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Properties

Label 12-310e6-1.1-c1e6-0-2
Degree $12$
Conductor $8.875\times 10^{14}$
Sign $1$
Analytic cond. $230.055$
Root an. cond. $1.57332$
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  + 6·2-s − 3-s + 21·4-s + 3·5-s − 6·6-s − 4·7-s + 56·8-s + 5·9-s + 18·10-s − 5·11-s − 21·12-s − 13-s − 24·14-s − 3·15-s + 126·16-s + 17-s + 30·18-s − 3·19-s + 63·20-s + 4·21-s − 30·22-s + 10·23-s − 56·24-s + 3·25-s − 6·26-s − 4·27-s − 84·28-s + ⋯
L(s)  = 1  + 4.24·2-s − 0.577·3-s + 21/2·4-s + 1.34·5-s − 2.44·6-s − 1.51·7-s + 19.7·8-s + 5/3·9-s + 5.69·10-s − 1.50·11-s − 6.06·12-s − 0.277·13-s − 6.41·14-s − 0.774·15-s + 63/2·16-s + 0.242·17-s + 7.07·18-s − 0.688·19-s + 14.0·20-s + 0.872·21-s − 6.39·22-s + 2.08·23-s − 11.4·24-s + 3/5·25-s − 1.17·26-s − 0.769·27-s − 15.8·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 31^{6}\)
Sign: $1$
Analytic conductor: \(230.055\)
Root analytic conductor: \(1.57332\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 31^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(41.87284379\)
\(L(\frac12)\) \(\approx\) \(41.87284379\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{6} \)
5 \( ( 1 - T + T^{2} )^{3} \)
31 \( 1 + 16 T + 152 T^{2} + 1045 T^{3} + 152 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
good3 \( 1 + T - 4 T^{2} - 5 T^{3} + 5 T^{4} + 4 T^{5} - 5 T^{6} + 4 p T^{7} + 5 p^{2} T^{8} - 5 p^{3} T^{9} - 4 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 4 T - 6 T^{2} - 18 T^{3} + 118 T^{4} + 130 T^{5} - 677 T^{6} + 130 p T^{7} + 118 p^{2} T^{8} - 18 p^{3} T^{9} - 6 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 5 T + 10 T^{2} + 17 T^{3} - 85 T^{4} - 610 T^{5} - 1997 T^{6} - 610 p T^{7} - 85 p^{2} T^{8} + 17 p^{3} T^{9} + 10 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + T - 12 T^{2} - 105 T^{3} - 59 T^{4} + 46 p T^{5} + 5509 T^{6} + 46 p^{2} T^{7} - 59 p^{2} T^{8} - 105 p^{3} T^{9} - 12 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - T - 44 T^{2} + 29 T^{3} + 1223 T^{4} - 424 T^{5} - 23519 T^{6} - 424 p T^{7} + 1223 p^{2} T^{8} + 29 p^{3} T^{9} - 44 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 42 T^{2} - 61 T^{3} + 69 p T^{4} + 726 T^{5} - 27501 T^{6} + 726 p T^{7} + 69 p^{3} T^{8} - 61 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 - 5 T + 3 T^{2} + 133 T^{3} + 3 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 - 2 T + 24 T^{2} + 31 T^{3} + 24 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 + 5 T - 40 T^{2} - 21 T^{3} + 955 T^{4} - 5960 T^{5} - 59315 T^{6} - 5960 p T^{7} + 955 p^{2} T^{8} - 21 p^{3} T^{9} - 40 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 3 T - 60 T^{2} - 447 T^{3} + 951 T^{4} + 10632 T^{5} + 34441 T^{6} + 10632 p T^{7} + 951 p^{2} T^{8} - 447 p^{3} T^{9} - 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 9 T + 36 T^{2} - 175 T^{3} - 945 T^{4} + 13014 T^{5} - 64605 T^{6} + 13014 p T^{7} - 945 p^{2} T^{8} - 175 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 6 T + 144 T^{2} + 555 T^{3} + 144 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 + 11 T + 28 T^{2} - 151 T^{3} - 2737 T^{4} - 22948 T^{5} - 155963 T^{6} - 22948 p T^{7} - 2737 p^{2} T^{8} - 151 p^{3} T^{9} + 28 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 10 T - 56 T^{2} + 734 T^{3} + 3662 T^{4} - 29662 T^{5} - 57557 T^{6} - 29662 p T^{7} + 3662 p^{2} T^{8} + 734 p^{3} T^{9} - 56 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
