↑

Properties

Label 12-209e6-1.1-c1e6-0-0
Degree $12$
Conductor $8.334\times 10^{13}$
Sign $1$
Analytic cond. $21.6043$
Root an. cond. $1.29184$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 6·5-s + 6·7-s − 9·8-s + 18·10-s − 3·11-s + 12·13-s + 18·14-s − 9·16-s − 3·17-s + 6·19-s − 9·22-s + 9·23-s + 27·25-s + 36·26-s − 10·27-s − 6·29-s − 3·31-s + 3·32-s − 9·34-s + 36·35-s − 18·37-s + 18·38-s − 54·40-s − 9·41-s − 12·43-s + 27·46-s + ⋯
L(s)  = 1  + 2.12·2-s + 2.68·5-s + 2.26·7-s − 3.18·8-s + 5.69·10-s − 0.904·11-s + 3.32·13-s + 4.81·14-s − 9/4·16-s − 0.727·17-s + 1.37·19-s − 1.91·22-s + 1.87·23-s + 27/5·25-s + 7.06·26-s − 1.92·27-s − 1.11·29-s − 0.538·31-s + 0.530·32-s − 1.54·34-s + 6.08·35-s − 2.95·37-s + 2.91·38-s − 8.53·40-s − 1.40·41-s − 1.82·43-s + 3.98·46-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(11^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(21.6043\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 11^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.008714844\)
\(L(\frac12)\) \(\approx\) \(9.008714844\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( ( 1 + T + T^{2} )^{3} \)
19 \( 1 - 6 T - 12 T^{2} + 169 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 - 3 T + 9 T^{2} - 9 p T^{3} + 9 p^{2} T^{4} - 57 T^{5} + 91 T^{6} - 57 p T^{7} + 9 p^{4} T^{8} - 9 p^{4} T^{9} + 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
3 \( 1 + 10 T^{3} + 73 T^{6} + 10 p^{3} T^{9} + p^{6} T^{12} \)
5 \( 1 - 6 T + 9 T^{2} + 27 T^{3} - 99 T^{4} - 33 T^{5} + 514 T^{6} - 33 p T^{7} - 99 p^{2} T^{8} + 27 p^{3} T^{9} + 9 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 6 T + 6 T^{2} - 10 T^{3} + 180 T^{4} - 324 T^{5} - 153 T^{6} - 324 p T^{7} + 180 p^{2} T^{8} - 10 p^{3} T^{9} + 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 12 T + 75 T^{2} - 301 T^{3} + 909 T^{4} - 2331 T^{5} + 7134 T^{6} - 2331 p T^{7} + 909 p^{2} T^{8} - 301 p^{3} T^{9} + 75 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 3 T - 9 T^{2} + 63 T^{3} + 216 T^{4} + 480 T^{5} + 8173 T^{6} + 480 p T^{7} + 216 p^{2} T^{8} + 63 p^{3} T^{9} - 9 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 9 T + 36 T^{2} + 54 T^{3} - 468 T^{4} + 711 T^{5} + 9649 T^{6} + 711 p T^{7} - 468 p^{2} T^{8} + 54 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 6 T + 108 T^{3} - 468 T^{4} - 132 p T^{5} + 25255 T^{6} - 132 p^{2} T^{7} - 468 p^{2} T^{8} + 108 p^{3} T^{9} + 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T - 48 T^{2} + 53 T^{3} + 1305 T^{4} - 4356 T^{5} - 46305 T^{6} - 4356 p T^{7} + 1305 p^{2} T^{8} + 53 p^{3} T^{9} - 48 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 9 T + 135 T^{2} + 685 T^{3} + 135 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 9 T + 36 T^{2} + 72 T^{3} - 657 T^{4} - 18369 T^{5} - 162935 T^{6} - 18369 p T^{7} - 657 p^{2} T^{8} + 72 p^{3} T^{9} + 36 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 12 T + 159 T^{2} + 1655 T^{3} + 14787 T^{4} + 113931 T^{5} + 812826 T^{6} + 113931 p T^{7} + 14787 p^{2} T^{8} + 1655 p^{3} T^{9} + 159 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 27 T + 324 T^{2} - 2034 T^{3} + 3888 T^{4} + 53811 T^{5} - 605051 T^{6} + 53811 p T^{7} + 3888 p^{2} T^{8} - 2034 p^{3} T^{9} + 324 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 18 T + 270 T^{2} - 2898 T^{3} + 26784 T^{4} - 217692 T^{5} + 1621135 T^{6} - 217692 p T^{7} + 26784 p^{2} T^{8} - 2898 p^{3} T^{9} + 270 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 6 T + 126 