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Properties

Label 12-5292e6-1.1-c1e6-0-2
Degree $12$
Conductor $2.196\times 10^{22}$
Sign $1$
Analytic cond. $5.69353\times 10^{9}$
Root an. cond. $6.50052$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s + 12·11-s − 3·13-s + 3·19-s + 12·23-s + 9·25-s − 15·29-s − 3·31-s + 3·37-s + 6·41-s − 3·43-s + 15·47-s − 18·53-s − 72·55-s + 3·59-s − 6·61-s + 18·65-s + 6·67-s + 30·71-s + 9·73-s − 3·79-s + 18·83-s − 6·89-s − 18·95-s − 15·97-s + 18·101-s − 48·103-s + ⋯
L(s)  = 1  − 2.68·5-s + 3.61·11-s − 0.832·13-s + 0.688·19-s + 2.50·23-s + 9/5·25-s − 2.78·29-s − 0.538·31-s + 0.493·37-s + 0.937·41-s − 0.457·43-s + 2.18·47-s − 2.47·53-s − 9.70·55-s + 0.390·59-s − 0.768·61-s + 2.23·65-s + 0.733·67-s + 3.56·71-s + 1.05·73-s − 0.337·79-s + 1.97·83-s − 0.635·89-s − 1.84·95-s − 1.52·97-s + 1.79·101-s − 4.72·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 3^{18} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(5.69353\times 10^{9}\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2566823575\)
\(L(\frac12)\) \(\approx\) \(0.2566823575\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 + 3 T + 9 T^{2} + 21 T^{3} + 9 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 6 T + 36 T^{2} - 123 T^{3} + 36 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( ( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} )^{3} \)
17 \( 1 - 18 T^{2} + 18 T^{3} + 18 T^{4} - 162 T^{5} + 4399 T^{6} - 162 p T^{7} + 18 p^{2} T^{8} + 18 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 - 3 T - 12 T^{2} + 67 T^{3} - 153 T^{4} - 54 T^{5} + 6315 T^{6} - 54 p T^{7} - 153 p^{2} T^{8} + 67 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 - 6 T + 54 T^{2} - 177 T^{3} + 54 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 + 15 T + 90 T^{2} + 411 T^{3} + 2205 T^{4} + 4110 T^{5} - 17723 T^{6} + 4110 p T^{7} + 2205 p^{2} T^{8} + 411 p^{3} T^{9} + 90 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T - 6 T^{2} - 221 T^{3} - 639 T^{4} + 2088 T^{5} + 61647 T^{6} + 2088 p T^{7} - 639 p^{2} T^{8} - 221 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T - 24 T^{2} + 131 T^{3} - 477 T^{4} - 576 T^{5} + 61917 T^{6} - 576 p T^{7} - 477 p^{2} T^{8} + 131 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 6 T - 90 T^{2} + 210 T^{3} + 7812 T^{4} - 8952 T^{5} - 340301 T^{6} - 8952 p T^{7} + 7812 p^{2} T^{8} + 210 p^{3} T^{9} - 90 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 3 T - 12 T^{2} + 11 p T^{3} + 153 T^{4} - 4176 T^{5} + 165435 T^{6} - 4176 p T^{7} + 153 p^{2} T^{8} + 11 p^{4} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 15 T + 48 T^{2} + 3 T^{3} + 3075 T^{4} - 18798 T^{5} + 4399 T^{6} - 18798 p T^{7} + 3075 p^{2} T^{8} + 3 p^{3} T^{9} + 48 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 18 T + 126 T^{2} + 522 T^{3} + 1332 T^{4} - 26064 T^{5} - 376733 T^{6} - 26064 p T^{7} + 1332 p^{2} T^{8} + 522 p^{3} T^{9} + 126 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 3 T - 114 T^{2} + 501 T^{3} + 6567 T^{4} - 20406 T^{5} - 323957 T^{6} - 20406 p T^{7} + 6567 p^{2} T^{8} + 501 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T - 102 T^{2} - 698 T^{3} + 6048 T^{4} + 26604 T^{5} - 259509 T^{6} + 26604 p T^{7} + 6048 p^{2} T^{8} - 698 p^{3} T^{9} - 102 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 6 T - 138 T^{2} + 446 T^{3} + 14148 T^{4} - 16668 T^{5} - 1033545 T^{6} - 16668 p T^{7} + 14148 p^{2} T^{8} + 446 p^{3} T^{9} - 138 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 15 T + 231 T^{2} - 1833 T^{3} + 231 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 9 T - 126 T^{2} + 607 T^{3} + 16623 T^{4} - 31608 T^{5} - 1256367 T^{6} - 31608 p T^{7} + 16623 p^{2} T^{8} + 607 p^{3} T^{9} - 126 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 3 T + 6 T^{2} + 1787 T^{3} + 2439 T^{4} + 14634 T^{5} + 1719519 T^{6} + 14634 p T^{7} + 2439 p^{2} T^{8} + 1787 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 18 T + 198 T^{3} + 23814 T^{4} - 132750 T^{5} - 711245 T^{6} - 132750 p T^{7} + 23814 p^{2} T^{8} + 198 p^{3} T^{9} - 18 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 6 T - 36 T^{2} + 474 T^{3} - 1098 T^{4} - 30882 T^{5} + 663847 T^{6} - 30882 p T^{7} - 1098 p^{2} T^{8} + 474 p^{3} T^{9} - 36 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 15 T - 84 T^{2} - 1139 T^{3} + 22203 T^{4} + 134028 T^{5} - 1218567 T^{6} + 134028 p T^{7} + 22203 p^{2} T^{8} - 1139 p^{3} T^{9} - 84 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.22055792520189903690333976031, −4.00641009323525280482813067246, −3.78507465238005754651264778714, −3.73634890672054281247008665592, −3.73179440559508033624848283899, −3.73147314999383044339009451492, −3.48545347357850242215428611018, −3.22942421214501387471052280383, −3.11270047690780848943929405874, −3.07410653736512197729621911717, −2.81806521092986637106103284695, −2.69721457119388169053207186701, −2.48917480237049661883844737002, −2.39069603414267197111122915595, −1.93470950486170070760017978102, −1.86814734203270373393875378983, −1.79599053176162652513505333738, −1.73720893534211311714368203832, −1.34143560596427332814770981096, −1.32293057485325334474538927963, −0.865524445919267820747148917664, −0.863869721723484250946110928983, −0.59332332823196204396166164482, −0.58152706192215813256038691537, −0.04890962472379066644826149087, 0.04890962472379066644826149087, 0.58152706192215813256038691537, 0.59332332823196204396166164482, 0.863869721723484250946110928983, 0.865524445919267820747148917664, 1.32293057485325334474538927963, 1.34143560596427332814770981096, 1.73720893534211311714368203832, 1.79599053176162652513505333738, 1.86814734203270373393875378983, 1.93470950486170070760017978102, 2.39069603414267197111122915595, 2.48917480237049661883844737002, 2.69721457119388169053207186701, 2.81806521092986637106103284695, 3.07410653736512197729621911717, 3.11270047690780848943929405874, 3.22942421214501387471052280383, 3.48545347357850242215428611018, 3.73147314999383044339009451492, 3.73179440559508033624848283899, 3.73634890672054281247008665592, 3.78507465238005754651264778714, 4.00641009323525280482813067246, 4.22055792520189903690333976031

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

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