Properties

Label 12-8470e6-1.1-c1e6-0-6
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s − 3-s + 21·4-s + 6·5-s − 6·6-s − 6·7-s + 56·8-s − 7·9-s + 36·10-s − 21·12-s + 9·13-s − 36·14-s − 6·15-s + 126·16-s + 9·17-s − 42·18-s + 12·19-s + 126·20-s + 6·21-s + 4·23-s − 56·24-s + 21·25-s + 54·26-s + 7·27-s − 126·28-s + 15·29-s − 36·30-s + ⋯
L(s)  = 1  + 4.24·2-s − 0.577·3-s + 21/2·4-s + 2.68·5-s − 2.44·6-s − 2.26·7-s + 19.7·8-s − 7/3·9-s + 11.3·10-s − 6.06·12-s + 2.49·13-s − 9.62·14-s − 1.54·15-s + 63/2·16-s + 2.18·17-s − 9.89·18-s + 2.75·19-s + 28.1·20-s + 1.30·21-s + 0.834·23-s − 11.4·24-s + 21/5·25-s + 10.5·26-s + 1.34·27-s − 23.8·28-s + 2.78·29-s − 6.57·30-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: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1529.312217\)
\(L(\frac12)\) \(\approx\) \(1529.312217\)
\(L(\frac{3}{2})\) 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 \( 1 + T + 8 T^{2} + 8 T^{3} + 13 p T^{4} + 14 p T^{5} + 139 T^{6} + 14 p^{2} T^{7} + 13 p^{3} T^{8} + 8 p^{3} T^{9} + 8 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 9 T + 93 T^{2} - 541 T^{3} + 3177 T^{4} - 13406 T^{5} + 55566 T^{6} - 13406 p T^{7} + 3177 p^{2} T^{8} - 541 p^{3} T^{9} + 93 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 9 T + 52 T^{2} - 184 T^{3} + 893 T^{4} - 4800 T^{5} + 24563 T^{6} - 4800 p T^{7} + 893 p^{2} T^{8} - 184 p^{3} T^{9} + 52 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 12 T + 137 T^{2} - 996 T^{3} + 6528 T^{4} - 34266 T^{5} + 162512 T^{6} - 34266 p T^{7} + 6528 p^{2} T^{8} - 996 p^{3} T^{9} + 137 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 4 T + 2 p T^{2} - 300 T^{3} + 1759 T^{4} - 7624 T^{5} + 53956 T^{6} - 7624 p T^{7} + 1759 p^{2} T^{8} - 300 p^{3} T^{9} + 2 p^{5} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 15 T + 209 T^{2} - 1923 T^{3} + 16085 T^{4} - 105266 T^{5} + 631966 T^{6} - 105266 p T^{7} + 16085 p^{2} T^{8} - 1923 p^{3} T^{9} + 209 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 8 T + 164 T^{2} - 928 T^{3} + 10963 T^{4} - 47424 T^{5} + 424160 T^{6} - 47424 p T^{7} + 10963 p^{2} T^{8} - 928 p^{3} T^{9} + 164 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 4 T + 160 T^{2} + 356 T^{3} + 11179 T^{4} + 14192 T^{5} + 492888 T^{6} + 14192 p T^{7} + 11179 p^{2} T^{8} + 356 p^{3} T^{9} + 160 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 4 T + 99 T^{2} - 830 T^{3} + 7420 T^{4} - 44674 T^{5} + 436500 T^{6} - 44674 p T^{7} + 7420 p^{2} T^{8} - 830 p^{3} T^{9} + 99 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 30 T + 513 T^{2} - 6248 T^{3} + 60360 T^{4} - 486492 T^{5} + 3407896 T^{6} - 486492 p T^{7} + 60360 p^{2} T^{8} - 6248 p^{3} T^{9} + 513 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 7 T + 243 T^{2} + 1385 T^{3} + 26383 T^{4} + 120370 T^{5} + 1609674 T^{6} + 120370 p T^{7} + 26383 p^{2} T^{8} + 1385 p^{3} T^{9} + 243 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 6 T + 188 T^{2} + 1522 T^{3} + 16811 T^{4} + 158560 T^{5} + 1022320 T^{6} + 158560 p T^{7} + 16811 p^{2} T^{8} + 1522 p^{3} T^{9} + 188 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 4 T + 209 T^{2} - 1104 T^{3} + 18708 T^{4} - 126174 T^{5} + 1162960 T^{6} - 126174 p T^{7} + 18708 p^{2} T^{8} - 1104 p^{3} T^{9} + 209 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 14 T + 262 T^{2} + 2678 T^{3} + 30483 T^{4} + 253628 T^{5} + 2216668 T^{6} + 253628 p T^{7} + 30483 p^{2} T^{8} + 2678 p^{3} T^{9} + 262 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 18 T + 417 T^{2} - 5348 T^{3} + 69572 T^{4} - 671932 T^{5} + 6180304 T^{6} - 671932 p T^{7} + 69572 p^{2} T^{8} - 5348 p^{3} T^{9} + 417 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 23 T + 493 T^{2} - 6365 T^{3} + 80081 T^{4} - 748358 T^{5} + 7129622 T^{6} - 748358 p T^{7} + 80081 p^{2} T^{8} - 6365 p^{3} T^{9} + 493 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 23 T + 608 T^{2} - 8876 T^{3} + 130169 T^{4} - 1331324 T^{5} + 13302699 T^{6} - 1331324 p T^{7} + 130169 p^{2} T^{8} - 8876 p^{3} T^{9} + 608 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 21 T + 477 T^{2} - 5959 T^{3} + 80833 T^{4} - 748962 T^{5} + 7830134 T^{6} - 748962 p T^{7} + 80833 p^{2} T^{8} - 5959 p^{3} T^{9} + 477 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 25 T + 576 T^{2} - 7832 T^{3} + 105627 T^{4} - 1043326 T^{5} + 10821879 T^{6} - 1043326 p T^{7} + 105627 p^{2} T^{8} - 7832 p^{3} T^{9} + 576 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 18 T + 499 T^{2} + 7566 T^{3} + 106924 T^{4} + 1309542 T^{5} + 12534832 T^{6} + 1309542 p T^{7} + 106924 p^{2} T^{8} + 7566 p^{3} T^{9} + 499 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 7 T + 232 T^{2} - 996 T^{3} + 21281 T^{4} - 3500 T^{5} + 1498955 T^{6} - 3500 p T^{7} + 21281 p^{2} T^{8} - 996 p^{3} T^{9} + 232 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
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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