T^{2} + 396 T^{3} + 9936 T^{4} + 26844 T^{5} + 645517 T^{6} + 26844 p T^{7} + 9936 p^{2} T^{8} + 396 p^{3} T^{9} + 126 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 21 T + 276 T^{2} + 3212 T^{3} + 30627 T^{4} + 262287 T^{5} + 2154501 T^{6} + 262287 p T^{7} + 30627 p^{2} T^{8} + 3212 p^{3} T^{9} + 276 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 12 T + 246 T^{2} - 2407 T^{3} + 31455 T^{4} - 255933 T^{5} + 2589657 T^{6} - 255933 p T^{7} + 31455 p^{2} T^{8} - 2407 p^{3} T^{9} + 246 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 42 T + 693 T^{2} + 4923 T^{3} - 3969 T^{4} - 410907 T^{5} - 4605290 T^{6} - 410907 p T^{7} - 3969 p^{2} T^{8} + 4923 p^{3} T^{9} + 693 p^{4} T^{10} + 42 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 15 T + 228 T^{2} + 2948 T^{3} + 30726 T^{4} + 320355 T^{5} + 2844447 T^{6} + 320355 p T^{7} + 30726 p^{2} T^{8} + 2948 p^{3} T^{9} + 228 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 9 T + 18 T^{2} + 560 T^{3} - 4077 T^{4} - 84915 T^{5} - 208113 T^{6} - 84915 p T^{7} - 4077 p^{2} T^{8} + 560 p^{3} T^{9} + 18 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 6 T - 114 T^{2} + 1530 T^{3} + 2760 T^{4} - 74184 T^{5} + 504295 T^{6} - 74184 p T^{7} + 2760 p^{2} T^{8} + 1530 p^{3} T^{9} - 114 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 21 T + 207 T^{2} - 1161 T^{3} + 13446 T^{4} - 257484 T^{5} + 3242665 T^{6} - 257484 p T^{7} + 13446 p^{2} T^{8} - 1161 p^{3} T^{9} + 207 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 27 T + 378 T^{2} - 4138 T^{3} + 46197 T^{4} - 470745 T^{5} + 4547829 T^{6} - 470745 p T^{7} + 46197 p^{2} T^{8} - 4138 p^{3} T^{9} + 378 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.93295005909695671024909872556, −6.47756181148488834194127251103, −6.04614368989806019716807755244, −6.00048155029988305561961940283, −5.94971436822779247623289480951, −5.80003586658514728371235040567, −5.48703852180423219489709731960, −5.36052644340071446066576825767, −5.18952814985292116923813858962, −5.08947409670719207392649138060, −4.90307114254482023278700128650, −4.81306250970171700443013101784, −4.45523798224992349543546290800, −4.31837441935652600246917496099, −3.96178803665317707022645952337, −3.71663853604243917382458143156, −3.55891066934152455849093122543, −3.54493958368815681251807289848, −2.91798988692377714922956049372, −2.82662684781981806462436135561, −2.31583992224896671881796508911, −2.05971847318127599786915505869, −1.53284915080034995178520983167, −1.35501342584562328238911062194, −1.30404639406222892841744389324, 1.30404639406222892841744389324, 1.35501342584562328238911062194, 1.53284915080034995178520983167, 2.05971847318127599786915505869, 2.31583992224896671881796508911, 2.82662684781981806462436135561, 2.91798988692377714922956049372, 3.54493958368815681251807289848, 3.55891066934152455849093122543, 3.71663853604243917382458143156, 3.96178803665317707022645952337, 4.31837441935652600246917496099, 4.45523798224992349543546290800, 4.81306250970171700443013101784, 4.90307114254482023278700128650, 5.08947409670719207392649138060, 5.18952814985292116923813858962, 5.36052644340071446066576825767, 5.48703852180423219489709731960, 5.80003586658514728371235040567, 5.94971436822779247623289480951, 6.00048155029988305561961940283, 6.04614368989806019716807755244, 6.47756181148488834194127251103, 6.93295005909695671024909872556

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

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