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Properties

Label 12-3344e6-1.1-c1e6-0-3
Degree $12$
Conductor $1.398\times 10^{21}$
Sign $1$
Analytic cond. $3.62460\times 10^{8}$
Root an. cond. $5.16739$
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  + 2·3-s − 2·7-s − 2·9-s − 6·11-s + 2·13-s + 12·17-s − 6·19-s − 4·21-s + 2·23-s − 2·25-s − 14·27-s + 4·29-s + 20·31-s − 12·33-s + 22·37-s + 4·39-s − 2·41-s − 10·43-s − 16·47-s + 5·49-s + 24·51-s + 12·53-s − 12·57-s + 14·59-s + 12·61-s + 4·63-s + 12·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s − 2/3·9-s − 1.80·11-s + 0.554·13-s + 2.91·17-s − 1.37·19-s − 0.872·21-s + 0.417·23-s − 2/5·25-s − 2.69·27-s + 0.742·29-s + 3.59·31-s − 2.08·33-s + 3.61·37-s + 0.640·39-s − 0.312·41-s − 1.52·43-s − 2.33·47-s + 5/7·49-s + 3.36·51-s + 1.64·53-s − 1.58·57-s + 1.82·59-s + 1.53·61-s + 0.503·63-s + 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 11^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(3.62460\times 10^{8}\)
Root analytic conductor: \(5.16739\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 11^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.06156822\)
\(L(\frac12)\) \(\approx\) \(14.06156822\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( ( 1 + T )^{6} \)
19 \( ( 1 + T )^{6} \)
good3 \( 1 - 2 T + 2 p T^{2} - 2 T^{3} + 7 T^{4} + 4 p T^{5} + 5 p T^{6} + 4 p^{2} T^{7} + 7 p^{2} T^{8} - 2 p^{3} T^{9} + 2 p^{5} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 2 T^{2} - 6 T^{3} + 43 T^{4} - 42 T^{5} + 37 T^{6} - 42 p T^{7} + 43 p^{2} T^{8} - 6 p^{3} T^{9} + 2 p^{4} T^{10} + p^{6} T^{12} \)
7 \( 1 + 2 T - T^{2} - 8 T^{3} + 78 T^{4} + 102 T^{5} + 138 T^{6} + 102 p T^{7} + 78 p^{2} T^{8} - 8 p^{3} T^{9} - p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 2 T + 29 T^{2} + 2 p T^{3} + 462 T^{4} + 300 T^{5} + 9066 T^{6} + 300 p T^{7} + 462 p^{2} T^{8} + 2 p^{4} T^{9} + 29 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 12 T + 84 T^{2} - 540 T^{3} + 3015 T^{4} - 14232 T^{5} + 61192 T^{6} - 14232 p T^{7} + 3015 p^{2} T^{8} - 540 p^{3} T^{9} + 84 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 2 T + 63 T^{2} - 110 T^{3} + 2079 T^{4} - 148 p T^{5} + 51730 T^{6} - 148 p^{2} T^{7} + 2079 p^{2} T^{8} - 110 p^{3} T^{9} + 63 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 4 T + 105 T^{2} - 14 p T^{3} + 5874 T^{4} - 19558 T^{5} + 210178 T^{6} - 19558 p T^{7} + 5874 p^{2} T^{8} - 14 p^{4} T^{9} + 105 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 20 T + 274 T^{2} - 2576 T^{3} + 20343 T^{4} - 132078 T^{5} + 786627 T^{6} - 132078 p T^{7} + 20343 p^{2} T^{8} - 2576 p^{3} T^{9} + 274 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 22 T + 289 T^{2} - 2614 T^{3} + 20331 T^{4} - 140172 T^{5} + 906342 T^{6} - 140172 p T^{7} + 20331 p^{2} T^{8} - 2614 p^{3} T^{9} + 289 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 2 T + 87 T^{2} - 34 T^{3} + 3864 T^{4} + 1352 T^{5} + 187108 T^{6} + 1352 p T^{7} + 3864 p^{2} T^{8} - 34 p^{3} T^{9} + 87 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 10 T + 233 T^{2} + 1680 T^{3} + 22528 T^{4} + 126170 T^{5} + 1239688 T^{6} + 126170 p T^{7} + 22528 p^{2} T^{8} + 1680 p^{3} T^{9} + 233 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 16 T + 186 T^{2} + 1264 T^{3} + 11055 T^{4} + 88096 T^{5} + 753964 T^{6} + 88096 p T^{7} + 11055 p^{2} T^{8} + 1264 p^{3} T^{9} + 186 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 12 T + 222 T^{2} - 1884 T^{3} + 22695 T^{4} - 150600 T^{5} + 1425124 T^{6} - 150600 p T^{7} + 22695 p^{2} T^{8} - 1884 p^{3} T^{9} + 222 