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Properties

Label 12-21e12-1.1-c1e6-0-6
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 + 4·3-s + 2·4-s − 5·5-s + 4·6-s − 8-s + 6·9-s − 5·10-s + 2·11-s + 8·12-s + 3·13-s − 20·15-s + 24·17-s + 6·18-s + 6·19-s − 10·20-s + 2·22-s − 4·24-s + 17·25-s + 3·26-s + 5·27-s − 29-s − 20·30-s − 3·31-s − 4·32-s + 8·33-s + 24·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.30·3-s + 4-s − 2.23·5-s + 1.63·6-s − 0.353·8-s + 2·9-s − 1.58·10-s + 0.603·11-s + 2.30·12-s + 0.832·13-s − 5.16·15-s + 5.82·17-s + 1.41·18-s + 1.37·19-s − 2.23·20-s + 0.426·22-s − 0.816·24-s + 17/5·25-s + 0.588·26-s + 0.962·27-s − 0.185·29-s − 3.65·30-s − 0.538·31-s − 0.707·32-s + 1.39·33-s + 4.11·34-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

\(L(1)\) \(\approx\) \(7.752489827\)
\(L(\frac12)\) \(\approx\) \(7.752489827\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 4 T + 10 T^{2} - 7 p T^{3} + 10 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
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 \( 1 + p T + 8 T^{2} + 7 T^{3} + 9 T^{4} - 62 T^{5} - 299 T^{6} - 62 p T^{7} + 9 p^{2} T^{8} + 7 p^{3} T^{9} + 8 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 - 2 T - 10 T^{2} - 34 T^{3} + 48 T^{4} + 416 T^{5} + 31 T^{6} + 416 p T^{7} + 48 p^{2} T^{8} - 34 p^{3} T^{9} - 10 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \)
17 \( ( 1 - 12 T + 90 T^{2} - 435 T^{3} + 90 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 - 3 T + 51 T^{2} - 107 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 36 T^{2} - 18 T^{3} + 468 T^{4} + 324 T^{5} - 5393 T^{6} + 324 p T^{7} + 468 p^{2} T^{8} - 18 p^{3} T^{9} - 36 p^{4} T^{10} + p^{6} T^{12} \)
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 \( ( 1 + 3 T + 57 T^{2} + 303 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 + 18 T + 234 T^{2} + 1917 T^{3} + 234 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 + 3 T + 51 T^{2} + 681 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 - 2 T + 116 T^{2} - 735 T^{3} + 116 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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

−6.01344381641018650133733615465, −5.65192831644491885038353965160, −5.62760298242770127860157479205, −5.50197879626265285477364267750, −5.27475152446301161997010811146, −5.26602796065890056232445390093, −4.84132637571162115249113343366, −4.60010295392169045779689100256, −4.52668515029490494701086907385, −4.44304334442599282446269216354, −3.67382935062646437201913784312, −3.62784330832784873183962070791, −3.61833017210131173535566320256, −3.55713903744432719498868437644, −3.42609858146325835422102536206, −3.13445257034930462443494352841, −3.03253701240543513256816078408, −2.98066055790184014548705809692, −2.77189612292971444974006859678, −2.51461064959421322489554493286, −1.59441615894190796634440062062, −1.56343346139272356099306735153, −1.46006388277482293689985435444, −1.31601575231585731486460766373, −0.46588058026455834887683551563, 0.46588058026455834887683551563, 1.31601575231585731486460766373, 1.46006388277482293689985435444, 1.56343346139272356099306735153, 1.59441615894190796634440062062, 2.51461064959421322489554493286, 2.77189612292971444974006859678, 2.98066055790184014548705809692, 3.03253701240543513256816078408, 3.13445257034930462443494352841, 3.42609858146325835422102536206, 3.55713903744432719498868437644, 3.61833017210131173535566320256, 3.62784330832784873183962070791, 3.67382935062646437201913784312, 4.44304334442599282446269216354, 4.52668515029490494701086907385, 4.60010295392169045779689100256, 4.84132637571162115249113343366, 5.26602796065890056232445390093, 5.27475152446301161997010811146, 5.50197879626265285477364267750, 5.62760298242770127860157479205, 5.65192831644491885038353965160, 6.01344381641018650133733615465

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

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