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Properties

Label 12-6270e6-1.1-c1e6-0-0
Degree $12$
Conductor $6.076\times 10^{22}$
Sign $1$
Analytic cond. $1.57495\times 10^{10}$
Root an. cond. $7.07574$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s − 6·3-s + 21·4-s − 6·5-s + 36·6-s − 56·8-s + 21·9-s + 36·10-s − 6·11-s − 126·12-s + 2·13-s + 36·15-s + 126·16-s − 4·17-s − 126·18-s + 6·19-s − 126·20-s + 36·22-s − 4·23-s + 336·24-s + 21·25-s − 12·26-s − 56·27-s + 2·29-s − 216·30-s + 4·31-s − 252·32-s + ⋯
L(s)  = 1  − 4.24·2-s − 3.46·3-s + 21/2·4-s − 2.68·5-s + 14.6·6-s − 19.7·8-s + 7·9-s + 11.3·10-s − 1.80·11-s − 36.3·12-s + 0.554·13-s + 9.29·15-s + 63/2·16-s − 0.970·17-s − 29.6·18-s + 1.37·19-s − 28.1·20-s + 7.67·22-s − 0.834·23-s + 68.5·24-s + 21/5·25-s − 2.35·26-s − 10.7·27-s + 0.371·29-s − 39.4·30-s + 0.718·31-s − 44.5·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 11^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(1.57495\times 10^{10}\)
Root analytic conductor: \(7.07574\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 11^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.01548174228\)
\(L(\frac12)\) \(\approx\) \(0.01548174228\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T )^{6} \)
5 \( ( 1 + T )^{6} \)
11 \( ( 1 + T )^{6} \)
19 \( ( 1 - T )^{6} \)
good7 \( 1 + 12 T^{2} - 6 T^{3} + 75 T^{4} - 62 T^{5} + 464 T^{6} - 62 p T^{7} + 75 p^{2} T^{8} - 6 p^{3} T^{9} + 12 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 - 2 T + 14 T^{2} - 62 T^{3} + 223 T^{4} - 456 T^{5} + 2868 T^{6} - 456 p T^{7} + 223 p^{2} T^{8} - 62 p^{3} T^{9} + 14 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 T + 2 p T^{2} + 232 T^{3} + 991 T^{4} + 244 p T^{5} + 24444 T^{6} + 244 p^{2} T^{7} + 991 p^{2} T^{8} + 232 p^{3} T^{9} + 2 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 4 T + 30 T^{2} + 78 T^{3} + 1047 T^{4} + 4434 T^{5} + 29620 T^{6} + 4434 p T^{7} + 1047 p^{2} T^{8} + 78 p^{3} T^{9} + 30 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 2 T + 74 T^{2} - 60 T^{3} + 3291 T^{4} - 2 T^{5} + 101476 T^{6} - 2 p T^{7} + 3291 p^{2} T^{8} - 60 p^{3} T^{9} + 74 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 4 T + 94 T^{2} - 152 T^{3} + 3463 T^{4} + 2908 T^{5} + 94196 T^{6} + 2908 p T^{7} + 3463 p^{2} T^{8} - 152 p^{3} T^{9} + 94 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 6 T + 2 p T^{2} + 366 T^{3} + 2919 T^{4} + 22980 T^{5} + 141196 T^{6} + 22980 p T^{7} + 2919 p^{2} T^{8} + 366 p^{3} T^{9} + 2 p^{5} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 4 T + 178 T^{2} + 712 T^{3} + 15343 T^{4} + 52052 T^{5} + 798108 T^{6} + 52052 p T^{7} + 15343 p^{2} T^{8} + 712 p^{3} T^{9} + 178 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 138 T^{2} - 48 T^{3} + 9975 T^{4} - 4144 T^{5} + 500396 T^{6} - 4144 p T^{7} + 9975 p^{2} T^{8} - 48 p^{3} T^{9} + 138 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 + 14 T + 194 T^{2} + 1176 T^{3} + 7143 T^{4} + 2066 T^{5} + 24940 T^{6} + 2066 p T^{7} + 7143 p^{2} T^{8} + 1176 p^{3} T^{9} + 194 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 126 T^{2} + 396 T^{3} + 6015 T^{4} + 60260 T^{5} + 227572 T^{6} + 60260 p T^{7} + 6015 p^{2} T^{8} + 396 p^{3} T^{9} + 126 p^{4} T^{10} + p^{6} T^{12} \)
