Properties

Label 12-570e6-1.1-c1e6-0-2
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 − 12·13-s + 6·17-s − 18·19-s − 3·23-s + 27-s + 6·29-s + 21·41-s − 24·43-s + 12·49-s − 24·53-s − 6·56-s − 24·61-s − 24·67-s + 12·71-s + 36·79-s − 12·89-s + 72·91-s − 6·97-s + 30·103-s − 12·104-s − 12·107-s + 48·113-s − 36·119-s + 30·121-s + 125-s + ⋯
L(s)  = 1  − 2.26·7-s + 0.353·8-s − 3.32·13-s + 1.45·17-s − 4.12·19-s − 0.625·23-s + 0.192·27-s + 1.11·29-s + 3.27·41-s − 3.65·43-s + 12/7·49-s − 3.29·53-s − 0.801·56-s − 3.07·61-s − 2.93·67-s + 1.42·71-s + 4.05·79-s − 1.27·89-s + 7.54·91-s − 0.609·97-s + 2.95·103-s − 1.17·104-s − 1.16·107-s + 4.51·113-s − 3.30·119-s + 2.72·121-s + 0.0894·125-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

\(L(1)\) \(\approx\) \(0.001706653947\)
\(L(\frac12)\) \(\approx\) \(0.001706653947\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + 24 T^{2} + 46 T^{3} - 18 T^{4} - 558 T^{5} - 1917 T^{6} - 558 p T^{7} - 18 p^{2} T^{8} + 46 p^{3} T^{9} + 24 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 30 T^{2} - 2 T^{3} + 570 T^{4} + 30 T^{5} - 7237 T^{6} + 30 p T^{7} + 570 p^{2} T^{8} - 2 p^{3} T^{9} - 30 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 12 T + 108 T^{2} + 714 T^{3} + 3888 T^{4} + 17760 T^{5} + 68975 T^{6} + 17760 p T^{7} + 3888 p^{2} T^{8} + 714 p^{3} T^{9} + 108 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 6 T + 48 T^{2} - 164 T^{3} + 1164 T^{4} - 4716 T^{5} + 27995 T^{6} - 4716 p T^{7} + 1164 p^{2} T^{8} - 164 p^{3} T^{9} + 48 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 3 T + 12 T^{2} - 38 T^{3} - 141 T^{4} - 3033 T^{5} - 3631 T^{6} - 3033 p T^{7} - 141 p^{2} T^{8} - 38 p^{3} T^{9} + 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 57 T^{2} - 144 T^{3} + 1482 T^{4} + 4104 T^{5} - 35953 T^{6} + 4104 p T^{7} + 1482 p^{2} T^{8} - 144 p^{3} T^{9} - 57 p^{4} T^{10} + p^{6} T^{12} \)
37 \( ( 1 + 36 T^{2} + 125 T^{3} + 36 p T^{4} + p^{3} T^{6} )^{2} \)
41 \( 1 - 21 T + 150 T^{2} + 10 T^{3} - 4287 T^{4} - 207 p T^{5} + 264917 T^{6} - 207 p^{2} T^{7} - 4287 p^{2} T^{8} + 10 p^{3} T^{9} + 150 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 24 T + 288 T^{2} + 2094 T^{3} + 8388 T^{4} - 1254 T^{5} - 193555 T^{6} - 1254 p T^{7} + 8388 p^{2} T^{8} + 2094 p^{3} T^{9} + 288 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 18 T^{2} - 140 T^{3} + 126 T^{4} + 13158 T^{5} + 64277 T^{6} + 13158 p T^{7} + 126 p^{2} T^{8} - 140 p^{3} T^{9} + 18 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 + 24 T + 237 T^{2} + 857 T^{3} - 3219 T^{4} - 55989 T^{5} - 422374 T^{6} - 55989 p T^{7} - 3219 p^{2} T^{8} + 857 p^{3} T^{9} + 237 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 54 T^{2} + 8 p T^{3} - 3546 T^{4} - 28998 T^{5} + 417329 T^{6} - 28998 p T^{7} - 3546 p^{2} T^{8} + 8 p^{4} T^{9} - 54 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 + 24 T + 192 T^{2} - 220 T^{3} - 13104 T^{4} - 76716 T^{5} - 269733 T^{6} - 76716 p T^{7} - 13104 p^{2} T^{8} - 220 p^{3} T^{9} + 192 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 24 T + 216 T^{2} + 438 T^{3} - 8208 T^{4} - 1668 p T^{5} - 997255 T^{6} - 1668 p^{2} T^{7} - 8208 p^{2} T^{8} + 438 p^{3} T^{9} + 216 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 12 T + 12 T^{2} + 1424 T^{3} - 10068 T^{4} - 51714 T^{5} + 1334165 T^{6} - 51714 p T^{7} - 10068 p^{2} T^{8} + 1424 p^{3} T^{9} + 12 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 358 T^{3} - 2052 T^{4} - 53514 T^{5} + 169095 T^{6} - 53514 p T^{7} - 2052 p^{2} T^{8} + 358 p^{3} T^{9} + p^{6} T^{12} \)
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 \( 1 - 165 T^{2} + 272 T^{3} + 13530 T^{4} - 22440 T^{5} - 1097605 T^{6} - 22440 p T^{7} + 13530 p^{2} T^{8} + 272 p^{3} T^{9} - 165 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 + 12 T + 219 T^{2} + 3319 T^{3} + 36993 T^{4} + 411093 T^{5} + 4455590 T^{6} + 411093 p T^{7} + 36993 p^{2} T^{8} + 3319 p^{3} T^{9} + 219 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 6 T + 12 T^{2} + 518 T^{3} - 2232 T^{4} - 106020 T^{5} - 917265 T^{6} - 106020 p T^{7} - 2232 p^{2} T^{8} + 518 p^{3} T^{9} + 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.96914968915346321409452967447, −5.65347400000297767898731927471, −5.47694967120535716531199448696, −5.17497427457877984665472579805, −4.81847585738440749317093586244, −4.75151752269083759825676487710, −4.69970912279363371544602658353, −4.64753387090124779296856441098, −4.60982342553592501594897116531, −4.18934181253212892707726585257, −3.90898589491712851559781885559, −3.84751792593567938056830707511, −3.52749990158142682845281874845, −3.14626901249478427120138495366, −3.11646708276415189487927309818, −3.02261030040728011136244267223, −2.95876885334111739301419248464, −2.43528869449694980255672504847, −2.31631145562151188860107344220, −1.97554648400314666676274647494, −1.84126797109781524679497901541, −1.83766894431369852570530587819, −1.06400991922615142310793182069, −0.51954211405115910775916284956, −0.01381130773038887264942458046, 0.01381130773038887264942458046, 0.51954211405115910775916284956, 1.06400991922615142310793182069, 1.83766894431369852570530587819, 1.84126797109781524679497901541, 1.97554648400314666676274647494, 2.31631145562151188860107344220, 2.43528869449694980255672504847, 2.95876885334111739301419248464, 3.02261030040728011136244267223, 3.11646708276415189487927309818, 3.14626901249478427120138495366, 3.52749990158142682845281874845, 3.84751792593567938056830707511, 3.90898589491712851559781885559, 4.18934181253212892707726585257, 4.60982342553592501594897116531, 4.64753387090124779296856441098, 4.69970912279363371544602658353, 4.75151752269083759825676487710, 4.81847585738440749317093586244, 5.17497427457877984665472579805, 5.47694967120535716531199448696, 5.65347400000297767898731927471, 5.96914968915346321409452967447

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

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