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Properties

Label 12-21e12-1.1-c1e6-0-3
Degree $12$
Conductor $7.356\times 10^{15}$
Sign $1$
Analytic cond. $1906.75$
Root an. cond. $1.87654$
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 − 2·3-s + 2·4-s + 10·5-s − 2·6-s − 8-s + 6·9-s + 10·10-s − 4·11-s − 4·12-s + 3·13-s − 20·15-s − 12·17-s + 6·18-s − 3·19-s + 20·20-s − 4·22-s + 2·24-s + 41·25-s + 3·26-s − 7·27-s − 29-s − 20·30-s − 3·31-s − 4·32-s + 8·33-s − 12·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 4-s + 4.47·5-s − 0.816·6-s − 0.353·8-s + 2·9-s + 3.16·10-s − 1.20·11-s − 1.15·12-s + 0.832·13-s − 5.16·15-s − 2.91·17-s + 1.41·18-s − 0.688·19-s + 4.47·20-s − 0.852·22-s + 0.408·24-s + 41/5·25-s + 0.588·26-s − 1.34·27-s − 0.185·29-s − 3.65·30-s − 0.538·31-s − 0.707·32-s + 1.39·33-s − 2.05·34-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1906.75\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.689777797\)
\(L(\frac12)\) \(\approx\) \(4.689777797\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 2 T - 2 T^{2} - p^{2} T^{3} - 2 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 \)
good2 \( 1 - T - T^{2} + p^{2} T^{3} - 3 T^{4} - p T^{5} + 13 T^{6} - p^{2} T^{7} - 3 p^{2} T^{8} + p^{5} T^{9} - p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
5 \( ( 1 - p T + 17 T^{2} - 39 T^{3} + 17 p T^{4} - p^{3} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 + 2 T + 14 T^{2} - 3 T^{3} + 14 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \)
17 \( 1 + 12 T + 54 T^{2} + 210 T^{3} + 1350 T^{4} + 5898 T^{5} + 19735 T^{6} + 5898 p T^{7} + 1350 p^{2} T^{8} + 210 p^{3} T^{9} + 54 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 42 T^{2} - 61 T^{3} + 69 p T^{4} + 726 T^{5} - 27501 T^{6} + 726 p T^{7} + 69 p^{3} T^{8} - 61 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 + 36 T^{2} - 9 T^{3} + 36 p T^{4} + p^{3} T^{6} )^{2} \)
29 \( 1 + T - 82 T^{2} - 31 T^{3} + 4425 T^{4} + 758 T^{5} - 148595 T^{6} + 758 p T^{7} + 4425 p^{2} T^{8} - 31 p^{3} T^{9} - 82 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T - 60 T^{2} - 219 T^{3} + 1983 T^{4} + 4746 T^{5} - 51289 T^{6} + 4746 p T^{7} + 1983 p^{2} T^{8} - 219 p^{3} T^{9} - 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T - 48 T^{2} + 435 T^{3} + 231 T^{4} - 8724 T^{5} + 60581 T^{6} - 8724 p T^{7} + 231 p^{2} T^{8} + 435 p^{3} T^{9} - 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 22 T + 206 T^{2} + 1802 T^{3} + 18432 T^{4} + 135116 T^{5} + 808243 T^{6} + 135116 p T^{7} + 18432 p^{2} T^{8} + 1802 p^{3} T^{9} + 206 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 13170 p T^{7} + 123 p^{2} T^{8} + 569 p^{3} T^{9} - 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 9 T - 6 T^{2} - 531 T^{3} - 2433 T^{4} + 3438 T^{5} + 104623 T^{6} + 3438 p T^{7} - 2433 p^{2} T^{8} - 531 p^{3} T^{9} - 6 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 18 T + 90 T^{2} - 378 T^{3} + 7848 T^{4} - 52668 T^{5} + 160459 T^{6} - 52668 p T^{7} + 7848 p^{2} T^{8} - 378 p^{3} T^{9} + 90 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 9 T - 90 T^{2} - 459 T^{3} + 10161 T^{4} + 20556 T^{5} - 598421 T^{6} + 20556 p T^{7} + 10161 p^{2} T^{8} - 459 p^{3} T^{9} - 90 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T - 126 T^{2} - 358 T^{3} + 12372 T^{4} + 11472 T^{5} - 838653 T^{6} + 11472 p T^{7} + 12372 p^{2} T^{8} - 358 p^{3} T^{9} - 126 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T^{2} + 1366 T^{3} + 438 T^{4} + 4098 T^{5} + 1065603 T^{6} + 4098 p T^{7} + 438 p^{2} T^{8} + 1366 p^{3} T^{9} + 6 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 9 T + 207 T^{2} - 1197 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 3 T - 42 T^{2} + 1209 T^{3} - 3165 T^{4} - 28380 T^{5} + 1003961 T^{6} - 28380 p T^{7} - 3165 p^{2} T^{8} + 1209 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 15 T + 36 T^{2} - 367 T^{3} - 3225 T^{4} - 51726 T^{5} - 676905 T^{6} - 51726 p T^{7} - 3225 p^{2} T^{8} - 367 p^{3} T^{9} + 36 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 12 T - 144 T^{2} - 582 T^{3} + 34812 T^{4} + 90444 T^{5} - 2656433 T^{6} + 90444 p T^{7} + 34812 p^{2} T^{8} - 582 p^{3} T^{9} - 144 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 2 T - 112 T^{2} - 1238 T^{3} + 1662 T^{4} + 59806 T^{5} + 720895 T^{6} + 59806 p T^{7} + 1662 p^{2} T^{8} - 1238 p^{3} T^{9} - 112 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 3 T - 168 T^{2} - 573 T^{3} + 14223 T^{4} + 78504 T^{5} - 1297807 T^{6} + 78504 p T^{7} + 14223 p^{2} T^{8} - 573 p^{3} T^{9} - 168 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.93349882772491475329813245730, −5.92076970703499884243732237081, −5.89007889710091500987780427085, −5.79583286830482968443290047630, −5.17475197656620499810485250944, −5.13130296208713931076443849505, −5.02859251282328740709304202641, −4.97829203422949946535368618935, −4.76563959683769665040461769596, −4.58384541765085720097359012416, −4.21601503882810233809749016942, −3.94948027167378425628095772862, −3.84545812492656520123080674831, −3.60169798908922343912562344231, −3.16426476275751591026397424429, −3.13067244034551400339363084194, −2.76001266267839688184204914914, −2.37389427422714551351038888063, −2.31642634150172268117133845466, −2.03758374673598009905404497514, −2.01438872698384705861912328315, −1.85291228192384056297959678623, −1.39721450364255502665417731466, −1.33369048171989525893683472762, −0.38111082988625489521535246033, 0.38111082988625489521535246033, 1.33369048171989525893683472762, 1.39721450364255502665417731466, 1.85291228192384056297959678623, 2.01438872698384705861912328315, 2.03758374673598009905404497514, 2.31642634150172268117133845466, 2.37389427422714551351038888063, 2.76001266267839688184204914914, 3.13067244034551400339363084194, 3.16426476275751591026397424429, 3.60169798908922343912562344231, 3.84545812492656520123080674831, 3.94948027167378425628095772862, 4.21601503882810233809749016942, 4.58384541765085720097359012416, 4.76563959683769665040461769596, 4.97829203422949946535368618935, 5.02859251282328740709304202641, 5.13130296208713931076443849505, 5.17475197656620499810485250944, 5.79583286830482968443290047630, 5.89007889710091500987780427085, 5.92076970703499884243732237081, 5.93349882772491475329813245730

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

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