Properties

Label 12-570e6-1.1-c1e6-0-8
Degree $12$
Conductor $3.430\times 10^{16}$
Sign $1$
Analytic cond. $8890.20$
Root an. cond. $2.13341$
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  − 6·7-s + 8-s + 12·11-s + 6·17-s − 18·19-s + 21·23-s − 27-s + 6·29-s + 24·37-s − 3·41-s + 24·47-s + 24·49-s + 24·53-s − 6·56-s + 12·59-s − 36·71-s + 24·73-s − 72·77-s − 36·79-s + 12·88-s + 48·89-s + 42·97-s + 24·101-s − 18·103-s − 36·107-s − 24·109-s − 36·119-s + ⋯
L(s)  = 1  − 2.26·7-s + 0.353·8-s + 3.61·11-s + 1.45·17-s − 4.12·19-s + 4.37·23-s − 0.192·27-s + 1.11·29-s + 3.94·37-s − 0.468·41-s + 3.50·47-s + 24/7·49-s + 3.29·53-s − 0.801·56-s + 1.56·59-s − 4.27·71-s + 2.80·73-s − 8.20·77-s − 4.05·79-s + 1.27·88-s + 5.08·89-s + 4.26·97-s + 2.38·101-s − 1.77·103-s − 3.48·107-s − 2.29·109-s − 3.30·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(8890.20\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.754108576\)
\(L(\frac12)\) \(\approx\) \(5.754108576\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
3 \( 1 + T^{3} + T^{6} \)
5 \( 1 - T^{3} + T^{6} \)
19 \( 1 + 18 T + 162 T^{2} + 883 T^{3} + 162 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
good7 \( 1 + 6 T + 12 T^{2} + 2 p T^{3} + 18 T^{4} - 162 T^{5} - 885 T^{6} - 162 p T^{7} + 18 p^{2} T^{8} + 2 p^{4} T^{9} + 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 12 T + 6 p T^{2} - 306 T^{3} + 1446 T^{4} - 5430 T^{5} + 17539 T^{6} - 5430 p T^{7} + 1446 p^{2} T^{8} - 306 p^{3} T^{9} + 6 p^{5} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - 89 T^{3} + p^{3} T^{6} )( 1 + 19 T^{3} + p^{3} T^{6} ) \)
17 \( 1 - 6 T + 36 T^{3} - 36 T^{4} - 12 p T^{5} + 2395 T^{6} - 12 p^{2} T^{7} - 36 p^{2} T^{8} + 36 p^{3} T^{9} - 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 21 T + 252 T^{2} - 2214 T^{3} + 693 p T^{4} - 96897 T^{5} + 503713 T^{6} - 96897 p T^{7} + 693 p^{3} T^{8} - 2214 p^{3} T^{9} + 252 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 6 T + 36 T^{2} - 18 T^{3} + 576 T^{4} + 2964 T^{5} - 2933 T^{6} + 2964 p T^{7} + 576 p^{2} T^{8} - 18 p^{3} T^{9} + 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 9 T^{2} - 592 T^{3} - 198 T^{4} + 2664 T^{5} + 144687 T^{6} + 2664 p T^{7} - 198 p^{2} T^{8} - 592 p^{3} T^{9} - 9 p^{4} T^{10} + p^{6} T^{12} \)
37 \( ( 1 - 12 T + 120 T^{2} - 815 T^{3} + 120 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 3 T + 54 T^{2} + 270 T^{3} + 1881 T^{4} + 25905 T^{5} + 88453 T^{6} + 25905 p T^{7} + 1881 p^{2} T^{8} + 270 p^{3} T^{9} + 54 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 178 T^{3} - 972 T^{4} + 10314 T^{5} + 71085 T^{6} + 10314 p T^{7} - 972 p^{2} T^{8} - 178 p^{3} T^{9} + p^{6} T^{12} \)
47 \( 1 - 24 T + 306 T^{2} - 2844 T^{3} + 25254 T^{4} - 215106 T^{5} + 1621333 T^{6} - 215106 p T^{7} + 25254 p^{2} T^{8} - 2844 p^{3} T^{9} + 306 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 24 T + 333 T^{2} - 3663 T^{3} + 34461 T^{4} - 278205 T^{5} + 2071162 T^{6} - 278205 p T^{7} + 34461 p^{2} T^{8} - 3663 p^{3} T^{9} + 333 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 12 T + 162 T^{2} - 792 T^{3} + 5418 T^{4} + 66 p T^{5} + 83809 T^{6} + 66 p^{2} T^{7} + 5418 p^{2} T^{8} - 792 p^{3} T^{9} + 162 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 92 T^{3} + 432 T^{4} + 25596 T^{5} + 54939 T^{6} + 25596 p T^{7} + 432 p^{2} T^{8} + 92 p^{3} T^{9} + p^{6} T^{12} \)