61 \( ( 1 - 7 T + 130 T^{2} - 731 T^{3} + 130 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 25 T + 233 T^{2} - 2238 T^{3} + 32875 T^{4} - 281177 T^{5} + 1796374 T^{6} - 281177 p T^{7} + 32875 p^{2} T^{8} - 2238 p^{3} T^{9} + 233 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 9 T - 126 T^{2} - 531 T^{3} + 17379 T^{4} + 28602 T^{5} - 1301969 T^{6} + 28602 p T^{7} + 17379 p^{2} T^{8} - 531 p^{3} T^{9} - 126 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 28 T + 354 T^{2} + 3402 T^{3} + 32578 T^{4} + 248542 T^{5} + 1756159 T^{6} + 248542 p T^{7} + 32578 p^{2} T^{8} + 3402 p^{3} T^{9} + 354 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 9 T - 126 T^{2} + 419 T^{3} + 16227 T^{4} + 1998 T^{5} - 1602633 T^{6} + 1998 p T^{7} + 16227 p^{2} T^{8} + 419 p^{3} T^{9} - 126 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 3 T - 36 T^{2} + 1815 T^{3} - 4635 T^{4} - 42132 T^{5} + 1864123 T^{6} - 42132 p T^{7} - 4635 p^{2} T^{8} + 1815 p^{3} T^{9} - 36 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 - 9 T + 213 T^{2} - 1359 T^{3} + 213 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( ( 1 + 16 T + 356 T^{2} + 3181 T^{3} + 356 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−6.58297136126241163913965912334, −5.83387166504566630126605671255, −5.82685993400179752080906665057, −5.66735372625740442481519548646, −5.49771247305071488628185757313, −5.49530319702499219127816498658, −5.41650651291282206154432900704, −5.13830706142591506417273308544, −4.91903886744062243511653115129, −4.60900652615228245871993456883, −4.39415873470671536746569920382, −4.34424279105768469830683603934, −4.32213924811143569335197446612, −3.78192281433900802862267445472, −3.61550687699726358468354524843, −3.42280701795866066121045842215, −3.37066606444371428173369390245, −2.99957245065516509932798163627, −2.75630516307116783471792168378, −2.63348240323449941819190372233, −2.20168968131516869419777700990, −2.14376951805353397508041541035, −1.80832211056484174261374324259, −1.47849941538790637519101730095, −0.926516895813215985039671683153, 0.926516895813215985039671683153, 1.47849941538790637519101730095, 1.80832211056484174261374324259, 2.14376951805353397508041541035, 2.20168968131516869419777700990, 2.63348240323449941819190372233, 2.75630516307116783471792168378, 2.99957245065516509932798163627, 3.37066606444371428173369390245, 3.42280701795866066121045842215, 3.61550687699726358468354524843, 3.78192281433900802862267445472, 4.32213924811143569335197446612, 4.34424279105768469830683603934, 4.39415873470671536746569920382, 4.60900652615228245871993456883, 4.91903886744062243511653115129, 5.13830706142591506417273308544, 5.41650651291282206154432900704, 5.49530319702499219127816498658, 5.49771247305071488628185757313, 5.66735372625740442481519548646, 5.82685993400179752080906665057, 5.83387166504566630126605671255, 6.58297136126241163913965912334

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

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