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Properties

Label 12-4840e6-1.1-c1e6-0-2
Degree $12$
Conductor $1.286\times 10^{22}$
Sign $1$
Analytic cond. $3.33222\times 10^{9}$
Root an. cond. $6.21671$
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 + 6·5-s + 4·7-s − 5·9-s − 12·15-s + 8·17-s + 12·19-s − 8·21-s − 8·23-s + 21·25-s + 10·27-s + 16·29-s − 4·31-s + 24·35-s + 8·37-s + 32·41-s − 4·43-s − 30·45-s − 6·47-s − 5·49-s − 16·51-s + 8·53-s − 24·57-s + 4·59-s + 16·61-s − 20·63-s − 2·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 2.68·5-s + 1.51·7-s − 5/3·9-s − 3.09·15-s + 1.94·17-s + 2.75·19-s − 1.74·21-s − 1.66·23-s + 21/5·25-s + 1.92·27-s + 2.97·29-s − 0.718·31-s + 4.05·35-s + 1.31·37-s + 4.99·41-s − 0.609·43-s − 4.47·45-s − 0.875·47-s − 5/7·49-s − 2.24·51-s + 1.09·53-s − 3.17·57-s + 0.520·59-s + 2.04·61-s − 2.51·63-s − 0.244·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 5^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(3.33222\times 10^{9}\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 5^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(38.83293306\)
\(L(\frac12)\) \(\approx\) \(38.83293306\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 - T )^{6} \)
11 \( 1 \)
good3 \( 1 + 2 T + p^{2} T^{2} + 2 p^{2} T^{3} + 50 T^{4} + 82 T^{5} + 181 T^{6} + 82 p T^{7} + 50 p^{2} T^{8} + 2 p^{5} T^{9} + p^{6} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 4 T + 3 p T^{2} - 8 p T^{3} + 218 T^{4} - 636 T^{5} + 1993 T^{6} - 636 p T^{7} + 218 p^{2} T^{8} - 8 p^{4} T^{9} + 3 p^{5} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 46 T^{2} - 16 T^{3} + 935 T^{4} - 560 T^{5} + 13172 T^{6} - 560 p T^{7} + 935 p^{2} T^{8} - 16 p^{3} T^{9} + 46 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 8 T + 86 T^{2} - 488 T^{3} + 191 p T^{4} - 14352 T^{5} + 70772 T^{6} - 14352 p T^{7} + 191 p^{3} T^{8} - 488 p^{3} T^{9} + 86 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 12 T + 142 T^{2} - 1028 T^{3} + 6983 T^{4} - 36232 T^{5} + 176372 T^{6} - 36232 p T^{7} + 6983 p^{2} T^{8} - 1028 p^{3} T^{9} + 142 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 8 T + 78 T^{2} + 464 T^{3} + 3647 T^{4} + 17448 T^{5} + 99796 T^{6} + 17448 p T^{7} + 3647 p^{2} T^{8} + 464 p^{3} T^{9} + 78 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 16 T + 158 T^{2} - 976 T^{3} + 3655 T^{4} - 5856 T^{5} - 10876 T^{6} - 5856 p T^{7} + 3655 p^{2} T^{8} - 976 p^{3} T^{9} + 158 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 4 T + 106 T^{2} + 580 T^{3} + 5471 T^{4} + 35008 T^{5} + 194684 T^{6} + 35008 p T^{7} + 5471 p^{2} T^{8} + 580 p^{3} T^{9} + 106 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 8 T + 142 T^{2} - 872 T^{3} + 9911 T^{4} - 52016 T^{5} + 449252 T^{6} - 52016 p T^{7} + 9911 p^{2} T^{8} - 872 p^{3} T^{9} + 142 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 32 T + 597 T^{2} - 7760 T^{3} + 78494 T^{4} - 647040 T^{5} + 4497049 T^{6} - 647040 p T^{7} + 78494 p^{2} T^{8} - 7760 p^{3} T^{9} + 597 p^{4} T^{10} - 32 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 4 T + 157 T^{2} + 160 T^{3} + 9938 T^{4} - 11060 T^{5} + 442193 T^{6} - 11060 p T^{7} + 9938 p^{2} T^{8} + 160 p^{3} T^{9} + 157 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 6 T + 217 T^{2} + 1142 T^{3} + 22146 T^{4} + 96502 T^{5} + 1330941 T^{6} + 96502 p T^{7} + 22146 p^{2} T^{8} + 1142 p^{3} T^{9} + 217 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 8 T + 142 T^{2} - 624 T^{3} + 8055 T^{4} - 33704 T^{5} + 443124 T^{6} - 33704 p T^{7} + 8055 p^{2} T^{8} - 624 p^{3} T^{9} + 142 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 4 T + 106 T^{2} - 804 T^{3} + 11415 T^{4} - 59440 T^{5} + 798588 T^{6} - 59440 p T^{7} + 11415 p^{2} T^{8} - 804 p^{3} T^{9} + 106 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 16 T + 245 T^{2} - 2240 T^{3} + 24374 T^{4} - 199632 T^{5} + 1856313 T^{6} - 199632 p T^{7} + 24374 p^{2} T^{8} - 2240 p^{3} T^{9} + 245 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 2 T + 49 T^{2} + 458 T^{3} + 11018 T^{4} + 49970 T^{5} + 322277 T^{6} + 49970 p T^{7} + 11018 p^{2} T^{8} + 458 p^{3} T^{9} + 49 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 28 T + 546 T^{2} + 7900 T^{3} + 98591 T^{4} + 1012032 T^{5} + 9200236 T^{6} + 1012032 p T^{7} + 98591 p^{2} T^{8} + 7900 p^{3} T^{9} + 546 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 16 T + 246 T^{2} - 2768 T^{3} + 28991 T^{4} - 286368 T^{5} + 2601844 T^{6} - 286368 p T^{7} + 28991 p^{2} T^{8} - 2768 p^{3} T^{9} + 246 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 342 T^{2} - 288 T^{3} + 55791 T^{4} - 50976 T^{5} + 5551220 T^{6} - 50976 p T^{7} + 55791 p^{2} T^{8} - 288 p^{3} T^{9} + 342 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 - 12 T + 210 T^{2} - 1028 T^{3} + 22311 T^{4} - 168072 T^{5} + 2881772 T^{6} - 168072 p T^{7} + 22311 p^{2} T^{8} - 1028 p^{3} T^{9} + 210 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 18 T + 253 T^{2} - 1798 T^{3} + 8058 T^{4} + 72934 T^{5} - 1088091 T^{6} + 72934 p T^{7} + 8058 p^{2} T^{8} - 1798 p^{3} T^{9} + 253 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 342 T^{2} + 1080 T^{3} + 57231 T^{4} + 248616 T^{5} + 6595652 T^{6} + 248616 p T^{7} + 57231 p^{2} T^{8} + 1080 p^{3} T^{9} + 342 p^{4} T^{10} + p^{6} T^{12} \)
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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

