Properties

Label 12-7098e6-1.1-c1e6-0-4
Degree $12$
Conductor $1.279\times 10^{23}$
Sign $1$
Analytic cond. $3.31496\times 10^{10}$
Root an. cond. $7.52846$
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 + 6·3-s + 21·4-s − 3·5-s − 36·6-s − 6·7-s − 56·8-s + 21·9-s + 18·10-s − 10·11-s + 126·12-s + 36·14-s − 18·15-s + 126·16-s + 2·17-s − 126·18-s − 2·19-s − 63·20-s − 36·21-s + 60·22-s + 4·23-s − 336·24-s − 4·25-s + 56·27-s − 126·28-s + 2·29-s + 108·30-s + ⋯
L(s)  = 1  − 4.24·2-s + 3.46·3-s + 21/2·4-s − 1.34·5-s − 14.6·6-s − 2.26·7-s − 19.7·8-s + 7·9-s + 5.69·10-s − 3.01·11-s + 36.3·12-s + 9.62·14-s − 4.64·15-s + 63/2·16-s + 0.485·17-s − 29.6·18-s − 0.458·19-s − 14.0·20-s − 7.85·21-s + 12.7·22-s + 0.834·23-s − 68.5·24-s − 4/5·25-s + 10.7·27-s − 23.8·28-s + 0.371·29-s + 19.7·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(3.31496\times 10^{10}\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7098} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( ( 1 + T )^{6} \)
3 \( ( 1 - T )^{6} \)
7 \( ( 1 + T )^{6} \)
13 \( 1 \)
good5 \( 1 + 3 T + 13 T^{2} + 36 T^{3} + 146 T^{4} + 296 T^{5} + 779 T^{6} + 296 p T^{7} + 146 p^{2} T^{8} + 36 p^{3} T^{9} + 13 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( ( 1 + 5 T + 39 T^{2} + 111 T^{3} + 39 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( ( 1 - T + 49 T^{2} - 33 T^{3} + 49 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 + 2 T + 73 T^{2} + 180 T^{3} + 2547 T^{4} + 6724 T^{5} + 57517 T^{6} + 6724 p T^{7} + 2547 p^{2} T^{8} + 180 p^{3} T^{9} + 73 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 4 T + 49 T^{2} - 258 T^{3} + 1767 T^{4} - 9730 T^{5} + 43695 T^{6} - 9730 p T^{7} + 1767 p^{2} T^{8} - 258 p^{3} T^{9} + 49 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 2 T + 73 T^{2} + 73 T^{3} + 2455 T^{4} + 10025 T^{5} + 62174 T^{6} + 10025 p T^{7} + 2455 p^{2} T^{8} + 73 p^{3} T^{9} + 73 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 9 T + 185 T^{2} + 1254 T^{3} + 14116 T^{4} + 73902 T^{5} + 580079 T^{6} + 73902 p T^{7} + 14116 p^{2} T^{8} + 1254 p^{3} T^{9} + 185 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 7 T + 119 T^{2} + 728 T^{3} + 8608 T^{4} + 42644 T^{5} + 376579 T^{6} + 42644 p T^{7} + 8608 p^{2} T^{8} + 728 p^{3} T^{9} + 119 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 11 T + 145 T^{2} + 1356 T^{3} + 11450 T^{4} + 2056 p T^{5} + 584365 T^{6} + 2056 p^{2} T^{7} + 11450 p^{2} T^{8} + 1356 p^{3} T^{9} + 145 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 5 T + 182 T^{2} + 644 T^{3} + 15513 T^{4} + 44827 T^{5} + 831288 T^{6} + 44827 p T^{7} + 15513 p^{2} T^{8} + 644 p^{3} T^{9} + 182 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 5 T + 119 T^{2} + 146 T^{3} + 7653 T^{4} + 14857 T^{5} + 492910 T^{6} + 14857 p T^{7} + 7653 p^{2} T^{8} + 146 p^{3} T^{9} + 119 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 6 T + 193 T^{2} + 945 T^{3} + 18905 T^{4} + 81825 T^{5} + 1231866 T^{6} + 81825 p T^{7} + 18905 p^{2} T^{8} + 945 p^{3} T^{9} + 193 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 28 T + 543 T^{2} + 7623 T^{3} + 89175 T^{4} + 862911 T^{5} + 7223994 T^{6} + 862911 p T^{7} + 89175 p^{2} T^{8} + 7623 p^{3} T^{9} + 543 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 23 T + 395 T^{2} - 4660 T^{3} + 48593 T^{4} - 420753 T^{5} + 3506886 T^{6} - 420753 p T^{7} + 48593 p^{2} T^{8} - 4660 p^{3} T^{9} + 395 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 10 T + 229 T^{2} - 1109 T^{3} + 14913 T^{4} + 4945 T^{5} + 545594 T^{6} + 4945 p T^{7} + 14913 p^{2} T^{8} - 1109 p^{3} T^{9} + 229 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 21 T + 458 T^{2} + 6314 T^{3} + 1175 p T^{4} + 832349 T^{5} + 7925792 T^{6} + 832349 p T^{7} + 1175 p^{3} T^{8} + 6314 p^{3} T^{9} + 458 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 7 T + 358 T^{2} - 1904 T^{3} + 55983 T^{4} - 230489 T^{5} + 5136092 T^{6} - 230489 p T^{7} + 55983 p^{2} T^{8} - 1904 p^{3} T^{9} + 358 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 14 T + 233 T^{2} + 3269 T^{3} + 36243 T^{4} + 360367 T^{5} + 3725982 T^{6} + 360367 p T^{7} + 36243 p^{2} T^{8} + 3269 p^{3} T^{9} + 233 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 17 T + 460 T^{2} + 6386 T^{3} + 91789 T^{4} + 1009741 T^{5} + 10035044 T^{6} + 1009741 p T^{7} + 91789 p^{2} T^{8} + 6386 p^{3} T^{9} + 460 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 17 T + 461 T^{2} + 5258 T^{3} + 87618 T^{4} + 785710 T^{5} + 9903763 T^{6} + 785710 p T^{7} + 87618 p^{2} T^{8} + 5258 p^{3} T^{9} + 461 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 297 T^{2} + 119 T^{3} + 54179 T^{4} + 18697 T^{5} + 6260030 T^{6} + 18697 p T^{7} + 54179 p^{2} T^{8} + 119 p^{3} T^{9} + 297 p^{4} T^{10} + p^{6} T^{12} \)
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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