−3.95932956884989273587366496775, −3.57403929389400225383922567404, −3.55867053297651267761807556094, −3.45217129864692571215120697000, −3.42403649967952192143118498615, −3.41886188988989925864085183041, −3.34337134544425258835176913886, −2.89832921033912134538241675131, −2.86740858107844579317177656622, −2.84553053769287833215163476179, −2.77541316228050721768010033430, −2.74231893219376353227023579841, −2.65862163912308979507733403485, −2.16967715942033118624295863563, −2.07222412732177558112603069991, −2.03638606984141410528219471478, −1.98931834315529945869751247918, −1.76474703321371671886084958456, −1.38695739158460627925737925942, −1.07834144182943712704518467806, −1.03338901485360479204673780015, −0.897657777852223124215823536650, −0.77828356348591121645581550133, −0.68861058607790811979049841117, −0.65704080240809037703148630982, 0.65704080240809037703148630982, 0.68861058607790811979049841117, 0.77828356348591121645581550133, 0.897657777852223124215823536650, 1.03338901485360479204673780015, 1.07834144182943712704518467806, 1.38695739158460627925737925942, 1.76474703321371671886084958456, 1.98931834315529945869751247918, 2.03638606984141410528219471478, 2.07222412732177558112603069991, 2.16967715942033118624295863563, 2.65862163912308979507733403485, 2.74231893219376353227023579841, 2.77541316228050721768010033430, 2.84553053769287833215163476179, 2.86740858107844579317177656622, 2.89832921033912134538241675131, 3.34337134544425258835176913886, 3.41886188988989925864085183041, 3.42403649967952192143118498615, 3.45217129864692571215120697000, 3.55867053297651267761807556094, 3.57403929389400225383922567404, 3.95932956884989273587366496775

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

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