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Properties

Label 12-3800e6-1.1-c1e6-0-8
Degree $12$
Conductor $3.011\times 10^{21}$
Sign $1$
Analytic cond. $7.80484\times 10^{8}$
Root an. cond. $5.50846$
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  + 2·3-s + 2·7-s − 4·9-s + 3·11-s − 13-s − 14·17-s − 6·19-s + 4·21-s + 12·23-s − 10·27-s + 9·29-s + 5·31-s + 6·33-s − 8·37-s − 2·39-s + 3·41-s + 15·43-s + 4·47-s − 8·49-s − 28·51-s + 13·53-s − 12·57-s + 9·61-s − 8·63-s − 3·67-s + 24·69-s + 19·71-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s − 4/3·9-s + 0.904·11-s − 0.277·13-s − 3.39·17-s − 1.37·19-s + 0.872·21-s + 2.50·23-s − 1.92·27-s + 1.67·29-s + 0.898·31-s + 1.04·33-s − 1.31·37-s − 0.320·39-s + 0.468·41-s + 2.28·43-s + 0.583·47-s − 8/7·49-s − 3.92·51-s + 1.78·53-s − 1.58·57-s + 1.15·61-s − 1.00·63-s − 0.366·67-s + 2.88·69-s + 2.25·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 5^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(7.80484\times 10^{8}\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 5^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.13882491\)
\(L(\frac12)\) \(\approx\) \(14.13882491\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( ( 1 + T )^{6} \)
good3 \( 1 - 2 T + 8 T^{2} - 14 T^{3} + 10 p T^{4} - 50 T^{5} + 29 p T^{6} - 50 p T^{7} + 10 p^{3} T^{8} - 14 p^{3} T^{9} + 8 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 2 T + 12 T^{2} - 22 T^{3} + 118 T^{4} - 18 p T^{5} + 779 T^{6} - 18 p^{2} T^{7} + 118 p^{2} T^{8} - 22 p^{3} T^{9} + 12 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T + 26 T^{2} - 26 T^{3} + 65 T^{4} + 749 T^{5} - 208 p T^{6} + 749 p T^{7} + 65 p^{2} T^{8} - 26 p^{3} T^{9} + 26 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + T + p T^{2} + 50 T^{3} + 38 T^{4} - 516 T^{5} + 1813 T^{6} - 516 p T^{7} + 38 p^{2} T^{8} + 50 p^{3} T^{9} + p^{5} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 14 T + 124 T^{2} + 848 T^{3} + 4842 T^{4} + 24018 T^{5} + 105929 T^{6} + 24018 p T^{7} + 4842 p^{2} T^{8} + 848 p^{3} T^{9} + 124 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 12 T + 112 T^{2} - 778 T^{3} + 4884 T^{4} - 25856 T^{5} + 128015 T^{6} - 25856 p T^{7} + 4884 p^{2} T^{8} - 778 p^{3} T^{9} + 112 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 9 T + 107 T^{2} - 600 T^{3} + 5158 T^{4} - 26640 T^{5} + 189919 T^{6} - 26640 p T^{7} + 5158 p^{2} T^{8} - 600 p^{3} T^{9} + 107 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 5 T + 50 T^{2} - 176 T^{3} + 771 T^{4} - 5843 T^{5} + 22284 T^{6} - 5843 p T^{7} + 771 p^{2} T^{8} - 176 p^{3} T^{9} + 50 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 8 T + 132 T^{2} + 743 T^{3} + 7797 T^{4} + 33802 T^{5} + 309503 T^{6} + 33802 p T^{7} + 7797 p^{2} T^{8} + 743 p^{3} T^{9} + 132 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 3 T + 66 T^{2} - 146 T^{3} + 4087 T^{4} - 5943 T^{5} + 137612 T^{6} - 5943 p T^{7} + 4087 p^{2} T^{8} - 146 p^{3} T^{9} + 66 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 15 T + 271 T^{2} - 2530 T^{3} + 26829 T^{4} - 185919 T^{5} + 1470270 T^{6} - 185919 p T^{7} + 26829 p^{2} T^{8} - 2530 p^{3} T^{9} + 271 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 4 T + 230 T^{2} - 749 T^{3} + 23811 T^{4} - 1352 p T^{5} + 1431531 T^{6} - 1352 p^{2} T^{7} + 23811 p^{2} T^{8} - 749 p^{3} T^{9} + 230 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 13 T + 126 T^{2} - 1554 T^{3} + 12921 T^{4} - 108347 T^{5} + 968797 T^{6} - 108347 p T^{7} + 12921 p^{2} T^{8} - 1554 p^{3} T^{9} + 126 