Properties

Label 12-7728e6-1.1-c1e6-0-1
Degree $12$
Conductor $2.130\times 10^{23}$
Sign $1$
Analytic cond. $5.52160\times 10^{10}$
Root an. cond. $7.85546$
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  + 6·3-s + 2·5-s + 6·7-s + 21·9-s + 3·11-s + 12·15-s + 6·17-s + 3·19-s + 36·21-s + 6·23-s − 8·25-s + 56·27-s + 5·29-s + 4·31-s + 18·33-s + 12·35-s + 37-s + 12·41-s + 6·43-s + 42·45-s + 6·47-s + 21·49-s + 36·51-s + 10·53-s + 6·55-s + 18·57-s + 14·59-s + ⋯
L(s)  = 1  + 3.46·3-s + 0.894·5-s + 2.26·7-s + 7·9-s + 0.904·11-s + 3.09·15-s + 1.45·17-s + 0.688·19-s + 7.85·21-s + 1.25·23-s − 8/5·25-s + 10.7·27-s + 0.928·29-s + 0.718·31-s + 3.13·33-s + 2.02·35-s + 0.164·37-s + 1.87·41-s + 0.914·43-s + 6.26·45-s + 0.875·47-s + 3·49-s + 5.04·51-s + 1.37·53-s + 0.809·55-s + 2.38·57-s + 1.82·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(5.52160\times 10^{10}\)
Root analytic conductor: \(7.85546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(360.5038128\)
\(L(\frac12)\) \(\approx\) \(360.5038128\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T )^{6} \)
7 \( ( 1 - T )^{6} \)
23 \( ( 1 - T )^{6} \)
good5 \( 1 - 2 T + 12 T^{2} - 33 T^{3} + 107 T^{4} - 233 T^{5} + 704 T^{6} - 233 p T^{7} + 107 p^{2} T^{8} - 33 p^{3} T^{9} + 12 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T + 25 T^{2} - 90 T^{3} + 571 T^{4} - 1571 T^{5} + 6806 T^{6} - 1571 p T^{7} + 571 p^{2} T^{8} - 90 p^{3} T^{9} + 25 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 12 T^{2} - 11 T^{3} + 197 T^{4} - 337 T^{5} + 3196 T^{6} - 337 p T^{7} + 197 p^{2} T^{8} - 11 p^{3} T^{9} + 12 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 6 T + 39 T^{2} - 226 T^{3} + 1303 T^{4} - 6056 T^{5} + 24418 T^{6} - 6056 p T^{7} + 1303 p^{2} T^{8} - 226 p^{3} T^{9} + 39 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T + 73 T^{2} - 210 T^{3} + 2859 T^{4} - 6971 T^{5} + 67286 T^{6} - 6971 p T^{7} + 2859 p^{2} T^{8} - 210 p^{3} T^{9} + 73 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 5 T + 99 T^{2} - 544 T^{3} + 5289 T^{4} - 27923 T^{5} + 6318 p T^{6} - 27923 p T^{7} + 5289 p^{2} T^{8} - 544 p^{3} T^{9} + 99 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 4 T + 49 T^{2} - 8 T^{3} + 415 T^{4} + 10892 T^{5} - 20898 T^{6} + 10892 p T^{7} + 415 p^{2} T^{8} - 8 p^{3} T^{9} + 49 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - T + 159 T^{2} - 28 T^{3} + 11545 T^{4} + 2949 T^{5} + 14054 p T^{6} + 2949 p T^{7} + 11545 p^{2} T^{8} - 28 p^{3} T^{9} + 159 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 12 T + 132 T^{2} - 652 T^{3} + 3972 T^{4} - 8476 T^{5} + 87366 T^{6} - 8476 p T^{7} + 3972 p^{2} T^{8} - 652 p^{3} T^{9} + 132 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 6 T + 128 T^{2} - 571 T^{3} + 8211 T^{4} - 26493 T^{5} + 385032 T^{6} - 26493 p T^{7} + 8211 p^{2} T^{8} - 571 p^{3} T^{9} + 128 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 6 T + 209 T^{2} - 800 T^{3} + 18535 T^{4} - 48802 T^{5} + 1031438 T^{6} - 48802 p T^{7} + 18535 p^{2} T^{8} - 800 p^{3} T^{9} + 209 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 10 T + 256 T^{2} - 37 p T^{3} + 27047 T^{4} - 169209 T^{5} + 1725728 T^{6} - 169209 p T^{7} + 27047 p^{2} T^{8} - 37 p^{4} T^{9} + 256 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 14 T + 298 T^{2} - 3351 T^{3} + 37621 T^{4} - 351297 T^{5} + 2779616 T^{6} - 351297 p T^{7} + 37621 p^{2} T^{8} - 