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Properties

Label 12-2415e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.984\times 10^{20}$
Sign $1$
Analytic cond. $5.14239\times 10^{7}$
Root an. cond. $4.39134$
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-s − 6·3-s − 3·4-s − 6·5-s − 6·6-s − 6·7-s − 3·8-s + 21·9-s − 6·10-s − 6·11-s + 18·12-s + 6·13-s − 6·14-s + 36·15-s + 16-s + 16·17-s + 21·18-s + 18·20-s + 36·21-s − 6·22-s + 6·23-s + 18·24-s + 21·25-s + 6·26-s − 56·27-s + 18·28-s − 4·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 3.46·3-s − 3/2·4-s − 2.68·5-s − 2.44·6-s − 2.26·7-s − 1.06·8-s + 7·9-s − 1.89·10-s − 1.80·11-s + 5.19·12-s + 1.66·13-s − 1.60·14-s + 9.29·15-s + 1/4·16-s + 3.88·17-s + 4.94·18-s + 4.02·20-s + 7.85·21-s − 1.27·22-s + 1.25·23-s + 3.67·24-s + 21/5·25-s + 1.17·26-s − 10.7·27-s + 3.40·28-s − 0.742·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(5.14239\times 10^{7}\)
Root analytic conductor: \(4.39134\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 23^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3074491811\)
\(L(\frac12)\) \(\approx\) \(0.3074491811\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T )^{6} \)
5 \( ( 1 + T )^{6} \)
7 \( ( 1 + T )^{6} \)
23 \( ( 1 - T )^{6} \)
good2 \( 1 - T + p^{2} T^{2} - p^{2} T^{3} + 3 p^{2} T^{4} - 11 T^{5} + 23 T^{6} - 11 p T^{7} + 3 p^{4} T^{8} - p^{5} T^{9} + p^{6} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 6 T + 39 T^{2} + 148 T^{3} + 662 T^{4} + 2190 T^{5} + 8546 T^{6} + 2190 p T^{7} + 662 p^{2} T^{8} + 148 p^{3} T^{9} + 39 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 6 T + 47 T^{2} - 140 T^{3} + 62 p T^{4} - 2018 T^{5} + 11858 T^{6} - 2018 p T^{7} + 62 p^{3} T^{8} - 140 p^{3} T^{9} + 47 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 16 T + 151 T^{2} - 1024 T^{3} + 5852 T^{4} - 29156 T^{5} + 129000 T^{6} - 29156 p T^{7} + 5852 p^{2} T^{8} - 1024 p^{3} T^{9} + 151 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 25 T^{2} - 42 T^{3} + 218 T^{4} + 106 T^{5} + 4924 T^{6} + 106 p T^{7} + 218 p^{2} T^{8} - 42 p^{3} T^{9} + 25 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 + 4 T + 30 T^{2} + 268 T^{3} + 67 p T^{4} + 11168 T^{5} + 51236 T^{6} + 11168 p T^{7} + 67 p^{3} T^{8} + 268 p^{3} T^{9} + 30 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 80 T^{2} - 8 T^{3} + 3655 T^{4} + 824 T^{5} + 131184 T^{6} + 824 p T^{7} + 3655 p^{2} T^{8} - 8 p^{3} T^{9} + 80 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 + 4 T + 171 T^{2} + 486 T^{3} + 13448 T^{4} + 30510 T^{5} + 633446 T^{6} + 30510 p T^{7} + 13448 p^{2} T^{8} + 486 p^{3} T^{9} + 171 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 4 T + 65 T^{2} - 434 T^{3} + 3698 T^{4} - 29294 T^{5} + 179000 T^{6} - 29294 p T^{7} + 3698 p^{2} T^{8} - 434 p^{3} T^{9} + 65 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 16 T + 161 T^{2} + 770 T^{3} + 2814 T^{4} - 190 T^{5} + 11528 T^{6} - 190 p T^{7} + 2814 p^{2} T^{8} + 770 p^{3} T^{9} + 161 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 38 T + 850 T^{2} - 13154 T^{3} + 156159 T^{4} - 31172 p T^{5} + 11151644 T^{6} - 31172 p^{2} T^{7} + 156159 p^{2} T^{8} - 13154 p^{3} T^{9} + 850 p^{4} T^{10} - 38 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 24 T + 384 T^{2} - 4608 T^{3} + 47759 T^{4} - 429080 T^{5} + 3357744 T^{6} - 429080 p T^{7} + 47759 p^{2} T^{8} - 4608 p^{3} T^{9} + 384 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 2 T + 125 T^{2} - 438 T^{3} + 