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Properties

Label 12-950e6-1.1-c1e6-0-4
Degree $12$
Conductor $7.351\times 10^{17}$
Sign $1$
Analytic cond. $190547.$
Root an. cond. $2.75423$
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 − 8-s + 21·9-s + 3·11-s − 15·13-s − 6·17-s + 6·19-s − 15·23-s + 6·24-s − 55·27-s + 24·29-s − 6·31-s − 18·33-s + 30·37-s + 90·39-s + 18·41-s + 3·43-s + 3·47-s + 18·49-s + 36·51-s − 12·53-s − 36·57-s + 21·59-s − 21·61-s + 9·67-s + 90·69-s + 15·71-s + ⋯
L(s)  = 1  − 3.46·3-s − 0.353·8-s + 7·9-s + 0.904·11-s − 4.16·13-s − 1.45·17-s + 1.37·19-s − 3.12·23-s + 1.22·24-s − 10.5·27-s + 4.45·29-s − 1.07·31-s − 3.13·33-s + 4.93·37-s + 14.4·39-s + 2.81·41-s + 0.457·43-s + 0.437·47-s + 18/7·49-s + 5.04·51-s − 1.64·53-s − 4.76·57-s + 2.73·59-s − 2.68·61-s + 1.09·67-s + 10.8·69-s + 1.78·71-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(190547.\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 5^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5751010215\)
\(L(\frac12)\) \(\approx\) \(0.5751010215\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{3} + T^{6} \)
5 \( 1 \)
19 \( 1 - 6 T - 12 T^{2} + 169 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
good3 \( 1 + 2 p T + 5 p T^{2} + 19 T^{3} + p T^{4} - 17 p T^{5} - 134 T^{6} - 17 p^{2} T^{7} + p^{3} T^{8} + 19 p^{3} T^{9} + 5 p^{5} T^{10} + 2 p^{6} T^{11} + p^{6} T^{12} \)
7 \( 1 - 18 T^{2} + 2 T^{3} + 198 T^{4} - 18 T^{5} - 1581 T^{6} - 18 p T^{7} + 198 p^{2} T^{8} + 2 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 3 T - 6 T^{2} + 81 T^{3} - 129 T^{4} - 318 T^{5} + 3067 T^{6} - 318 p T^{7} - 129 p^{2} T^{8} + 81 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 15 T + 120 T^{2} + 698 T^{3} + 3303 T^{4} + 13401 T^{5} + 49605 T^{6} + 13401 p T^{7} + 3303 p^{2} T^{8} + 698 p^{3} T^{9} + 120 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 6 T + 27 T^{2} + 207 T^{3} + 801 T^{4} + 219 p T^{5} + 1286 p T^{6} + 219 p^{2} T^{7} + 801 p^{2} T^{8} + 207 p^{3} T^{9} + 27 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 15 T + 90 T^{2} + 162 T^{3} - 1017 T^{4} - 10101 T^{5} - 54485 T^{6} - 10101 p T^{7} - 1017 p^{2} T^{8} + 162 p^{3} T^{9} + 90 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 24 T + 252 T^{2} - 1494 T^{3} + 4356 T^{4} + 8364 T^{5} - 131345 T^{6} + 8364 p T^{7} + 4356 p^{2} T^{8} - 1494 p^{3} T^{9} + 252 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 6 T - 42 T^{2} - 130 T^{3} + 1872 T^{4} + 72 T^{5} - 74217 T^{6} + 72 p T^{7} + 1872 p^{2} T^{8} - 130 p^{3} T^{9} - 42 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 15 T + 150 T^{2} - 983 T^{3} + 150 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 18 T + 144 T^{2} - 720 T^{3} + 4068 T^{4} - 31752 T^{5} + 224083 T^{6} - 31752 p T^{7} + 4068 p^{2} T^{8} - 720 p^{3} T^{9} + 144 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T + 3 T^{2} - 121 T^{3} - 252 T^{4} + 10710 T^{5} - 79635 T^{6} + 10710 p T^{7} - 252 p^{2} T^{8} - 121 p^{3} T^{9} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 3 T + 36 T^{2} - 648 T^{3} + 3330 T^{4} - 19857 T^{5} + 286237 T^{6} - 19857 p T^{7} + 3330 p^{2} T^{8} - 648 p^{3} T^{9} + 36 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 12 T + 54 T^{2} - 27 T^{3} - 171 T^{4} + 18363 T^{5} + 156889 T^{6} + 18363 p T^{7} - 171 p^{2} T^{8} - 27 p^{3} T^{9} + 54 