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Properties

Label 12-2268e6-1.1-c1e6-0-3
Degree $12$
Conductor $1.361\times 10^{20}$
Sign $1$
Analytic cond. $3.52793\times 10^{7}$
Root an. cond. $4.25559$
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  + 2·5-s − 4·7-s + 10·11-s + 2·13-s + 4·17-s − 3·19-s + 28·23-s − 3·25-s + 5·29-s + 2·31-s − 8·35-s − 12·41-s + 9·43-s + 9·47-s + 2·49-s − 6·53-s + 20·55-s + 5·59-s − 7·61-s + 4·65-s + 16·67-s + 22·71-s + 73-s − 40·77-s + 8·79-s − 17·83-s + 8·85-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.51·7-s + 3.01·11-s + 0.554·13-s + 0.970·17-s − 0.688·19-s + 5.83·23-s − 3/5·25-s + 0.928·29-s + 0.359·31-s − 1.35·35-s − 1.87·41-s + 1.37·43-s + 1.31·47-s + 2/7·49-s − 0.824·53-s + 2.69·55-s + 0.650·59-s − 0.896·61-s + 0.496·65-s + 1.95·67-s + 2.61·71-s + 0.117·73-s − 4.55·77-s + 0.900·79-s − 1.86·83-s + 0.867·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 3^{24} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(3.52793\times 10^{7}\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 3^{24} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.58435428\)
\(L(\frac12)\) \(\approx\) \(14.58435428\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + 4 T + 2 p T^{2} + 55 T^{3} + 2 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( ( 1 - T + 3 T^{2} - 19 T^{3} + 3 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 5 T + 21 T^{2} - 47 T^{3} + 21 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 2 T - 24 T^{2} + 6 T^{3} + 358 T^{4} + 250 T^{5} - 5513 T^{6} + 250 p T^{7} + 358 p^{2} T^{8} + 6 p^{3} T^{9} - 24 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 4 T - 20 T^{2} + 146 T^{3} + 104 T^{4} - 1480 T^{5} + 4195 T^{6} - 1480 p T^{7} + 104 p^{2} T^{8} + 146 p^{3} T^{9} - 20 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 54 p T^{7} - 153 p^{2} T^{8} - 67 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 - 14 T + 108 T^{2} - 581 T^{3} + 108 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 - 5 T - 50 T^{2} + 79 T^{3} + 2315 T^{4} + 520 T^{5} - 83843 T^{6} + 520 p T^{7} + 2315 p^{2} T^{8} + 79 p^{3} T^{9} - 50 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 2 T - 78 T^{2} + 42 T^{3} + 3976 T^{4} - 200 T^{5} - 142097 T^{6} - 200 p T^{7} + 3976 p^{2} T^{8} + 42 p^{3} T^{9} - 78 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 54 T^{2} - 274 T^{3} + 918 T^{4} + 7398 T^{5} + 12183 T^{6} + 7398 p T^{7} + 918 p^{2} T^{8} - 274 p^{3} T^{9} - 54 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 + 12 T + 12 T^{2} - 222 T^{3} + 1176 T^{4} + 9480 T^{5} - 3197 T^{6} + 9480 p T^{7} + 1176 p^{2} T^{8} - 222 p^{3} T^{9} + 12 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 9 T - 18 T^{2} + 95 T^{3} + 1107 T^{4} + 11178 T^{5} - 169149 T^{6} + 11178 p T^{7} + 1107 p^{2} T^{8} + 95 p^{3} T^{9} - 18 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 9 T - 6 T^{2} + 9 p T^{3} - 1947 T^{4} + 1260 T^{5} + 36583 T^{6} + 1260 p T^{7} - 1947 p^{2} T^{8} + 9 p^{4} T^{9} - 6 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 6 T - 24 T^{2} + 222 T^{3} + 6 T^{4} - 17166 T^{5} + 10591 T^{6} - 17166 p T^{7} + 6 p^{2} T^{8} + 222 p^{3} T^{9} - 24 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 5 T - 134 T^{2} + 223 T^{3} + 13355 T^{4} - 