13648 T^{4} - 37492 T^{5} + 904930 T^{6} - 37492 p T^{7} + 13648 p^{2} T^{8} - 438 p^{3} T^{9} + 125 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 4 T + 141 T^{2} + 802 T^{3} + 15914 T^{4} + 70530 T^{5} + 1131724 T^{6} + 70530 p T^{7} + 15914 p^{2} T^{8} + 802 p^{3} T^{9} + 141 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 30 T + 9 p T^{2} + 8702 T^{3} + 103946 T^{4} + 1040144 T^{5} + 9134428 T^{6} + 1040144 p T^{7} + 103946 p^{2} T^{8} + 8702 p^{3} T^{9} + 9 p^{5} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 6 T + 174 T^{2} + 1434 T^{3} + 16735 T^{4} + 149732 T^{5} + 1337860 T^{6} + 149732 p T^{7} + 16735 p^{2} T^{8} + 1434 p^{3} T^{9} + 174 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 38 T + 795 T^{2} - 10960 T^{3} + 112170 T^{4} - 945366 T^{5} + 7782578 T^{6} - 945366 p T^{7} + 112170 p^{2} T^{8} - 10960 p^{3} T^{9} + 795 p^{4} T^{10} - 38 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 14 T + 436 T^{2} + 4982 T^{3} + 82631 T^{4} + 749304 T^{5} + 8588072 T^{6} + 749304 p T^{7} + 82631 p^{2} T^{8} + 4982 p^{3} T^{9} + 436 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 22 T + 285 T^{2} - 2196 T^{3} + 21536 T^{4} - 252786 T^{5} + 2933656 T^{6} - 252786 p T^{7} + 21536 p^{2} T^{8} - 2196 p^{3} T^{9} + 285 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 20 T + 526 T^{2} - 76 p T^{3} + 105711 T^{4} - 1023872 T^{5} + 11959396 T^{6} - 1023872 p T^{7} + 105711 p^{2} T^{8} - 76 p^{4} T^{9} + 526 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 4 T + 290 T^{2} - 180 T^{3} + 38031 T^{4} + 93304 T^{5} + 3614428 T^{6} + 93304 p T^{7} + 38031 p^{2} T^{8} - 180 p^{3} T^{9} + 290 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.94939135345399888064979589682, −4.41659103912430819824236927106, −4.25578055879879623099008729996, −4.15037372500162828368787838593, −4.11825698821944310413665899212, −4.06436970888776145739909648036, −3.86217022615093919348419445518, −3.56739021502311175519561526306, −3.55118449290751029380214282970, −3.54352421118107154155658767173, −3.26149323535511773965572093661, −3.17826590912320134068982090655, −2.95893982783557810435372776369, −2.78834143081207331667366058893, −2.46356460733964575322142628518, −2.41130578451561721222589735356, −2.02930442811941917931887286565, −1.64805958985137228332250926669, −1.50576794834911577385317012964, −1.08190853195154388967198164855, −0.808677558757411918142903402311, −0.74383651182353871892617623803, −0.63767837520693802226342322899, −0.57509925469793233611339881646, −0.19314499332035778169378202981, 0.19314499332035778169378202981, 0.57509925469793233611339881646, 0.63767837520693802226342322899, 0.74383651182353871892617623803, 0.808677558757411918142903402311, 1.08190853195154388967198164855, 1.50576794834911577385317012964, 1.64805958985137228332250926669, 2.02930442811941917931887286565, 2.41130578451561721222589735356, 2.46356460733964575322142628518, 2.78834143081207331667366058893, 2.95893982783557810435372776369, 3.17826590912320134068982090655, 3.26149323535511773965572093661, 3.54352421118107154155658767173, 3.55118449290751029380214282970, 3.56739021502311175519561526306, 3.86217022615093919348419445518, 4.06436970888776145739909648036, 4.11825698821944310413665899212, 4.15037372500162828368787838593, 4.25578055879879623099008729996, 4.41659103912430819824236927106, 4.94939135345399888064979589682

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

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