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Properties

Label 12-3800e6-1.1-c0e6-0-5
Degree 1212
Conductor 3.011×10213.011\times 10^{21}
Sign 11
Analytic cond. 46.520446.5204
Root an. cond. 1.377111.37711
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 8-s + 21·9-s − 6·24-s − 55·27-s + 6·41-s − 3·49-s − 3·59-s + 3·67-s + 21·72-s + 3·73-s + 120·81-s + 3·97-s + 6·107-s − 36·123-s + 127-s + 131-s + 137-s + 139-s + 18·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 18·177-s + ⋯
L(s)  = 1  − 6·3-s + 8-s + 21·9-s − 6·24-s − 55·27-s + 6·41-s − 3·49-s − 3·59-s + 3·67-s + 21·72-s + 3·73-s + 120·81-s + 3·97-s + 6·107-s − 36·123-s + 127-s + 131-s + 137-s + 139-s + 18·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 18·177-s + ⋯

Functional equation

Λ(s)=((218⋅512⋅196)s/2 ΓC(s)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((218⋅512⋅196)s/2 ΓC(s)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 218⋅512⋅1962^{18} \cdot 5^{12} \cdot 19^{6}
Sign: 11
Analytic conductor: 46.520446.5204
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 218⋅512⋅196, ( :[0]6), 1)(12,\ 2^{18} \cdot 5^{12} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.43735672000.4373567200
L(12)L(\frac12) ≈\approx 0.43735672000.4373567200
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1−T3+T6 1 - T^{3} + T^{6}
5 1 1
19 1+T3+T6 1 + T^{3} + T^{6}
good3 (1+T)6(1−T3+T6) ( 1 + T )^{6}( 1 - T^{3} + T^{6} )
7 (1−T+T2)3(1+T+T2)3 ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3}
11 (1+T3+T6)2 ( 1 + T^{3} + T^{6} )^{2}
13 (1−T3+T6)(1+T3+T6) ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} )
17 (1−T3+T6)2 ( 1 - T^{3} + T^{6} )^{2}
23 (1−T3+T6)(1+T3+T6) ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} )
29 (1−T3+T6)(1+T3+T6) ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} )
31 (1−T+T2)3(1+T+T2)3 ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3}
37 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
41 (1−T)6(1+T3+T6) ( 1 - T )^{6}( 1 + T^{3} + T^{6} )
43 (1−T3+T6)2 ( 1 - T^{3} + T^{6} )^{2}
47 (1−T3+T6)(1+T3+T6) ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} )
53 (1−T3+T6)(1+T3+T6) ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} )
59 (1+T+T2)3(1+T3+T6) ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} )
61 (1−T3+T6)(1+T3+T6) ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} )
67 (1−T+T2)3(1−T3+T6) ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} )
71 (1−T3+T6)(1+T3+T6) ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} )
73 (1−T+T2)3(1−T3+T6) ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} )
79 (1−T3+T6)(1+T3+T6) ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} )
83 (1−T3+T6)2 ( 1 - T^{3} + T^{6} )^{2}
89 (1+T3+T6)2 ( 1 + T^{3} + T^{6} )^{2}
97 (1−T+T2)3(1−T3+T6) ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} )
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(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

−4.65588623403090266911680122197, −4.53440605318215059845970878353, −4.45452884919507975297115695901, −4.38417401003492059285509340524, −4.30105730120281373904677862784, −4.08611119879870587017480992804, −4.04082048391047313774880645067, −3.61280674962931583743874246866, −3.37659520587155956345329470465, −3.37616031694706102480738190385, −3.32494972547290781847087915164, −3.28108941571175036622815203704, −2.85705350107394372737111342598, −2.26514056137762521908302021417, −2.24588040020576349407323000242, −2.15419658212726388349431128931, −2.05552982932751691187882542096, −1.83259384862959404810141391154, −1.67424212192426977264743411018, −1.42759487613721642894819643118, −1.15853950373466430246170786952, −0.856843530204907589373333334017, −0.75136841781040699102854412938, −0.73053505432368097083646628390, −0.61065719934470144284918960642, 0.61065719934470144284918960642, 0.73053505432368097083646628390, 0.75136841781040699102854412938, 0.856843530204907589373333334017, 1.15853950373466430246170786952, 1.42759487613721642894819643118, 1.67424212192426977264743411018, 1.83259384862959404810141391154, 2.05552982932751691187882542096, 2.15419658212726388349431128931, 2.24588040020576349407323000242, 2.26514056137762521908302021417, 2.85705350107394372737111342598, 3.28108941571175036622815203704, 3.32494972547290781847087915164, 3.37616031694706102480738190385, 3.37659520587155956345329470465, 3.61280674962931583743874246866, 4.04082048391047313774880645067, 4.08611119879870587017480992804, 4.30105730120281373904677862784, 4.38417401003492059285509340524, 4.45452884919507975297115695901, 4.53440605318215059845970878353, 4.65588623403090266911680122197

Graph of the ZZ-function along the critical line

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