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Properties

Label 12-945e6-1.1-c1e6-0-3
Degree $12$
Conductor $7.122\times 10^{17}$
Sign $1$
Analytic cond. $184609.$
Root an. cond. $2.74697$
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  + 2-s + 2·4-s − 3·5-s − 2·7-s + 3·8-s − 3·10-s − 11-s − 4·13-s − 2·14-s + 6·16-s + 11·17-s + 13·19-s − 6·20-s − 22-s − 10·23-s + 3·25-s − 4·26-s − 4·28-s + 14·29-s − 8·31-s + 8·32-s + 11·34-s + 6·35-s − 2·37-s + 13·38-s − 9·40-s − 20·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 4-s − 1.34·5-s − 0.755·7-s + 1.06·8-s − 0.948·10-s − 0.301·11-s − 1.10·13-s − 0.534·14-s + 3/2·16-s + 2.66·17-s + 2.98·19-s − 1.34·20-s − 0.213·22-s − 2.08·23-s + 3/5·25-s − 0.784·26-s − 0.755·28-s + 2.59·29-s − 1.43·31-s + 1.41·32-s + 1.88·34-s + 1.01·35-s − 0.328·37-s + 2.10·38-s − 1.42·40-s − 3.12·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(184609.\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.159220447\)
\(L(\frac12)\) \(\approx\) \(8.159220447\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 + T + T^{2} )^{3} \)
7 \( 1 + 2 T - 4 T^{2} - 13 T^{3} - 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 - T - T^{2} - T^{4} + p T^{5} + 5 T^{6} + p^{2} T^{7} - p^{2} T^{8} - p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + T - 2 p T^{2} - 39 T^{3} + 245 T^{4} + 328 T^{5} - 2317 T^{6} + 328 p T^{7} + 245 p^{2} T^{8} - 39 p^{3} T^{9} - 2 p^{5} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 + 2 T + 30 T^{2} + 33 T^{3} + 30 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 11 T + 47 T^{2} - 120 T^{3} + 401 T^{4} - 1181 T^{5} + 2174 T^{6} - 1181 p T^{7} + 401 p^{2} T^{8} - 120 p^{3} T^{9} + 47 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 13 T + 66 T^{2} - 253 T^{3} + 1593 T^{4} - 9740 T^{5} + 45987 T^{6} - 9740 p T^{7} + 1593 p^{2} T^{8} - 253 p^{3} T^{9} + 66 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 10 T + 20 T^{2} - 114 T^{3} - 10 T^{4} + 5050 T^{5} + 33083 T^{6} + 5050 p T^{7} - 10 p^{2} T^{8} - 114 p^{3} T^{9} + 20 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 - 7 T + 79 T^{2} - 325 T^{3} + 79 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 8 T - 26 T^{2} - 254 T^{3} + 1414 T^{4} + 6878 T^{5} - 15853 T^{6} + 6878 p T^{7} + 1414 p^{2} T^{8} - 254 p^{3} T^{9} - 26 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 2 T + 10 T^{2} + 826 T^{3} + 1096 T^{4} + 6140 T^{5} + 270059 T^{6} + 6140 p T^{7} + 1096 p^{2} T^{8} + 826 p^{3} T^{9} + 10 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 10 T + 152 T^{2} + 841 T^{3} + 152 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 6 T + 61 T^{2} - 4 p T^{3} + 61 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 17 T + 132 T^{2} - 571 T^{3} - 563 T^{4} + 48640 T^{5} - 488857 T^{6} + 48640 p T^{7} - 563 p^{2} T^{8} - 571 p^{3} T^{9} + 132 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 7 T - 70 T^{2} - 225 T^{3} + 4151 T^{4} - 5684 T^{5} - 310579 T^{6} - 5684 p T^{7} + 4151 p^{2} T^{8} - 225 p^{3} T^{9} - 70 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 17 T + 96 T^{2} - 367 T^{3} + 1093 T^{4} + 46780 T^{5} - 670093 T^{6} + 46780 p T^{7} + 1093 p^{2} T^{8} - 367 p^{3} T^{9} + 96 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - T - 158 T^{2} - 5 T^{3} + 15505 T^{4} + 3434 T^{5} - 1085971 T^{6} + 3434 p T^{7} + 15505 p^{2} T^{8} - 5 p^{3} T^{9} - 158 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 15 T - 10 T^{2} + 133 T^{3} + 12045 T^{4} - 24800 T^{5} - 763333 T^{6} - 24800 p T^{7} + 12045 p^{2} T^{8} + 133 p^{3} T^{9} - 10 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 2 T + 109 T^{2} + 404 T^{3} + 109 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 112 T^{2} - 362 T^{3} + 4368 T^{4} + 20272 T^{5} - 104917 T^{6} + 20272 p T^{7} + 4368 p^{2} T^{8} - 362 p^{3} T^{9} - 112 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 - 10 T - 150 T^{2} + 662 T^{3} + 26940 T^{4} - 62660 T^{5} - 2075961 T^{6} - 62660 p T^{7} + 26940 p^{2} T^{8} + 662 p^{3} T^{9} - 150 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 16 T + 301 T^{2} + 2632 T^{3} + 301 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 24 T + 164 T^{2} - 1194 T^{3} + 28748 T^{4} - 221004 T^{5} + 636635 T^{6} - 221004 p T^{7} + 28748 p^{2} T^{8} - 1194 p^{3} T^{9} + 164 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 5 T + 247 T^{2} - 9 p T^{3} + 247 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.35739521332271963033651682707, −5.22132827528763683552470889641, −5.01986573415333547670131585430, −4.75271066513199096912033459592, −4.75072952014114020081227860779, −4.48474223429822463157029016157, −4.41830650388738499416246476860, −4.09733703037626012651899368006, −3.90726261910517058626686695421, −3.65141510001634765062817744432, −3.52368975619582912080612209974, −3.46281083264283431509065645299, −3.36411707593412970650312495278, −3.21503584334065052081647597045, −3.08657477731727225241295183044, −2.66482930399043414091326082051, −2.42211678377244039420185313409, −2.37463051491271589778597346711, −2.26975316215828453899083648524, −1.82625965015367733718899125953, −1.52267211990137044559038583453, −1.14482532571163980019222925199, −1.10212301426456341047344902154, −0.57791342936689660333643752631, −0.55195897698756283828349737205, 0.55195897698756283828349737205, 0.57791342936689660333643752631, 1.10212301426456341047344902154, 1.14482532571163980019222925199, 1.52267211990137044559038583453, 1.82625965015367733718899125953, 2.26975316215828453899083648524, 2.37463051491271589778597346711, 2.42211678377244039420185313409, 2.66482930399043414091326082051, 3.08657477731727225241295183044, 3.21503584334065052081647597045, 3.36411707593412970650312495278, 3.46281083264283431509065645299, 3.52368975619582912080612209974, 3.65141510001634765062817744432, 3.90726261910517058626686695421, 4.09733703037626012651899368006, 4.41830650388738499416246476860, 4.48474223429822463157029016157, 4.75072952014114020081227860779, 4.75271066513199096912033459592, 5.01986573415333547670131585430, 5.22132827528763683552470889641, 5.35739521332271963033651682707

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

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