−4.29528004026213863479368256959, −4.19354326007236006951223042329, −4.12574007100460366029484904264, −3.59525854738450272153750697130, −3.58484249415974404845911413849, −3.55137170052535091085224908438, −3.51097364398601086691112301382, −2.98773988818005823291079902888, −2.94222623190381881790941156852, −2.93354658570206065776158624175, −2.78639681382574888451728858589, −2.73309333576716047461195451532, −2.67387234298055596892021370119, −2.10908648679711057998186919618, −2.05453474023109268457619947334, −1.96349845074531796692548607354, −1.93978671001539915946718073739, −1.65940070019316151238944338334, −1.64060028833432289707609540998, −1.18851638592805071124050620636, −0.865543980655330629100108664435, −0.831593847679191202559858382604, −0.817225901353903119133286008740, −0.63790633295046160347830288329, −0.49270137886943292314371979562, 0.49270137886943292314371979562, 0.63790633295046160347830288329, 0.817225901353903119133286008740, 0.831593847679191202559858382604, 0.865543980655330629100108664435, 1.18851638592805071124050620636, 1.64060028833432289707609540998, 1.65940070019316151238944338334, 1.93978671001539915946718073739, 1.96349845074531796692548607354, 2.05453474023109268457619947334, 2.10908648679711057998186919618, 2.67387234298055596892021370119, 2.73309333576716047461195451532, 2.78639681382574888451728858589, 2.93354658570206065776158624175, 2.94222623190381881790941156852, 2.98773988818005823291079902888, 3.51097364398601086691112301382, 3.55137170052535091085224908438, 3.58484249415974404845911413849, 3.59525854738450272153750697130, 4.12574007100460366029484904264, 4.19354326007236006951223042329, 4.29528004026213863479368256959

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

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