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Properties

Label 12-549e6-1.1-c1e6-0-3
Degree $12$
Conductor $2.738\times 10^{16}$
Sign $1$
Analytic cond. $7097.35$
Root an. cond. $2.09374$
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  − 4-s − 2·5-s + 2·7-s + 2·8-s + 8·11-s + 6·13-s − 2·16-s − 10·17-s + 8·19-s + 2·20-s − 3·25-s − 2·28-s + 10·29-s − 2·32-s − 4·35-s − 4·37-s − 4·40-s + 10·41-s + 4·43-s − 8·44-s + 4·47-s − 13·49-s − 6·52-s + 2·53-s − 16·55-s + 4·56-s + 16·59-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s + 0.755·7-s + 0.707·8-s + 2.41·11-s + 1.66·13-s − 1/2·16-s − 2.42·17-s + 1.83·19-s + 0.447·20-s − 3/5·25-s − 0.377·28-s + 1.85·29-s − 0.353·32-s − 0.676·35-s − 0.657·37-s − 0.632·40-s + 1.56·41-s + 0.609·43-s − 1.20·44-s + 0.583·47-s − 1.85·49-s − 0.832·52-s + 0.274·53-s − 2.15·55-s + 0.534·56-s + 2.08·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 61^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 61^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 61^{6}\)
Sign: $1$
Analytic conductor: \(7097.35\)
Root analytic conductor: \(2.09374\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 61^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.079561294\)
\(L(\frac12)\) \(\approx\) \(4.079561294\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
61 \( ( 1 + T )^{6} \)
good2 \( 1 + T^{2} - p T^{3} + 3 T^{4} - p T^{5} + 3 T^{6} - p^{2} T^{7} + 3 p^{2} T^{8} - p^{4} T^{9} + p^{4} T^{10} + p^{6} T^{12} \)
5 \( 1 + 2 T + 7 T^{2} + 22 T^{3} + 59 T^{4} + 32 p T^{5} + 346 T^{6} + 32 p^{2} T^{7} + 59 p^{2} T^{8} + 22 p^{3} T^{9} + 7 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 2 T + 17 T^{2} - 10 T^{3} + 163 T^{4} - 152 T^{5} + 1590 T^{6} - 152 p T^{7} + 163 p^{2} T^{8} - 10 p^{3} T^{9} + 17 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 8 T + 61 T^{2} - 30 p T^{3} + 1527 T^{4} - 6058 T^{5} + 21498 T^{6} - 6058 p T^{7} + 1527 p^{2} T^{8} - 30 p^{4} T^{9} + 61 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 6 T + 55 T^{2} - 274 T^{3} + 1507 T^{4} - 6080 T^{5} + 24378 T^{6} - 6080 p T^{7} + 1507 p^{2} T^{8} - 274 p^{3} T^{9} + 55 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 10 T + 90 T^{2} + 482 T^{3} + 2835 T^{4} + 12484 T^{5} + 60164 T^{6} + 12484 p T^{7} + 2835 p^{2} T^{8} + 482 p^{3} T^{9} + 90 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 8 T + 54 T^{2} - 104 T^{3} + 263 T^{4} + 1280 T^{5} + 28 p T^{6} + 1280 p T^{7} + 263 p^{2} T^{8} - 104 p^{3} T^{9} + 54 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 93 T^{2} - 2 T^{3} + 4215 T^{4} - 346 T^{5} + 119626 T^{6} - 346 p T^{7} + 4215 p^{2} T^{8} - 2 p^{3} T^{9} + 93 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 - 10 T + 118 T^{2} - 618 T^{3} + 151 p T^{4} - 15748 T^{5} + 114652 T^{6} - 15748 p T^{7} + 151 p^{3} T^{8} - 618 p^{3} T^{9} + 118 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 78 T^{2} + 256 T^{3} + 3039 T^{4} + 15680 T^{5} + 105380 T^{6} + 15680 p T^{7} + 3039 p^{2} T^{8} + 256 p^{3} T^{9} + 78 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 + 4 T + 98 T^{2} + 324 T^{3} + 4279 T^{4} + 13768 T^{5} + 142140 T^{6} + 13768 p T^{7} + 4279 p^{2} T^{8} + 324 p^{3} T^{9} + 98 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 10 T + 191 T^{2} - 1350 T^{3} + 16051 T^{4} - 89568 T^{5} + 814746 T^{6} - 89568 p T^{7} + 16051 p^{2} T^{8} - 1350 p^{3} T^{9} + 191 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 4 T + 206 T^{2} - 684 T^{3} + 19303 T^{4} - 52984 T^{5} + 1058436 T^{6} - 52984 p T^{7} + 19303 p^{2} T^{8} - 684 p^{3} T^{9} + 206 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 4 T + 178 T^{2} - 908 T^{3} + 