3351 p^{3} T^{9} + 298 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 4 T + 272 T^{2} - 953 T^{3} + 35101 T^{4} - 104871 T^{5} + 2702852 T^{6} - 104871 p T^{7} + 35101 p^{2} T^{8} - 953 p^{3} T^{9} + 272 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 7 T + 213 T^{2} + 1641 T^{3} + 28203 T^{4} + 178550 T^{5} + 2328126 T^{6} + 178550 p T^{7} + 28203 p^{2} T^{8} + 1641 p^{3} T^{9} + 213 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 7 T + 255 T^{2} + 1599 T^{3} + 34771 T^{4} + 192504 T^{5} + 3014298 T^{6} + 192504 p T^{7} + 34771 p^{2} T^{8} + 1599 p^{3} T^{9} + 255 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 10 T + 235 T^{2} - 946 T^{3} + 18759 T^{4} + 2748 T^{5} + 1094522 T^{6} + 2748 p T^{7} + 18759 p^{2} T^{8} - 946 p^{3} T^{9} + 235 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 14 T + 437 T^{2} + 4660 T^{3} + 81019 T^{4} + 669726 T^{5} + 8334494 T^{6} + 669726 p T^{7} + 81019 p^{2} T^{8} + 4660 p^{3} T^{9} + 437 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 14 T + 277 T^{2} + 2124 T^{3} + 31819 T^{4} + 202230 T^{5} + 2904830 T^{6} + 202230 p T^{7} + 31819 p^{2} T^{8} + 2124 p^{3} T^{9} + 277 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 25 T + 541 T^{2} - 7201 T^{3} + 90491 T^{4} - 869630 T^{5} + 8970830 T^{6} - 869630 p T^{7} + 90491 p^{2} T^{8} - 7201 p^{3} T^{9} + 541 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 11 T + 355 T^{2} - 1540 T^{3} + 38305 T^{4} + 70111 T^{5} + 2600022 T^{6} + 70111 p T^{7} + 38305 p^{2} T^{8} - 1540 p^{3} T^{9} + 355 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.08045016674930086524201433496, −3.65351657022409757064871374440, −3.56071387109352125623172186186, −3.55562003804754493785107543530, −3.50192685545371827038647299079, −3.47188849760518258688070294402, −3.25504298334978458034956427917, −2.92726095925159181928255394494, −2.75299072604980445573256052700, −2.72094837910258982318021663860, −2.63039496883098828518661489211, −2.42584800887570587652722268481, −2.42055117258442146790351930221, −2.22959391727995647894241412268, −2.09487490023616780842556930740, −1.78659491045548428330746173993, −1.73222407040235872236063795864, −1.70263354059082544307214306915, −1.58883348292688686936527117726, −1.26941385472485703981730464892, −0.966685214681881007894364630013, −0.908682071045697644274408534152, −0.902678039315383940439711508014, −0.792833659183510084243003157604, −0.41908913781131534970223846344, 0.41908913781131534970223846344, 0.792833659183510084243003157604, 0.902678039315383940439711508014, 0.908682071045697644274408534152, 0.966685214681881007894364630013, 1.26941385472485703981730464892, 1.58883348292688686936527117726, 1.70263354059082544307214306915, 1.73222407040235872236063795864, 1.78659491045548428330746173993, 2.09487490023616780842556930740, 2.22959391727995647894241412268, 2.42055117258442146790351930221, 2.42584800887570587652722268481, 2.63039496883098828518661489211, 2.72094837910258982318021663860, 2.75299072604980445573256052700, 2.92726095925159181928255394494, 3.25504298334978458034956427917, 3.47188849760518258688070294402, 3.50192685545371827038647299079, 3.55562003804754493785107543530, 3.56071387109352125623172186186, 3.65351657022409757064871374440, 4.08045016674930086524201433496

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

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