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 21 T + 81 T^{2} + 1647 T^{3} - 17208 T^{4} - 28362 T^{5} + 1122445 T^{6} - 28362 p T^{7} - 17208 p^{2} T^{8} + 1647 p^{3} T^{9} + 81 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 21 T + 348 T^{2} + 4526 T^{3} + 50787 T^{4} + 478683 T^{5} + 3989277 T^{6} + 478683 p T^{7} + 50787 p^{2} T^{8} + 4526 p^{3} T^{9} + 348 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 9 T - 18 T^{2} + 956 T^{3} - 6723 T^{4} - 42849 T^{5} + 795909 T^{6} - 42849 p T^{7} - 6723 p^{2} T^{8} + 956 p^{3} T^{9} - 18 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 15 T + 252 T^{2} - 2682 T^{3} + 27171 T^{4} - 243771 T^{5} + 1964377 T^{6} - 243771 p T^{7} + 27171 p^{2} T^{8} - 2682 p^{3} T^{9} + 252 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 21 T + 183 T^{2} - 985 T^{3} + 1242 T^{4} + 12420 T^{5} + 19041 T^{6} + 12420 p T^{7} + 1242 p^{2} T^{8} - 985 p^{3} T^{9} + 183 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 236 T^{3} - 437343 T^{6} + 236 p^{3} T^{9} + p^{6} T^{12} \)
83 \( 1 + 3 T - 186 T^{2} - 513 T^{3} + 20193 T^{4} + 30612 T^{5} - 1773101 T^{6} + 30612 p T^{7} + 20193 p^{2} T^{8} - 513 p^{3} T^{9} - 186 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 24 T + 234 T^{2} + 261 T^{3} - 11691 T^{4} - 1437 p T^{5} - 9991 p T^{6} - 1437 p^{2} T^{7} - 11691 p^{2} T^{8} + 261 p^{3} T^{9} + 234 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 9 T + 306 T^{2} - 1330 T^{3} + 39717 T^{4} - 49707 T^{5} + 3717309 T^{6} - 49707 p T^{7} + 39717 p^{2} T^{8} - 1330 p^{3} T^{9} + 306 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.44427710331010282534582682908, −4.91974915019523378491520520755, −4.88884332359401057420411507677, −4.82520034040204097469775237405, −4.72235019019138543758878661699, −4.67732367274961209611686945715, −4.66292938715104482267434864567, −4.30641788396469775418394665661, −4.07378079074184803716039615344, −3.87862405086338231314709236589, −3.79887191859816050695133503353, −3.72765913070352164911717821225, −3.48396929562351125461412620185, −2.60078925963567896361863843603, −2.59999006526048220180331402134, −2.58852272060552552503421653704, −2.42964512276168824677715903279, −2.41262480930829977642039064246, −2.37204227994717184539877770402, −1.49267974661624035835195918239, −1.37766273060003358992027104895, −1.03598318037926179044281777944, −0.912251847920778446333640195242, −0.40969699519537779752012905334, −0.36337090286967728143408864980, 0.36337090286967728143408864980, 0.40969699519537779752012905334, 0.912251847920778446333640195242, 1.03598318037926179044281777944, 1.37766273060003358992027104895, 1.49267974661624035835195918239, 2.37204227994717184539877770402, 2.41262480930829977642039064246, 2.42964512276168824677715903279, 2.58852272060552552503421653704, 2.59999006526048220180331402134, 2.60078925963567896361863843603, 3.48396929562351125461412620185, 3.72765913070352164911717821225, 3.79887191859816050695133503353, 3.87862405086338231314709236589, 4.07378079074184803716039615344, 4.30641788396469775418394665661, 4.66292938715104482267434864567, 4.67732367274961209611686945715, 4.72235019019138543758878661699, 4.82520034040204097469775237405, 4.88884332359401057420411507677, 4.91974915019523378491520520755, 5.44427710331010282534582682908

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

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