4526 T^{5} - 908765 T^{6} - 4526 p T^{7} + 13355 p^{2} T^{8} + 223 p^{3} T^{9} - 134 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 7 T - 45 T^{2} + 84 T^{3} + 1093 T^{4} - 27251 T^{5} - 184994 T^{6} - 27251 p T^{7} + 1093 p^{2} T^{8} + 84 p^{3} T^{9} - 45 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 16 T - 4 T^{2} + 234 T^{3} + 15028 T^{4} - 85604 T^{5} - 250049 T^{6} - 85604 p T^{7} + 15028 p^{2} T^{8} + 234 p^{3} T^{9} - 4 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 11 T + 141 T^{2} - 1373 T^{3} + 141 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - T - 136 T^{2} + 477 T^{3} + 8461 T^{4} - 27386 T^{5} - 503183 T^{6} - 27386 p T^{7} + 8461 p^{2} T^{8} + 477 p^{3} T^{9} - 136 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 8 T - 18 T^{2} + 126 T^{3} - 2882 T^{4} + 21658 T^{5} + 208339 T^{6} + 21658 p T^{7} - 2882 p^{2} T^{8} + 126 p^{3} T^{9} - 18 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 17 T - 44 T^{2} - 181 T^{3} + 33593 T^{4} + 150080 T^{5} - 1518893 T^{6} + 150080 p T^{7} + 33593 p^{2} T^{8} - 181 p^{3} T^{9} - 44 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 3 T - 33 T^{2} - 1704 T^{3} + 393 T^{4} + 31803 T^{5} + 1804174 T^{6} + 31803 p T^{7} + 393 p^{2} T^{8} - 1704 p^{3} T^{9} - 33 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 14 T - 55 T^{2} - 1806 T^{3} + 1414 T^{4} + 117670 T^{5} + 945685 T^{6} + 117670 p T^{7} + 1414 p^{2} T^{8} - 1806 p^{3} T^{9} - 55 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.55774062002564963002138793691, −4.55184989250896081922894726263, −4.28562605482244374936905049612, −4.23170354092425264510658564698, −4.12009058311257065226577899423, −3.86880750355639836279290855191, −3.81401975377687896298570346889, −3.71599524623527955100136116559, −3.33004189500800532882821924079, −3.17884899542477397601388240637, −3.12303375875826804180996785833, −3.01489500017206919175607318442, −2.96385298606395440552360633520, −2.79770734212208197651167622244, −2.47309816128636559499470223219, −2.24364134413256668075074533013, −2.19713329212690146024655493417, −1.77941403127079059610127080050, −1.62857791726058252989425496255, −1.28538984972666784261469635172, −1.21372064478813345024539015140, −1.10834280904450524236722028953, −1.06494186289986028728163044591, −0.49663260210324974916237500422, −0.45285688780812373793170563951, 0.45285688780812373793170563951, 0.49663260210324974916237500422, 1.06494186289986028728163044591, 1.10834280904450524236722028953, 1.21372064478813345024539015140, 1.28538984972666784261469635172, 1.62857791726058252989425496255, 1.77941403127079059610127080050, 2.19713329212690146024655493417, 2.24364134413256668075074533013, 2.47309816128636559499470223219, 2.79770734212208197651167622244, 2.96385298606395440552360633520, 3.01489500017206919175607318442, 3.12303375875826804180996785833, 3.17884899542477397601388240637, 3.33004189500800532882821924079, 3.71599524623527955100136116559, 3.81401975377687896298570346889, 3.86880750355639836279290855191, 4.12009058311257065226577899423, 4.23170354092425264510658564698, 4.28562605482244374936905049612, 4.55184989250896081922894726263, 4.55774062002564963002138793691

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

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