15631 T^{4} - 1784 p T^{5} + 881340 T^{6} - 1784 p^{2} T^{7} + 15631 p^{2} T^{8} - 908 p^{3} T^{9} + 178 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 2 T + 146 T^{2} - 322 T^{3} + 10283 T^{4} - 28788 T^{5} + 545108 T^{6} - 28788 p T^{7} + 10283 p^{2} T^{8} - 322 p^{3} T^{9} + 146 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 16 T + 389 T^{2} - 4106 T^{3} + 56583 T^{4} - 440906 T^{5} + 4375002 T^{6} - 440906 p T^{7} + 56583 p^{2} T^{8} - 4106 p^{3} T^{9} + 389 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 2 T + 161 T^{2} + 606 T^{3} + 16899 T^{4} + 67908 T^{5} + 1369430 T^{6} + 67908 p T^{7} + 16899 p^{2} T^{8} + 606 p^{3} T^{9} + 161 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 18 T + 438 T^{2} - 5406 T^{3} + 76731 T^{4} - 709420 T^{5} + 7223660 T^{6} - 709420 p T^{7} + 76731 p^{2} T^{8} - 5406 p^{3} T^{9} + 438 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 30 T + 719 T^{2} - 11614 T^{3} + 160203 T^{4} - 1739636 T^{5} + 16506762 T^{6} - 1739636 p T^{7} + 160203 p^{2} T^{8} - 11614 p^{3} T^{9} + 719 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 6 T + 401 T^{2} - 1650 T^{3} + 68699 T^{4} - 203164 T^{5} + 6836870 T^{6} - 203164 p T^{7} + 68699 p^{2} T^{8} - 1650 p^{3} T^{9} + 401 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 12 T + 234 T^{2} - 2388 T^{3} + 34631 T^{4} - 291352 T^{5} + 3439916 T^{6} - 291352 p T^{7} + 34631 p^{2} T^{8} - 2388 p^{3} T^{9} + 234 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 26 T + 550 T^{2} + 7986 T^{3} + 104323 T^{4} + 1107204 T^{5} + 11288476 T^{6} + 1107204 p T^{7} + 104323 p^{2} T^{8} + 7986 p^{3} T^{9} + 550 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 16 T + 402 T^{2} - 5552 T^{3} + 76159 T^{4} - 885888 T^{5} + 9052124 T^{6} - 885888 p T^{7} + 76159 p^{2} T^{8} - 5552 p^{3} T^{9} + 402 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.85392596076079248716261027132, −5.75002566909106858928358953105, −5.23737508409146616759613007244, −5.22517069300163741598722515620, −5.08136757932618374962067450741, −4.94960537263257400698560354551, −4.70287241209355048159564174179, −4.44342547469520007809416202157, −4.33278265774652469734710640548, −4.18521803090260859881304562880, −3.99275926580474509139711953616, −3.91688341228181469496621522422, −3.70132921800541195944185754683, −3.48903214165376976403830790023, −3.41417722470120207877197022945, −3.02089945690969016236117313036, −2.84586198646766296517158255473, −2.35199857032856776179868129347, −2.26531470472546980051390925608, −1.97042961572849279502382844784, −1.90968587964786898377356030839, −1.15462171270501754712954674420, −1.13360905762976589563213604202, −1.09941435987076577792880098992, −0.48584183943761213093177238401, 0.48584183943761213093177238401, 1.09941435987076577792880098992, 1.13360905762976589563213604202, 1.15462171270501754712954674420, 1.90968587964786898377356030839, 1.97042961572849279502382844784, 2.26531470472546980051390925608, 2.35199857032856776179868129347, 2.84586198646766296517158255473, 3.02089945690969016236117313036, 3.41417722470120207877197022945, 3.48903214165376976403830790023, 3.70132921800541195944185754683, 3.91688341228181469496621522422, 3.99275926580474509139711953616, 4.18521803090260859881304562880, 4.33278265774652469734710640548, 4.44342547469520007809416202157, 4.70287241209355048159564174179, 4.94960537263257400698560354551, 5.08136757932618374962067450741, 5.22517069300163741598722515620, 5.23737508409146616759613007244, 5.75002566909106858928358953105, 5.85392596076079248716261027132

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

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