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Properties

Label 12-8470e6-1.1-c1e6-0-10
Degree $12$
Conductor $3.692\times 10^{23}$
Sign $1$
Analytic cond. $9.57112\times 10^{10}$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s − 5·3-s + 21·4-s − 6·5-s − 30·6-s + 6·7-s + 56·8-s + 9·9-s − 36·10-s − 105·12-s + 36·14-s + 30·15-s + 126·16-s − 15·17-s + 54·18-s − 13·19-s − 126·20-s − 30·21-s − 2·23-s − 280·24-s + 21·25-s − 7·27-s + 126·28-s − 4·29-s + 180·30-s − 2·31-s + 252·32-s + ⋯
L(s)  = 1  + 4.24·2-s − 2.88·3-s + 21/2·4-s − 2.68·5-s − 12.2·6-s + 2.26·7-s + 19.7·8-s + 3·9-s − 11.3·10-s − 30.3·12-s + 9.62·14-s + 7.74·15-s + 63/2·16-s − 3.63·17-s + 12.7·18-s − 2.98·19-s − 28.1·20-s − 6.54·21-s − 0.417·23-s − 57.1·24-s + 21/5·25-s − 1.34·27-s + 23.8·28-s − 0.742·29-s + 32.8·30-s − 0.359·31-s + 44.5·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(9.57112\times 10^{10}\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{6} \)
5 \( ( 1 + T )^{6} \)
7 \( ( 1 - T )^{6} \)
11 \( 1 \)
good3 \( 1 + 5 T + 16 T^{2} + 14 p T^{3} + 97 T^{4} + 196 T^{5} + 359 T^{6} + 196 p T^{7} + 97 p^{2} T^{8} + 14 p^{4} T^{9} + 16 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 40 T^{2} - 34 T^{3} + 875 T^{4} - 1070 T^{5} + 13068 T^{6} - 1070 p T^{7} + 875 p^{2} T^{8} - 34 p^{3} T^{9} + 40 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 15 T + 138 T^{2} + 970 T^{3} + 5707 T^{4} + 28520 T^{5} + 124645 T^{6} + 28520 p T^{7} + 5707 p^{2} T^{8} + 970 p^{3} T^{9} + 138 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 13 T + 154 T^{2} + 1132 T^{3} + 7763 T^{4} + 40096 T^{5} + 197123 T^{6} + 40096 p T^{7} + 7763 p^{2} T^{8} + 1132 p^{3} T^{9} + 154 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 2 T + 58 T^{2} + 254 T^{3} + 2239 T^{4} + 7596 T^{5} + 69484 T^{6} + 7596 p T^{7} + 2239 p^{2} T^{8} + 254 p^{3} T^{9} + 58 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 4 T + 122 T^{2} + 510 T^{3} + 7151 T^{4} + 27354 T^{5} + 257848 T^{6} + 27354 p T^{7} + 7151 p^{2} T^{8} + 510 p^{3} T^{9} + 122 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 2 T + 70 T^{2} + 48 T^{3} + 2507 T^{4} + 2554 T^{5} + 82896 T^{6} + 2554 p T^{7} + 2507 p^{2} T^{8} + 48 p^{3} T^{9} + 70 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 160 T^{2} - 22 T^{3} + 12535 T^{4} - 2030 T^{5} + 583972 T^{6} - 2030 p T^{7} + 12535 p^{2} T^{8} - 22 p^{3} T^{9} + 160 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 + 17 T + 292 T^{2} + 3128 T^{3} + 32117 T^{4} + 245654 T^{5} + 1786223 T^{6} + 245654 p T^{7} + 32117 p^{2} T^{8} + 3128 p^{3} T^{9} + 292 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 15 T + 294 T^{2} - 3028 T^{3} + 33531 T^{4} - 252386 T^{5} + 1955439 T^{6} - 252386 p T^{7} + 33531 p^{2} T^{8} - 3028 p^{3} T^{9} + 294 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 18 T + 6 p T^{2} + 3342 T^{3} + 33743 T^{4} + 282684 T^{5} + 2116396 T^{6} + 282684 p T^{7} + 33743 p^{2} T^{8} + 3342 p^{3} T^{9} + 6 p^{5} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 22 T + 326 T^{2} - 3856 T^{3} + 36371 T^{4} - 301158 T^{5} + 2323416 T^{6} - 301158 p T^{7} + 36371 p^{2} T^{8} - 3856 p^{3} T^{9} + 326 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 7 T + 134 T^{2} + 868 T^{3} + 11523 T^{4} + 64404 T^{5} + 783443 T^{6} + 64404 p T^{7} + 11523 p^{2} T^{8} + 868 p^{3} T^{9} + 134 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 