Properties

Label 12-1110e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.870\times 10^{18}$
Sign $1$
Analytic cond. $484842.$
Root an. cond. $2.97714$
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  + 6·3-s − 3·4-s − 6·7-s + 21·9-s + 6·11-s − 18·12-s + 6·16-s − 36·21-s − 3·25-s + 56·27-s + 18·28-s + 36·33-s − 63·36-s − 2·37-s − 12·41-s − 18·44-s − 24·47-s + 36·48-s + 5·49-s + 6·53-s − 126·63-s − 10·64-s − 20·71-s − 26·73-s − 18·75-s − 36·77-s + 126·81-s + ⋯
L(s)  = 1  + 3.46·3-s − 3/2·4-s − 2.26·7-s + 7·9-s + 1.80·11-s − 5.19·12-s + 3/2·16-s − 7.85·21-s − 3/5·25-s + 10.7·27-s + 3.40·28-s + 6.26·33-s − 10.5·36-s − 0.328·37-s − 1.87·41-s − 2.71·44-s − 3.50·47-s + 5.19·48-s + 5/7·49-s + 0.824·53-s − 15.8·63-s − 5/4·64-s − 2.37·71-s − 3.04·73-s − 2.07·75-s − 4.10·77-s + 14·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(484842.\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1110} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 37^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.184196804\)
\(L(\frac12)\) \(\approx\) \(2.184196804\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{3} \)
3 \( ( 1 - T )^{6} \)
5 \( ( 1 + T^{2} )^{3} \)
37 \( 1 + 2 T + 31 T^{2} + 116 T^{3} + 31 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
good7 \( ( 1 + 3 T + 11 T^{2} + 26 T^{3} + 11 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 3 T + 23 T^{2} - 58 T^{3} + 23 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 25 T^{2} + 119 T^{4} + 1026 T^{6} + 119 p^{2} T^{8} - 25 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 19 T^{2} + 771 T^{4} + 8674 T^{6} + 771 p^{2} T^{8} + 19 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 - 61 T^{2} + 1727 T^{4} - 35238 T^{6} + 1727 p^{2} T^{8} - 61 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 - 25 T^{2} + 419 T^{4} - 774 T^{6} + 419 p^{2} T^{8} - 25 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \)
31 \( 1 - 66 T^{2} + 2111 T^{4} - 63356 T^{6} + 2111 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 + 2 T + p T^{2} )^{6} \)
43 \( 1 - 94 T^{2} + 6055 T^{4} - 331108 T^{6} + 6055 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \)
47 \( ( 1 + 12 T + 137 T^{2} + 952 T^{3} + 137 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 - 3 T + 149 T^{2} - 310 T^{3} + 149 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 170 T^{2} + 15207 T^{4} - 990668 T^{6} + 15207 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} \)
61 \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{3} \)
67 \( ( 1 - 7 T^{2} - 256 T^{3} - 7 p T^{4} + p^{3} T^{6} )^{2} \)
71 \( ( 1 + 10 T + 21 T^{2} - 468 T^{3} + 21 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 13 T + 191 T^{2} + 1446 T^{3} + 191 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 226 T^{2} + 28063 T^{4} - 2568124 T^{6} + 28063 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \)
83 \( ( 1 - 7 T + 181 T^{2} - 810 T^{3} + 181 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 225 T^{2} + 27615 T^{4} - 2438334 T^{6} + 27615 p^{2} T^{8} - 225 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 - 430 T^{2} + 84863 T^{4} - 10184484 T^{6} + 84863 p^{2} T^{8} - 430 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.15865534989914161451649026893, −4.83304020450057745203569416331, −4.80265329687410360886994779318, −4.65028642070425191317479203836, −4.35748155591622308533367233526, −4.25342075985463205402851253593, −4.21670720821100285176876464842, −3.79409821659877342865488685514, −3.77142648111712192071763491712, −3.73213394971237129634747897045, −3.48534724967726944665300173346, −3.27456400561402474595884393913, −3.27379169299567488510848209138, −3.01963307213901217309815872341, −3.00128140812606589667031366775, −2.74710474826647008075535668407, −2.55380178145259060831776332519, −2.21650449288896423544958415927, −1.97403564587966968325803893320, −1.63653399655713109092775493772, −1.60928711298272813010075349699, −1.48933829900666965542041427950, −1.16310788075165819611705531178, −0.63660432305458623611981268130, −0.16752176211543677570180018239, 0.16752176211543677570180018239, 0.63660432305458623611981268130, 1.16310788075165819611705531178, 1.48933829900666965542041427950, 1.60928711298272813010075349699, 1.63653399655713109092775493772, 1.97403564587966968325803893320, 2.21650449288896423544958415927, 2.55380178145259060831776332519, 2.74710474826647008075535668407, 3.00128140812606589667031366775, 3.01963307213901217309815872341, 3.27379169299567488510848209138, 3.27456400561402474595884393913, 3.48534724967726944665300173346, 3.73213394971237129634747897045, 3.77142648111712192071763491712, 3.79409821659877342865488685514, 4.21670720821100285176876464842, 4.25342075985463205402851253593, 4.35748155591622308533367233526, 4.65028642070425191317479203836, 4.80265329687410360886994779318, 4.83304020450057745203569416331, 5.15865534989914161451649026893

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

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