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Properties

Label 12-882e6-1.1-c1e6-0-7
Degree $12$
Conductor $4.708\times 10^{17}$
Sign $1$
Analytic cond. $122032.$
Root an. cond. $2.65382$
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 − 4·5-s − 7·7-s − 4·10-s − 10·11-s + 12·13-s − 7·14-s + 6·17-s + 26·19-s − 10·22-s + 22·23-s + 5·25-s + 12·26-s + 2·29-s − 14·31-s + 6·34-s + 28·35-s − 25·37-s + 26·38-s + 18·43-s + 22·46-s − 30·47-s + 28·49-s + 5·50-s + 32·53-s + 40·55-s + 2·58-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.78·5-s − 2.64·7-s − 1.26·10-s − 3.01·11-s + 3.32·13-s − 1.87·14-s + 1.45·17-s + 5.96·19-s − 2.13·22-s + 4.58·23-s + 25-s + 2.35·26-s + 0.371·29-s − 2.51·31-s + 1.02·34-s + 4.73·35-s − 4.10·37-s + 4.21·38-s + 2.74·43-s + 3.24·46-s − 4.37·47-s + 4·49-s + 0.707·50-s + 4.39·53-s + 5.39·55-s + 0.262·58-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(122032.\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.534606611\)
\(L(\frac12)\) \(\approx\) \(3.534606611\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
3 \( 1 \)
7 \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + 3 p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} \)
good5 \( 1 + 4 T + 11 T^{2} + 24 T^{3} + 69 T^{4} + 198 T^{5} + 559 T^{6} + 198 p T^{7} + 69 p^{2} T^{8} + 24 p^{3} T^{9} + 11 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 10 T + 3 p T^{2} + 38 T^{3} + 45 T^{4} + 74 T^{5} - 343 T^{6} + 74 p T^{7} + 45 p^{2} T^{8} + 38 p^{3} T^{9} + 3 p^{5} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 12 T + 47 T^{2} + 12 T^{3} - 461 T^{4} - 42 p T^{5} + 9143 T^{6} - 42 p^{2} T^{7} - 461 p^{2} T^{8} + 12 p^{3} T^{9} + 47 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 6 T - 9 T^{2} + 212 T^{3} - 531 T^{4} - 1636 T^{5} + 15483 T^{6} - 1636 p T^{7} - 531 p^{2} T^{8} + 212 p^{3} T^{9} - 9 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( ( 1 - 13 T + 111 T^{2} - 565 T^{3} + 111 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 22 T + 181 T^{2} - 648 T^{3} + 1805 T^{4} - 20186 T^{5} + 149345 T^{6} - 20186 p T^{7} + 1805 p^{2} T^{8} - 648 p^{3} T^{9} + 181 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 2 T - 25 T^{2} + 402 T^{3} - 23 p T^{4} - 5326 T^{5} + 84679 T^{6} - 5326 p T^{7} - 23 p^{3} T^{8} + 402 p^{3} T^{9} - 25 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 7 T + 37 T^{2} + 133 T^{3} + 37 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 + 25 T + 294 T^{2} + 2288 T^{3} + 15977 T^{4} + 118671 T^{5} + 801760 T^{6} + 118671 p T^{7} + 15977 p^{2} T^{8} + 2288 p^{3} T^{9} + 294 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - p T^{2} + 294 T^{3} - 1343 T^{4} - 11326 T^{5} + 120191 T^{6} - 11326 p T^{7} - 1343 p^{2} T^{8} + 294 p^{3} T^{9} - p^{5} T^{10} + p^{6} T^{12} \)
43 \( 1 - 18 T + 113 T^{2} - 476 T^{3} + 4199 T^{4} - 28612 T^{5} + 142365 T^{6} - 28612 p T^{7} + 4199 p^{2} T^{8} - 476 p^{3} T^{9} + 113 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 30 T + 405 T^{2} + 2746 T^{3} + 1353 T^{4} - 169112 T^{5} - 1755951 T^{6} - 169112 p T^{7} + 1353 p^{2} T^{8} + 2746 p^{3} T^{9} + 405 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 32 T + 411 T^{2} - 2314 T^{3} - 4519 T^{4} + 211810 T^{5} - 2117045 T^{6} + 211810 p T^{7} - 4519 p^{2} T^{8} - 2314 p^{3} T^{9} + 411 p^{4} T^{10} - 32 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 32 T + 517 T^{2} - 6312 T^{3} + 67881 T^{4} - 636524 T^{5} + 5192685 T^{6} - 636524 p T^{7} + 67881 p^{2} T^{8} - 6312 p^{3} T^{9} + 517 p^{4} T^{10} - 32 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 17 T + 18 T^{2} - 1592 T^{3} - 12559 T^{4} + 56943 T^{5} + 1306360 T^{6} + 56943 p T^{7} - 12559 p^{2} T^{8} - 1592 p^{3} T^{9} + 18 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