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 161 T^{2} + 255 T^{3} + 12605 T^{4} + 44407 T^{5} + 744938 T^{6} + 44407 p T^{7} + 12605 p^{2} T^{8} + 255 p^{3} T^{9} + 161 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 - 9 T + 234 T^{2} - 2302 T^{3} + 30205 T^{4} - 3881 p T^{5} + 2409968 T^{6} - 3881 p^{2} T^{7} + 30205 p^{2} T^{8} - 2302 p^{3} T^{9} + 234 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 3 T + 203 T^{2} + 820 T^{3} + 23862 T^{4} + 100928 T^{5} + 1840467 T^{6} + 100928 p T^{7} + 23862 p^{2} T^{8} + 820 p^{3} T^{9} + 203 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 19 T + 351 T^{2} - 3254 T^{3} + 31709 T^{4} - 169223 T^{5} + 1652142 T^{6} - 169223 p T^{7} + 31709 p^{2} T^{8} - 3254 p^{3} T^{9} + 351 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 3 T + 133 T^{2} - 274 T^{3} + 18744 T^{4} - 31092 T^{5} + 1356111 T^{6} - 31092 p T^{7} + 18744 p^{2} T^{8} - 274 p^{3} T^{9} + 133 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 16 T + 189 T^{2} + 1815 T^{3} + 19499 T^{4} + 139691 T^{5} + 1154286 T^{6} + 139691 p T^{7} + 19499 p^{2} T^{8} + 1815 p^{3} T^{9} + 189 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 31 T + 824 T^{2} - 13802 T^{3} + 205891 T^{4} - 2327315 T^{5} + 23897248 T^{6} - 2327315 p T^{7} + 205891 p^{2} T^{8} - 13802 p^{3} T^{9} + 824 p^{4} T^{10} - 31 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 14 T + 467 T^{2} - 5503 T^{3} + 93933 T^{4} - 916667 T^{5} + 10735038 T^{6} - 916667 p T^{7} + 93933 p^{2} T^{8} - 5503 p^{3} T^{9} + 467 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 11 T + 374 T^{2} + 3862 T^{3} + 74175 T^{4} + 650639 T^{5} + 8888572 T^{6} + 650639 p T^{7} + 74175 p^{2} T^{8} + 3862 p^{3} T^{9} + 374 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.38910394051291516032485622433, −4.16731602832766746146318495392, −4.13553955410216834044883967172, −3.92612465456304242242161633044, −3.76237981341429066971916659832, −3.68736162001612522648546472761, −3.64822751435999229273070736384, −3.24856094615183467759307554723, −3.06211752652064591989601183392, −2.99314187427319959733149592097, −2.81805441937401220809702193758, −2.71447115400875784857741886769, −2.69761580071139637903617648115, −2.49775783671414909583897896506, −2.13585591453109941088560000511, −2.10241663233512773714292129046, −2.09438419404373246788619638319, −1.81764494964424915577628759079, −1.81450089913514348343164485599, −1.35375753140645687279696808282, −1.09070139763871223489935896851, −0.856980305515190160249224023033, −0.64836615162937481688212384811, −0.53123580245396223865020900895, −0.35630216519218056329676458585, 0.35630216519218056329676458585, 0.53123580245396223865020900895, 0.64836615162937481688212384811, 0.856980305515190160249224023033, 1.09070139763871223489935896851, 1.35375753140645687279696808282, 1.81450089913514348343164485599, 1.81764494964424915577628759079, 2.09438419404373246788619638319, 2.10241663233512773714292129046, 2.13585591453109941088560000511, 2.49775783671414909583897896506, 2.69761580071139637903617648115, 2.71447115400875784857741886769, 2.81805441937401220809702193758, 2.99314187427319959733149592097, 3.06211752652064591989601183392, 3.24856094615183467759307554723, 3.64822751435999229273070736384, 3.68736162001612522648546472761, 3.76237981341429066971916659832, 3.92612465456304242242161633044, 4.13553955410216834044883967172, 4.16731602832766746146318495392, 4.38910394051291516032485622433

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

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