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Properties

Label 12-9800e6-1.1-c1e6-0-1
Degree $12$
Conductor $8.858\times 10^{23}$
Sign $1$
Analytic cond. $2.29624\times 10^{11}$
Root an. cond. $8.84609$
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-s − 5·9-s + 11-s + 7·13-s + 4·17-s + 5·19-s + 6·23-s − 4·27-s − 3·29-s + 2·31-s + 33-s + 9·37-s + 7·39-s − 6·41-s − 3·43-s + 27·47-s + 4·51-s − 5·53-s + 5·57-s + 24·59-s − 9·61-s + 17·67-s + 6·69-s + 4·71-s + 18·73-s − 22·79-s + 7·81-s + ⋯
L(s)  = 1  + 0.577·3-s − 5/3·9-s + 0.301·11-s + 1.94·13-s + 0.970·17-s + 1.14·19-s + 1.25·23-s − 0.769·27-s − 0.557·29-s + 0.359·31-s + 0.174·33-s + 1.47·37-s + 1.12·39-s − 0.937·41-s − 0.457·43-s + 3.93·47-s + 0.560·51-s − 0.686·53-s + 0.662·57-s + 3.12·59-s − 1.15·61-s + 2.07·67-s + 0.722·69-s + 0.474·71-s + 2.10·73-s − 2.47·79-s + 7/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 5^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(2.29624\times 10^{11}\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 5^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(37.39939527\)
\(L(\frac12)\) \(\approx\) \(37.39939527\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 - T + 2 p T^{2} - 7 T^{3} + 26 T^{4} - 11 p T^{5} + 94 T^{6} - 11 p^{2} T^{7} + 26 p^{2} T^{8} - 7 p^{3} T^{9} + 2 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - T + 29 T^{2} + 14 T^{3} + 483 T^{4} + 235 T^{5} + 6782 T^{6} + 235 p T^{7} + 483 p^{2} T^{8} + 14 p^{3} T^{9} + 29 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 7 T + 68 T^{2} - 355 T^{3} + 2007 T^{4} - 8250 T^{5} + 33848 T^{6} - 8250 p T^{7} + 2007 p^{2} T^{8} - 355 p^{3} T^{9} + 68 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 4 T + 27 T^{2} - 96 T^{3} + 891 T^{4} - 2668 T^{5} + 14962 T^{6} - 2668 p T^{7} + 891 p^{2} T^{8} - 96 p^{3} T^{9} + 27 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 5 T + 67 T^{2} - 174 T^{3} + 1749 T^{4} - 2193 T^{5} + 32374 T^{6} - 2193 p T^{7} + 1749 p^{2} T^{8} - 174 p^{3} T^{9} + 67 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T + 119 T^{2} - 582 T^{3} + 6310 T^{4} - 24766 T^{5} + 188355 T^{6} - 24766 p T^{7} + 6310 p^{2} T^{8} - 582 p^{3} T^{9} + 119 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T + 69 T^{2} + 400 T^{3} + 3087 T^{4} + 19485 T^{5} + 104422 T^{6} + 19485 p T^{7} + 3087 p^{2} T^{8} + 400 p^{3} T^{9} + 69 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 2 T + 71 T^{2} - 6 p T^{3} + 2171 T^{4} - 5960 T^{5} + 55754 T^{6} - 5960 p T^{7} + 2171 p^{2} T^{8} - 6 p^{4} T^{9} + 71 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 9 T + 147 T^{2} - 1118 T^{3} + 10827 T^{4} - 70557 T^{5} + 496562 T^{6} - 70557 p T^{7} + 10827 p^{2} T^{8} - 1118 p^{3} T^{9} + 147 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 6 T + 146 T^{2} + 964 T^{3} + 11686 T^{4} + 62770 T^{5} + 608006 T^{6} + 62770 p T^{7} + 11686 p^{2} T^{8} + 964 p^{3} T^{9} + 146 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 3 T + 107 T^{2} + 442 T^{3} + 8753 T^{4} + 31575 T^{5} + 420350 T^{6} + 31575 p T^{7} + 8753 p^{2} T^{8} + 442 p^{3} T^{9} + 107 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 27 T + 555 T^{2} - 7616 T^{3} + 87063 T^{4} - 776961 T^{5} + 5938186 T^{6} - 776961 p T^{7} + 87063 p^{2} T^{8} - 7616 p^{3} T^{9} + 555 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 5 T + 241 T^{2} + 1058 T^{3} + 26841 T^{4} + 99265 T^{5} + 1789306 T^{6} + 99265 p T^{7} + 26841 p^{2} T^{8} + 1058 p^{3} T^{9} + 241 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 24 T + 447 T^{2} - 5038 T^{3} + 50343 T^{4} - 380622 T^{5} + 3114514 T^{6} - 380622 p T^{7} + 50343 p^{2} T^{8} - 5038 p^{3} T^{9} + 447 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 9 T + 194 T^{2} + 1195 T^{3} + 19136 T^{4} + 87421 T^{5} + 1239000 T^{6} + 87421 p T^{7} + 19136 p^{2} T^{8} + 1195 p^{3} T^{9} + 194 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 17 T + 440 T^{2} - 5299 T^{3} + 76318 T^{4} - 685417 T^{5} + 6881660 T^{6} - 685417 p T^{7} + 76318 p^{2} T^{8} - 5299 p^{3} T^{9} + 440 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 4 T + 154 T^{2} - 140 T^{3} + 10351 T^{4} + 37464 T^{5} + 449740 T^{6} + 37464 p T^{7} + 10351 p^{2} T^{8} - 140 p^{3} T^{9} + 154 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 18 T + 359 T^{2} - 2572 T^{3} + 21527 T^{4} + 46910 T^{5} - 84318 T^{6} + 46910 p T^{7} + 21527 p^{2} T^{8} - 2572 p^{3} T^{9} + 359 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 22 T + 535 T^{2} + 7818 T^{3} + 111171 T^{4} + 1168164 T^{5} + 11872954 T^{6} + 1168164 p T^{7} + 111171 p^{2} T^{8} + 7818 p^{3} T^{9} + 535 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 9 T + 395 T^{2} - 2346 T^{3} + 64909 T^{4} - 272593 T^{5} + 6476022 T^{6} - 272593 p T^{7} + 64909 p^{2} T^{8} - 2346 p^{3} T^{9} + 395 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 15 T + 266 T^{2} + 2177 T^{3} + 12040 T^{4} + 30051 T^{5} - 545360 T^{6} + 30051 p T^{7} + 12040 p^{2} T^{8} + 2177 p^{3} T^{9} + 266 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 12 T + 374 T^{2} - 4220 T^{3} + 74255 T^{4} - 677816 T^{5} + 9184692 T^{6} - 677816 p T^{7} + 74255 p^{2} T^{8} - 4220 p^{3} T^{9} + 374 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
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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