20 T + 326 T^{2} + 2938 T^{3} + 23295 T^{4} + 112874 T^{5} + 847776 T^{6} + 112874 p T^{7} + 23295 p^{2} T^{8} + 2938 p^{3} T^{9} + 326 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 29 T + 572 T^{2} + 8250 T^{3} + 98265 T^{4} + 979044 T^{5} + 8617941 T^{6} + 979044 p T^{7} + 98265 p^{2} T^{8} + 8250 p^{3} T^{9} + 572 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 2 T + 258 T^{2} + 560 T^{3} + 31551 T^{4} + 66622 T^{5} + 2594952 T^{6} + 66622 p T^{7} + 31551 p^{2} T^{8} + 560 p^{3} T^{9} + 258 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - T + 172 T^{2} - 576 T^{3} + 22731 T^{4} - 59064 T^{5} + 1971113 T^{6} - 59064 p T^{7} + 22731 p^{2} T^{8} - 576 p^{3} T^{9} + 172 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 12 T + 390 T^{2} + 3858 T^{3} + 70439 T^{4} + 554166 T^{5} + 7197976 T^{6} + 554166 p T^{7} + 70439 p^{2} T^{8} + 3858 p^{3} T^{9} + 390 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 35 T + 846 T^{2} + 14848 T^{3} + 212607 T^{4} + 2493634 T^{5} + 24781859 T^{6} + 2493634 p T^{7} + 212607 p^{2} T^{8} + 14848 p^{3} T^{9} + 846 p^{4} T^{10} + 35 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 29 T + 654 T^{2} + 9894 T^{3} + 136223 T^{4} + 1508924 T^{5} + 15667235 T^{6} + 1508924 p T^{7} + 136223 p^{2} T^{8} + 9894 p^{3} T^{9} + 654 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 19 T + 262 T^{2} + 2172 T^{3} + 28259 T^{4} + 364562 T^{5} + 4760381 T^{6} + 364562 p T^{7} + 28259 p^{2} T^{8} + 2172 p^{3} T^{9} + 262 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.45888000436320847807961721725, −4.33124130547277825933337546245, −4.16509218884189341552647559557, −4.08737443061898819103000287647, −4.06921294592902058476246831899, −3.92486906127401289644781623735, −3.87579090431928980764120734117, −3.47543950305502998503998600622, −3.45154993010908848864252007163, −3.09945970804122573884860295251, −3.08813781158215951019043592965, −3.03935349662536666867092087927, −3.02602786217034420862124818489, −2.46844559134977890564393720408, −2.43608495234153353703544069169, −2.42557439155520253820831286658, −2.37934078168086316022645722123, −2.02786599958810357895669064796, −2.00707143299679323085460828136, −1.78987209666502174251391932816, −1.56175834493196827513806965033, −1.35841165360969547260480573950, −1.18191539225703612469118377861, −1.14676749598389816483448881821, −1.14357868511710481442022382815, 0, 0, 0, 0, 0, 0, 1.14357868511710481442022382815, 1.14676749598389816483448881821, 1.18191539225703612469118377861, 1.35841165360969547260480573950, 1.56175834493196827513806965033, 1.78987209666502174251391932816, 2.00707143299679323085460828136, 2.02786599958810357895669064796, 2.37934078168086316022645722123, 2.42557439155520253820831286658, 2.43608495234153353703544069169, 2.46844559134977890564393720408, 3.02602786217034420862124818489, 3.03935349662536666867092087927, 3.08813781158215951019043592965, 3.09945970804122573884860295251, 3.45154993010908848864252007163, 3.47543950305502998503998600622, 3.87579090431928980764120734117, 3.92486906127401289644781623735, 4.06921294592902058476246831899, 4.08737443061898819103000287647, 4.16509218884189341552647559557, 4.33124130547277825933337546245, 4.45888000436320847807961721725

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

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