075 T^{6} - 22914 p T^{7} + 1005 p^{2} T^{8} - 45 p^{3} T^{9} - 24 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 84 T^{2} + 38 T^{3} + 2100 T^{4} - 1596 T^{5} - 5185 T^{6} - 1596 p T^{7} + 2100 p^{2} T^{8} + 38 p^{3} T^{9} - 84 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 + 12 T - 3 T^{2} - 1212 T^{3} - 5214 T^{4} + 48180 T^{5} + 814133 T^{6} + 48180 p T^{7} - 5214 p^{2} T^{8} - 1212 p^{3} T^{9} - 3 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 189 T^{2} - 16 T^{3} + 23058 T^{4} + 1512 T^{5} - 1791717 T^{6} + 1512 p T^{7} + 23058 p^{2} T^{8} - 16 p^{3} T^{9} - 189 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 + 9 T + 123 T^{2} + 477 T^{3} + 123 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 6 T - 192 T^{2} - 386 T^{3} + 30078 T^{4} + 32202 T^{5} - 2439513 T^{6} + 32202 p T^{7} + 30078 p^{2} T^{8} - 386 p^{3} T^{9} - 192 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 3 T - 114 T^{2} + 653 T^{3} + 3879 T^{4} - 23274 T^{5} - 4161 T^{6} - 23274 p T^{7} + 3879 p^{2} T^{8} + 653 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 15 T + 267 T^{2} + 2223 T^{3} + 267 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 15 T - 114 T^{2} + 477 T^{3} + 43035 T^{4} - 156444 T^{5} - 2866295 T^{6} - 156444 p T^{7} + 43035 p^{2} T^{8} + 477 p^{3} T^{9} - 114 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 45 T + 963 T^{2} - 12059 T^{3} + 963 p T^{4} - 45 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−7.918299842298887121279945495677, −7.85019668045001615297489333699, −7.71602567121731371041422831253, −7.53400516902111194902452304018, −7.44229780624515942392292025020, −7.37264035611548788024806866697, −7.17900965003447893967802479464, −6.66134489220317241257638117196, −6.43881623493255313144542609699, −6.25652013537764870443900986256, −6.04866766317231801552341841296, −5.99534773471710112325488696400, −5.71478197185815595071563491412, −5.07780975599190330713244507662, −4.96514643585683507956602522047, −4.73633986828674671963355072377, −4.41215307715792759859193151860, −4.20596048612112708995179278306, −3.79479374736086033284608987479, −3.75229953747646392894346503755, −3.38396963030364195163891180328, −2.96800190682158495625039596705, −2.31197962326812811421567525558, −1.79744230101110718502144436565, −1.13021797604519816026999231821, 1.13021797604519816026999231821, 1.79744230101110718502144436565, 2.31197962326812811421567525558, 2.96800190682158495625039596705, 3.38396963030364195163891180328, 3.75229953747646392894346503755, 3.79479374736086033284608987479, 4.20596048612112708995179278306, 4.41215307715792759859193151860, 4.73633986828674671963355072377, 4.96514643585683507956602522047, 5.07780975599190330713244507662, 5.71478197185815595071563491412, 5.99534773471710112325488696400, 6.04866766317231801552341841296, 6.25652013537764870443900986256, 6.43881623493255313144542609699, 6.66134489220317241257638117196, 7.17900965003447893967802479464, 7.37264035611548788024806866697, 7.44229780624515942392292025020, 7.53400516902111194902452304018, 7.71602567121731371041422831253, 7.85019668045001615297489333699, 7.918299842298887121279945495677

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

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