−4.43496149250196233737291559771, −3.89329223800587564592655483534, −3.88370001566702896396144556471, −3.85558359062283771052041389534, −3.63884544792054286722307415143, −3.61828844920933141606611597551, −3.58646285925628898942347596251, −3.28369404549406216232662574420, −3.09603950672117069465681999398, −3.08635063210991462563996116440, −3.02624309303735036376971597378, −2.97955454978843168009570220574, −2.74837673376280236409416220452, −2.44620484750367778742196299500, −2.33805981068471164879952558858, −2.31695117112569373924307533165, −2.30260493663583147737363002950, −2.13193394247118709772878554531, −2.04193524717527508343077136088, −1.50532166815047045812582980333, −1.48407580225627986325243838214, −1.41863045474004995985773967114, −1.10232727334361255550279761249, −1.08188952923419062436962184983, −1.02631367090756010867941940444, 0, 0, 0, 0, 0, 0, 1.02631367090756010867941940444, 1.08188952923419062436962184983, 1.10232727334361255550279761249, 1.41863045474004995985773967114, 1.48407580225627986325243838214, 1.50532166815047045812582980333, 2.04193524717527508343077136088, 2.13193394247118709772878554531, 2.30260493663583147737363002950, 2.31695117112569373924307533165, 2.33805981068471164879952558858, 2.44620484750367778742196299500, 2.74837673376280236409416220452, 2.97955454978843168009570220574, 3.02624309303735036376971597378, 3.08635063210991462563996116440, 3.09603950672117069465681999398, 3.28369404549406216232662574420, 3.58646285925628898942347596251, 3.61828844920933141606611597551, 3.63884544792054286722307415143, 3.85558359062283771052041389534, 3.88370001566702896396144556471, 3.89329223800587564592655483534, 4.43496149250196233737291559771

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

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