59 \( 1 + 8 T + 306 T^{2} + 1924 T^{3} + 41103 T^{4} + 207324 T^{5} + 3139756 T^{6} + 207324 p T^{7} + 41103 p^{2} T^{8} + 1924 p^{3} T^{9} + 306 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 10 T + 142 T^{2} - 1614 T^{3} + 14079 T^{4} - 110256 T^{5} + 1073268 T^{6} - 110256 p T^{7} + 14079 p^{2} T^{8} - 1614 p^{3} T^{9} + 142 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T + 198 T^{2} + 962 T^{3} + 23895 T^{4} + 97988 T^{5} + 1892180 T^{6} + 97988 p T^{7} + 23895 p^{2} T^{8} + 962 p^{3} T^{9} + 198 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 18 T + 266 T^{2} + 3280 T^{3} + 34647 T^{4} + 315334 T^{5} + 2874780 T^{6} + 315334 p T^{7} + 34647 p^{2} T^{8} + 3280 p^{3} T^{9} + 266 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 2 T + 226 T^{2} + 852 T^{3} + 21787 T^{4} + 128650 T^{5} + 1544100 T^{6} + 128650 p T^{7} + 21787 p^{2} T^{8} + 852 p^{3} T^{9} + 226 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 6 T + 292 T^{2} - 744 T^{3} + 35755 T^{4} - 13482 T^{5} + 3036768 T^{6} - 13482 p T^{7} + 35755 p^{2} T^{8} - 744 p^{3} T^{9} + 292 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 28 T + 688 T^{2} + 10894 T^{3} + 154723 T^{4} + 1716242 T^{5} + 17345640 T^{6} + 1716242 p T^{7} + 154723 p^{2} T^{8} + 10894 p^{3} T^{9} + 688 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 6 T + 352 T^{2} + 2760 T^{3} + 58823 T^{4} + 486914 T^{5} + 6310000 T^{6} + 486914 p T^{7} + 58823 p^{2} T^{8} + 2760 p^{3} T^{9} + 352 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 8 T + 430 T^{2} + 2292 T^{3} + 77943 T^{4} + 290308 T^{5} + 8862260 T^{6} + 290308 p T^{7} + 77943 p^{2} T^{8} + 2292 p^{3} T^{9} + 430 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.29425516787544787621384833886, −3.69417594404908596877586651696, −3.65555738969101328393114188519, −3.64691907162484875680319810513, −3.60281342780219755353919695145, −3.48909576805090513767552076886, −3.35227428145007250777102620581, −2.90066699226022145458674756513, −2.77060983156850717415805605136, −2.72061555580207576840975418627, −2.58677983921679293472107681215, −2.54131921452066419455119866253, −2.51672435351064905576830157868, −1.73616865379689341722631619487, −1.66309608269956713929272346426, −1.64876545105342405782773028757, −1.61735060907610039550379628544, −1.51946422362523121628762332776, −1.33653370694939551540906654299, −0.970862178543351963197604762064, −0.60152415948064867737394383275, −0.58369157732805271908525635048, −0.40515936304219511704377406296, −0.38830305548881813631028457167, −0.12793705972227599469674594090, 0.12793705972227599469674594090, 0.38830305548881813631028457167, 0.40515936304219511704377406296, 0.58369157732805271908525635048, 0.60152415948064867737394383275, 0.970862178543351963197604762064, 1.33653370694939551540906654299, 1.51946422362523121628762332776, 1.61735060907610039550379628544, 1.64876545105342405782773028757, 1.66309608269956713929272346426, 1.73616865379689341722631619487, 2.51672435351064905576830157868, 2.54131921452066419455119866253, 2.58677983921679293472107681215, 2.72061555580207576840975418627, 2.77060983156850717415805605136, 2.90066699226022145458674756513, 3.35227428145007250777102620581, 3.48909576805090513767552076886, 3.60281342780219755353919695145, 3.64691907162484875680319810513, 3.65555738969101328393114188519, 3.69417594404908596877586651696, 4.29425516787544787621384833886

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

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