−3.90165943423040093042246214858, −3.67633518763919815680332640703, −3.51025767285191508250855778991, −3.38810844715147819428642860181, −3.33383910431295756625557369296, −3.31075681704893219382233204954, −3.28630901841633105734209570550, −2.81885356316534144163888389832, −2.74873963315517334455180852127, −2.70351424545678503523346973640, −2.57476692311276386338466612650, −2.52703873501999586784596716070, −2.42740104322483852796194659974, −2.16293979448063845646457284546, −1.86867163414217188730784675070, −1.76463652398962010646662689249, −1.68083531314568127030566353276, −1.45982578440862350481469101254, −1.27087540805290005891484034757, −1.17053015083395442087841755255, −0.874265369384653401955771264989, −0.77803734556415126579721069007, −0.57132320258953828564636696000, −0.47771142977067613208334508559, −0.41912620588610110479933159851, 0.41912620588610110479933159851, 0.47771142977067613208334508559, 0.57132320258953828564636696000, 0.77803734556415126579721069007, 0.874265369384653401955771264989, 1.17053015083395442087841755255, 1.27087540805290005891484034757, 1.45982578440862350481469101254, 1.68083531314568127030566353276, 1.76463652398962010646662689249, 1.86867163414217188730784675070, 2.16293979448063845646457284546, 2.42740104322483852796194659974, 2.52703873501999586784596716070, 2.57476692311276386338466612650, 2.70351424545678503523346973640, 2.74873963315517334455180852127, 2.81885356316534144163888389832, 3.28630901841633105734209570550, 3.31075681704893219382233204954, 3.33383910431295756625557369296, 3.38810844715147819428642860181, 3.51025767285191508250855778991, 3.67633518763919815680332640703, 3.90165943423040093042246214858

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

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