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Properties

Label 12-670e6-1.1-c1e6-0-1
Degree $12$
Conductor $9.046\times 10^{16}$
Sign $1$
Analytic cond. $23448.3$
Root an. cond. $2.31300$
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  + 3·2-s + 2·3-s + 3·4-s + 6·5-s + 6·6-s + 2·7-s − 2·8-s − 9-s + 18·10-s − 12·11-s + 6·12-s + 2·13-s + 6·14-s + 12·15-s − 9·16-s − 8·17-s − 3·18-s − 10·19-s + 18·20-s + 4·21-s − 36·22-s + 8·23-s − 4·24-s + 21·25-s + 6·26-s + 6·27-s + 6·28-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.15·3-s + 3/2·4-s + 2.68·5-s + 2.44·6-s + 0.755·7-s − 0.707·8-s − 1/3·9-s + 5.69·10-s − 3.61·11-s + 1.73·12-s + 0.554·13-s + 1.60·14-s + 3.09·15-s − 9/4·16-s − 1.94·17-s − 0.707·18-s − 2.29·19-s + 4.02·20-s + 0.872·21-s − 7.67·22-s + 1.66·23-s − 0.816·24-s + 21/5·25-s + 1.17·26-s + 1.15·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 67^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 67^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 67^{6}\)
Sign: $1$
Analytic conductor: \(23448.3\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 67^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(16.75787374\)
\(L(\frac12)\) \(\approx\) \(16.75787374\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{3} \)
5 \( ( 1 - T )^{6} \)
67 \( 1 + 15 T - 30 T^{2} - 1447 T^{3} - 30 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
good3 \( ( 1 - T + 2 T^{2} - 7 T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
7 \( 1 - 2 T - 11 T^{2} + 10 T^{3} + 80 T^{4} + 22 T^{5} - 685 T^{6} + 22 p T^{7} + 80 p^{2} T^{8} + 10 p^{3} T^{9} - 11 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( ( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{3} \)
13 \( 1 - 2 T - 19 T^{2} + 2 p T^{3} + 158 T^{4} + 30 T^{5} - 1863 T^{6} + 30 p T^{7} + 158 p^{2} T^{8} + 2 p^{4} T^{9} - 19 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 8 T + 7 T^{2} + 456 T^{4} - 196 T^{5} - 11919 T^{6} - 196 p T^{7} + 456 p^{2} T^{8} + 7 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 10 T + 71 T^{2} + 218 T^{3} - 70 T^{4} - 6366 T^{5} - 37065 T^{6} - 6366 p T^{7} - 70 p^{2} T^{8} + 218 p^{3} T^{9} + 71 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 8 T - 19 T^{2} + 64 T^{3} + 2068 T^{4} - 3900 T^{5} - 35473 T^{6} - 3900 p T^{7} + 2068 p^{2} T^{8} + 64 p^{3} T^{9} - 19 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 2 T - 15 T^{2} + 178 T^{3} - 14 p T^{4} - 1530 T^{5} + 46609 T^{6} - 1530 p T^{7} - 14 p^{3} T^{8} + 178 p^{3} T^{9} - 15 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 15 T + 91 T^{2} - 392 T^{3} + 1667 T^{4} + 1799 T^{5} - 58798 T^{6} + 1799 p T^{7} + 1667 p^{2} T^{8} - 392 p^{3} T^{9} + 91 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - T - 45 T^{2} - 284 T^{3} + 533 T^{4} + 7509 T^{5} + 35806 T^{6} + 7509 p T^{7} + 533 p^{2} T^{8} - 284 p^{3} T^{9} - 45 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 3 T - 89 T^{2} + 112 T^{3} + 4977 T^{4} - 829 T^{5} - 229842 T^{6} - 829 p T^{7} + 4977 p^{2} T^{8} + 112 p^{3} T^{9} - 89 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( ( 1 + 4 T + p T^{2} )^{6} \)
47 \( 1 - 28 T + 389 T^{2} - 4300 T^{3} + 42820 T^{4} - 357312 T^{5} + 2567015 T^{6} - 357312 p T^{7} + 42820 p^{2} T^{8} - 4300 p^{3} T^{9} + 389 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 3 T + 6 T^{2} - 291 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( ( 1 - 2 T + 149 T^{2} - 244 T^{3} + 149 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( 1 + 12 T + 67 T^{2} + 548 T^{3} - 1024 T^{4} - 53600 T^{5} - 408187 T^{6} - 53600 p T^{7} - 1024 p^{2} T^{8} + 548 p^{3} T^{9} + 67 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 13 T + 11 T^{2} - 1376 T^{3} - 8549 T^{4} + 44835 T^{5} + 1032674 T^{6} + 44835 p T^{7} - 8549 p^{2} T^{8} - 1376 p^{3} T^{9} + 11 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 8 T - 121 T^{2} + 320 T^{3} + 13520 T^{4} + 6948 T^{5} - 1251447 T^{6} + 6948 p T^{7} + 13520 p^{2} T^{8} + 320 p^{3} T^{9} - 121 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 81 T^{2} - 256 T^{3} + 162 T^{4} + 10368 T^{5} + 484143 T^{6} + 10368 p T^{7} + 162 p^{2} T^{8} - 256 p^{3} T^{9} - 81 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 + 13 T + 47 T^{2} - 452 T^{3} - 8489 T^{4} - 49401 T^{5} - 217678 T^{6} - 49401 p T^{7} - 8489 p^{2} T^{8} - 452 p^{3} T^{9} + 47 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 + 25 T + 414 T^{2} + 4341 T^{3} + 414 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 + 6 T - 155 T^{2} - 1006 T^{3} + 11498 T^{4} + 39646 T^{5} - 894259 T^{6} + 39646 p T^{7} + 11498 p^{2} T^{8} - 1006 p^{3} T^{9} - 155 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.61385733124902559763466653692, −5.47721960319881564522440034835, −5.27851675437723944315602039570, −4.99723605396183947142471410691, −4.97706417389587619339615081198, −4.87350520211993468108619792551, −4.47468948498605609140933374904, −4.41445351652433965560718690218, −4.36679221018499154479145937131, −4.20858938532067516266792255551, −4.19775664039105472473519965056, −3.59549658250935952564045527986, −3.34338176373669104276474895281, −3.17497422151215613540859571721, −2.80504400255658514049715686250, −2.68261075957153917046498885239, −2.67697170400291633178365338819, −2.64916966730529240752113311068, −2.52101476111860323610630110533, −2.38600852777455654129628756125, −1.82406914291028556060649460825, −1.59216395561569687032276779444, −1.54442845858244255029643721241, −0.77562865053076373122657692367, −0.44456184679367295362881142514, 0.44456184679367295362881142514, 0.77562865053076373122657692367, 1.54442845858244255029643721241, 1.59216395561569687032276779444, 1.82406914291028556060649460825, 2.38600852777455654129628756125, 2.52101476111860323610630110533, 2.64916966730529240752113311068, 2.67697170400291633178365338819, 2.68261075957153917046498885239, 2.80504400255658514049715686250, 3.17497422151215613540859571721, 3.34338176373669104276474895281, 3.59549658250935952564045527986, 4.19775664039105472473519965056, 4.20858938532067516266792255551, 4.36679221018499154479145937131, 4.41445351652433965560718690218, 4.47468948498605609140933374904, 4.87350520211993468108619792551, 4.97706417389587619339615081198, 4.99723605396183947142471410691, 5.27851675437723944315602039570, 5.47721960319881564522440034835, 5.61385733124902559763466653692

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

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