67 \( ( 1 - 13 T + 241 T^{2} - 1729 T^{3} + 241 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( 1 + 28 T + 209 T^{2} - 1092 T^{3} - 18451 T^{4} + 69734 T^{5} + 2044741 T^{6} + 69734 p T^{7} - 18451 p^{2} T^{8} - 1092 p^{3} T^{9} + 209 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 2 T - 41 T^{2} - 256 T^{3} + 3209 T^{4} + 38042 T^{5} - 151033 T^{6} + 38042 p T^{7} + 3209 p^{2} T^{8} - 256 p^{3} T^{9} - 41 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 + 13 T + 221 T^{2} + 1887 T^{3} + 221 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 16 T + 61 T^{2} - 180 T^{3} - 55 T^{4} + 111496 T^{5} - 1687195 T^{6} + 111496 p T^{7} - 55 p^{2} T^{8} - 180 p^{3} T^{9} + 61 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 22 T + 171 T^{2} + 310 T^{3} - 16467 T^{4} + 106498 T^{5} - 363229 T^{6} + 106498 p T^{7} - 16467 p^{2} T^{8} + 310 p^{3} T^{9} + 171 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 + 11 T + 217 T^{2} + 2147 T^{3} + 217 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.39666878489589344419670866051, −5.19896346393926594775671016981, −5.10799335907503063013304508786, −5.06181995475910074901960166383, −4.85809260368076666069171695713, −4.73408697604434178656317281719, −4.19308830603866237939807845396, −4.16642535793103971184828472065, −3.70386252898511077740895845967, −3.66833334602414622209177018216, −3.58975747289867535344867831356, −3.48992077716765451526691597139, −3.28123295357519941510105013803, −3.27852413617958650940490894429, −3.08847445742321912692240746400, −2.93358320369328873032552812912, −2.91775133053700739295534337510, −2.46270824244381794261319830957, −2.25143137584160037815814090437, −1.79021636047020125506491647106, −1.27647773549378750299303462505, −1.09839770270572987625466609588, −1.08020113940707389492009246057, −0.65588732252602116238850462336, −0.40648194210481450987576461414, 0.40648194210481450987576461414, 0.65588732252602116238850462336, 1.08020113940707389492009246057, 1.09839770270572987625466609588, 1.27647773549378750299303462505, 1.79021636047020125506491647106, 2.25143137584160037815814090437, 2.46270824244381794261319830957, 2.91775133053700739295534337510, 2.93358320369328873032552812912, 3.08847445742321912692240746400, 3.27852413617958650940490894429, 3.28123295357519941510105013803, 3.48992077716765451526691597139, 3.58975747289867535344867831356, 3.66833334602414622209177018216, 3.70386252898511077740895845967, 4.16642535793103971184828472065, 4.19308830603866237939807845396, 4.73408697604434178656317281719, 4.85809260368076666069171695713, 5.06181995475910074901960166383, 5.10799335907503063013304508786, 5.19896346393926594775671016981, 5.39666878489589344419670866051

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

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