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 14 T + 273 T^{2} - 2918 T^{3} + 33999 T^{4} - 296864 T^{5} + 2553550 T^{6} - 296864 p T^{7} + 33999 p^{2} T^{8} - 2918 p^{3} T^{9} + 273 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 12 T + 152 T^{2} - 1172 T^{3} + 14159 T^{4} - 116176 T^{5} + 1159904 T^{6} - 116176 p T^{7} + 14159 p^{2} T^{8} - 1172 p^{3} T^{9} + 152 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 12 T + 218 T^{2} - 1720 T^{3} + 23423 T^{4} - 187358 T^{5} + 2036891 T^{6} - 187358 p T^{7} + 23423 p^{2} T^{8} - 1720 p^{3} T^{9} + 218 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 30 T + 678 T^{2} - 10974 T^{3} + 145239 T^{4} - 1578396 T^{5} + 14506351 T^{6} - 1578396 p T^{7} + 145239 p^{2} T^{8} - 10974 p^{3} T^{9} + 678 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 24 T + 386 T^{2} - 3464 T^{3} + 15071 T^{4} + 58784 T^{5} - 1353316 T^{6} + 58784 p T^{7} + 15071 p^{2} T^{8} - 3464 p^{3} T^{9} + 386 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 104 T^{2} + 1184 T^{3} + 8831 T^{4} + 48736 T^{5} + 1512080 T^{6} + 48736 p T^{7} + 8831 p^{2} T^{8} + 1184 p^{3} T^{9} + 104 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 - 14 T + 363 T^{2} - 4364 T^{3} + 66390 T^{4} - 642158 T^{5} + 6985258 T^{6} - 642158 p T^{7} + 66390 p^{2} T^{8} - 4364 p^{3} T^{9} + 363 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 14 T + 361 T^{2} + 4970 T^{3} + 71459 T^{4} + 741128 T^{5} + 8430454 T^{6} + 741128 p T^{7} + 71459 p^{2} T^{8} + 4970 p^{3} T^{9} + 361 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 46 T + 1373 T^{2} + 28598 T^{3} + 470895 T^{4} + 6187836 T^{5} + 67342614 T^{6} + 6187836 p T^{7} + 470895 p^{2} T^{8} + 28598 p^{3} T^{9} + 1373 p^{4} T^{10} + 46 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.58414524396110825970601775052, −4.04126952809553522203288574102, −4.02345283148272429690748755803, −3.94887585684137193636159324073, −3.86077193420406117378507007735, −3.68543056357397597204409449016, −3.61189693602015652583701060110, −3.45347743597300010618274802882, −3.22697873759890274399305199497, −2.94504671304132314976762988574, −2.85927969076945514074356729167, −2.85777714163330582435021888006, −2.73289805680166479042187673309, −2.44696049089228524954824809622, −2.39184867897748383865875444586, −2.31693685794062428601630602627, −2.09932245557274451815775135064, −1.97958108537885702832940058949, −1.49189262791119483110151022276, −1.29945929374827035025862930804, −1.24745544838439043486173497037, −0.877402217181564622264380039451, −0.63189858483081410786641183464, −0.49749908859948295515717762693, −0.43926498014944920094336599693, 0.43926498014944920094336599693, 0.49749908859948295515717762693, 0.63189858483081410786641183464, 0.877402217181564622264380039451, 1.24745544838439043486173497037, 1.29945929374827035025862930804, 1.49189262791119483110151022276, 1.97958108537885702832940058949, 2.09932245557274451815775135064, 2.31693685794062428601630602627, 2.39184867897748383865875444586, 2.44696049089228524954824809622, 2.73289805680166479042187673309, 2.85777714163330582435021888006, 2.85927969076945514074356729167, 2.94504671304132314976762988574, 3.22697873759890274399305199497, 3.45347743597300010618274802882, 3.61189693602015652583701060110, 3.68543056357397597204409449016, 3.86077193420406117378507007735, 3.94887585684137193636159324073, 4.02345283148272429690748755803, 4.04126952809553522203288574102, 4.58414524396110825970601775052

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

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