67 \( 1 + 216 T^{2} + 758 T^{3} + 20736 T^{4} + 143100 T^{5} + 1418169 T^{6} + 143100 p T^{7} + 20736 p^{2} T^{8} + 758 p^{3} T^{9} + 216 p^{4} T^{10} + p^{6} T^{12} \)
71 \( 1 + 36 T + 540 T^{2} + 4104 T^{3} + 7884 T^{4} - 205434 T^{5} - 2772683 T^{6} - 205434 p T^{7} + 7884 p^{2} T^{8} + 4104 p^{3} T^{9} + 540 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 24 T + 240 T^{2} - 946 T^{3} - 5940 T^{4} + 123174 T^{5} - 1187865 T^{6} + 123174 p T^{7} - 5940 p^{2} T^{8} - 946 p^{3} T^{9} + 240 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 36 T + 720 T^{2} + 10676 T^{3} + 133056 T^{4} + 1425600 T^{5} + 13446993 T^{6} + 1425600 p T^{7} + 133056 p^{2} T^{8} + 10676 p^{3} T^{9} + 720 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 213 T^{2} + 144 T^{3} + 27690 T^{4} - 15336 T^{5} - 2616869 T^{6} - 15336 p T^{7} + 27690 p^{2} T^{8} + 144 p^{3} T^{9} - 213 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 48 T + 1035 T^{2} - 13545 T^{3} + 123993 T^{4} - 908751 T^{5} + 7190038 T^{6} - 908751 p T^{7} + 123993 p^{2} T^{8} - 13545 p^{3} T^{9} + 1035 p^{4} T^{10} - 48 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 42 T + 924 T^{2} - 14482 T^{3} + 180792 T^{4} - 1932012 T^{5} + 19226511 T^{6} - 1932012 p T^{7} + 180792 p^{2} T^{8} - 14482 p^{3} T^{9} + 924 p^{4} T^{10} - 42 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.02848574787213229734622923271, −5.58245821246959647419293582659, −5.49493137974040945958901498448, −5.37734398070317283508263707249, −4.99458270285813387130876238775, −4.85352919546511728205953365283, −4.54695525907449977564774229488, −4.49857022739937508458948792373, −4.28403891394398948134451282158, −3.97549521280550000102841580202, −3.95384429530793840134949572664, −3.92776683667664695126528704876, −3.75251137402945864965678354884, −3.32104113619295812173599101957, −3.27040862525733456523789223764, −2.99528263254116291022937265793, −2.58192407301158852374975642740, −2.52752548512281226359857221160, −2.41222456189140128218598906236, −2.29449859182110137903216335871, −1.63850319801149132434811023687, −1.30155696186020984279144278043, −1.02241324143229531504490359387, −0.76895633756127334468015259787, −0.75013417158504110099771127527, 0.75013417158504110099771127527, 0.76895633756127334468015259787, 1.02241324143229531504490359387, 1.30155696186020984279144278043, 1.63850319801149132434811023687, 2.29449859182110137903216335871, 2.41222456189140128218598906236, 2.52752548512281226359857221160, 2.58192407301158852374975642740, 2.99528263254116291022937265793, 3.27040862525733456523789223764, 3.32104113619295812173599101957, 3.75251137402945864965678354884, 3.92776683667664695126528704876, 3.95384429530793840134949572664, 3.97549521280550000102841580202, 4.28403891394398948134451282158, 4.49857022739937508458948792373, 4.54695525907449977564774229488, 4.85352919546511728205953365283, 4.99458270285813387130876238775, 5.37734398070317283508263707249, 5.49493137974040945958901498448, 5.58245821246959647419293582659, 6.02848574787213229734622923271

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

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