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Properties

Label 12-3584e6-1.1-c1e6-0-7
Degree $12$
Conductor $2.119\times 10^{21}$
Sign $1$
Analytic cond. $5.49377\times 10^{8}$
Root an. cond. $5.34961$
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  + 4·3-s + 6·7-s + 4·9-s + 8·11-s + 20·19-s + 24·21-s − 14·25-s − 4·27-s + 4·29-s − 8·31-s + 32·33-s + 20·37-s − 4·41-s + 24·43-s + 8·47-s + 21·49-s + 4·53-s + 80·57-s + 12·59-s − 8·61-s + 24·63-s + 16·67-s − 8·71-s + 16·73-s − 56·75-s + 48·77-s − 24·79-s + ⋯
L(s)  = 1  + 2.30·3-s + 2.26·7-s + 4/3·9-s + 2.41·11-s + 4.58·19-s + 5.23·21-s − 2.79·25-s − 0.769·27-s + 0.742·29-s − 1.43·31-s + 5.57·33-s + 3.28·37-s − 0.624·41-s + 3.65·43-s + 1.16·47-s + 3·49-s + 0.549·53-s + 10.5·57-s + 1.56·59-s − 1.02·61-s + 3.02·63-s + 1.95·67-s − 0.949·71-s + 1.87·73-s − 6.46·75-s + 5.47·77-s − 2.70·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{54} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{54} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{54} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(5.49377\times 10^{8}\)
Root analytic conductor: \(5.34961\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{54} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(116.5516047\)
\(L(\frac12)\) \(\approx\) \(116.5516047\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( ( 1 - T )^{6} \)
good3 \( 1 - 4 T + 4 p T^{2} - 28 T^{3} + 61 T^{4} - 112 T^{5} + 200 T^{6} - 112 p T^{7} + 61 p^{2} T^{8} - 28 p^{3} T^{9} + 4 p^{5} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 14 T^{2} + 16 T^{3} + 101 T^{4} + 32 p T^{5} + 592 T^{6} + 32 p^{2} T^{7} + 101 p^{2} T^{8} + 16 p^{3} T^{9} + 14 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 8 T + 64 T^{2} - 312 T^{3} + 1503 T^{4} - 496 p T^{5} + 20304 T^{6} - 496 p^{2} T^{7} + 1503 p^{2} T^{8} - 312 p^{3} T^{9} + 64 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 54 T^{2} - 32 T^{3} + 1333 T^{4} - 1200 T^{5} + 20768 T^{6} - 1200 p T^{7} + 1333 p^{2} T^{8} - 32 p^{3} T^{9} + 54 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 54 T^{2} + 32 T^{3} + 1535 T^{4} + 1504 T^{5} + 29908 T^{6} + 1504 p T^{7} + 1535 p^{2} T^{8} + 32 p^{3} T^{9} + 54 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 - 20 T + 244 T^{2} - 2172 T^{3} + 799 p T^{4} - 86880 T^{5} + 413880 T^{6} - 86880 p T^{7} + 799 p^{3} T^{8} - 2172 p^{3} T^{9} + 244 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 62 T^{2} + 96 T^{3} + 2003 T^{4} + 5792 T^{5} + 48028 T^{6} + 5792 p T^{7} + 2003 p^{2} T^{8} + 96 p^{3} T^{9} + 62 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 - 4 T + 98 T^{2} - 452 T^{3} + 5303 T^{4} - 21352 T^{5} + 191644 T^{6} - 21352 p T^{7} + 5303 p^{2} T^{8} - 452 p^{3} T^{9} + 98 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 8 T + 114 T^{2} + 728 T^{3} + 6319 T^{4} + 37968 T^{5} + 244028 T^{6} + 37968 p T^{7} + 6319 p^{2} T^{8} + 728 p^{3} T^{9} + 114 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 20 T + 338 T^{2} - 3764 T^{3} + 36903 T^{4} - 280776 T^{5} + 1906812 T^{6} - 280776 p T^{7} + 36903 p^{2} T^{8} - 3764 p^{3} T^{9} + 338 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 4 T + 162 T^{2} + 404 T^{3} + 12255 T^{4} + 22024 T^{5} + 603804 T^{6} + 22024 p T^{7} + 12255 p^{2} T^{8} + 404 p^{3} T^{9} + 162 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 24 T + 400 T^{2} - 4552 T^{3} + 42463 T^{4} - 328720 T^{5} + 2285616 T^{6} - 328720 p T^{7} + 42463 p^{2} T^{8} - 4552 p^{3} T^{9} + 400 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 8 T + 130 T^{2} - 984 T^{3} + 13135 T^{4} - 75984 T^{5} + 688796 T^{6} - 75984 p T^{7} + 13135 p^{2} T^{8} - 984 p^{3} T^{9} + 130 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 4 T + 50 T^{2} - 484 T^{3} + 1703 T^{4} - 552 T^{5} + 148476 T^{6} - 552 p T^{7} + 1703 p^{2} T^{8} - 484 p^{3} T^{9} + 50 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 12 T + 244 T^{2} - 1764 T^{3} + 23101 T^{4} - 120736 T^{5} + 1453496 T^{6} - 120736 p T^{7} + 23101 p^{2} T^{8} - 1764 p^{3} T^{9} + 244 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 8 T + 174 T^{2} + 24 p T^{3} + 14933 T^{4} + 157792 T^{5} + 1030544 T^{6} + 157792 p T^{7} + 14933 p^{2} T^{8} + 24 p^{4} T^{9} + 174 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 16 T + 352 T^{2} - 4144 T^{3} + 52527 T^{4} - 477408 T^{5} + 4484240 T^{6} - 477408 p T^{7} + 52527 p^{2} T^{8} - 4144 p^{3} T^{9} + 352 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 8 T + 178 T^{2} + 2040 T^{3} + 21071 T^{4} + 222640 T^{5} + 1862108 T^{6} + 222640 p T^{7} + 21071 p^{2} T^{8} + 2040 p^{3} T^{9} + 178 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 16 T + 198 T^{2} - 432 T^{3} - 4401 T^{4} + 142912 T^{5} - 1075596 T^{6} + 142912 p T^{7} - 4401 p^{2} T^{8} - 432 p^{3} T^{9} + 198 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 24 T + 538 T^{2} + 7784 T^{3} + 107647 T^{4} + 1125840 T^{5} + 11257356 T^{6} + 1125840 p T^{7} + 107647 p^{2} T^{8} + 7784 p^{3} T^{9} + 538 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 12 T + 252 T^{2} - 3156 T^{3} + 41885 T^{4} - 390384 T^{5} + 4354696 T^{6} - 390384 p T^{7} + 41885 p^{2} T^{8} - 3156 p^{3} T^{9} + 252 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 16 T + 502 T^{2} - 6064 T^{3} + 105327 T^{4} - 991680 T^{5} + 12211732 T^{6} - 991680 p T^{7} + 105327 p^{2} T^{8} - 6064 p^{3} T^{9} + 502 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 16 T + 486 T^{2} - 5296 T^{3} + 98847 T^{4} - 852544 T^{5} + 12088692 T^{6} - 852544 p T^{7} + 98847 p^{2} T^{8} - 5296 p^{3} T^{9} + 486 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.29474896545374864105875344450, −4.22457961494760466367756335231, −4.05027785401275436766158143598, −3.94300003859926790170257595191, −3.76254153196801637727819110987, −3.68143907344264713087657617683, −3.63548085238467453620666447056, −3.38725957997512161583357820721, −3.11687025998740701178126882441, −3.00055368222021784216793825379, −2.98221322719782709029408396427, −2.85084897630968451669196805470, −2.75216891583685003515458931185, −2.30806935305158460201916583023, −2.18542074243260781325767670564, −2.14863845191009468116856904080, −1.96221455155191577318408024751, −1.94209406993531021956637166832, −1.66221930314456178350626692860, −1.30056911147779192196949751612, −1.29384980931160922868569903515, −0.848723383181589804544182245659, −0.825496096786516613207998089471, −0.70415775934132938734479857739, −0.65952272735912067124283381652, 0.65952272735912067124283381652, 0.70415775934132938734479857739, 0.825496096786516613207998089471, 0.848723383181589804544182245659, 1.29384980931160922868569903515, 1.30056911147779192196949751612, 1.66221930314456178350626692860, 1.94209406993531021956637166832, 1.96221455155191577318408024751, 2.14863845191009468116856904080, 2.18542074243260781325767670564, 2.30806935305158460201916583023, 2.75216891583685003515458931185, 2.85084897630968451669196805470, 2.98221322719782709029408396427, 3.00055368222021784216793825379, 3.11687025998740701178126882441, 3.38725957997512161583357820721, 3.63548085238467453620666447056, 3.68143907344264713087657617683, 3.76254153196801637727819110987, 3.94300003859926790170257595191, 4.05027785401275436766158143598, 4.22457961494760466367756335231, 4.29474896545374864105875344450

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

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