Properties

Label 12-1680e6-1.1-c1e6-0-3
Degree $12$
Conductor $2.248\times 10^{19}$
Sign $1$
Analytic cond. $5.82798\times 10^{6}$
Root an. cond. $3.66263$
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  + 3·3-s + 3·5-s + 3·9-s − 3·11-s − 6·13-s + 9·15-s − 6·17-s + 3·19-s − 3·23-s + 3·25-s − 2·27-s + 12·29-s + 12·31-s − 9·33-s − 3·37-s − 18·39-s − 18·41-s + 9·45-s + 3·47-s + 6·49-s − 18·51-s + 15·53-s − 9·55-s + 9·57-s + 30·59-s − 18·65-s − 9·69-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.34·5-s + 9-s − 0.904·11-s − 1.66·13-s + 2.32·15-s − 1.45·17-s + 0.688·19-s − 0.625·23-s + 3/5·25-s − 0.384·27-s + 2.22·29-s + 2.15·31-s − 1.56·33-s − 0.493·37-s − 2.88·39-s − 2.81·41-s + 1.34·45-s + 0.437·47-s + 6/7·49-s − 2.52·51-s + 2.06·53-s − 1.21·55-s + 1.19·57-s + 3.90·59-s − 2.23·65-s − 1.08·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(5.82798\times 10^{6}\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(11.23202547\)
\(L(\frac12)\) \(\approx\) \(11.23202547\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T + T^{2} )^{3} \)
5 \( ( 1 - T + T^{2} )^{3} \)
7 \( 1 - 6 T^{2} + 4 T^{3} - 6 p T^{4} + p^{3} T^{6} \)
good11 \( 1 + 3 T + 12 T^{2} + 111 T^{3} + 222 T^{4} + 741 T^{5} + 5074 T^{6} + 741 p T^{7} + 222 p^{2} T^{8} + 111 p^{3} T^{9} + 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 + 3 T + 24 T^{2} + 81 T^{3} + 24 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 6 T + 39 T^{2} + 214 T^{3} + 756 T^{4} + 2694 T^{5} + 12725 T^{6} + 2694 p T^{7} + 756 p^{2} T^{8} + 214 p^{3} T^{9} + 39 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( ( 1 - 8 T + p T^{2} )^{3}( 1 + 7 T + p T^{2} )^{3} \)
23 \( 1 + 3 T - 36 T^{2} - 97 T^{3} + 642 T^{4} + 597 T^{5} - 13546 T^{6} + 597 p T^{7} + 642 p^{2} T^{8} - 97 p^{3} T^{9} - 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 - 6 T + 81 T^{2} - 300 T^{3} + 81 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 12 T + 78 T^{2} - 204 T^{3} - 1218 T^{4} + 18168 T^{5} - 127402 T^{6} + 18168 p T^{7} - 1218 p^{2} T^{8} - 204 p^{3} T^{9} + 78 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 3 T - 15 T^{2} + 390 T^{3} + 165 T^{4} - 5037 T^{5} + 92954 T^{6} - 5037 p T^{7} + 165 p^{2} T^{8} + 390 p^{3} T^{9} - 15 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 9 T + 3 p T^{2} + 680 T^{3} + 3 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 90 T^{2} - 88 T^{3} + 90 p T^{4} + p^{3} T^{6} )^{2} \)
47 \( 1 - 3 T - 60 T^{2} + 637 T^{3} + 252 T^{4} - 16287 T^{5} + 142082 T^{6} - 16287 p T^{7} + 252 p^{2} T^{8} + 637 p^{3} T^{9} - 60 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 15 T + 6 T^{2} + 7 T^{3} + 11664 T^{4} - 54399 T^{5} - 166888 T^{6} - 54399 p T^{7} + 11664 p^{2} T^{8} + 7 p^{3} T^{9} + 6 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 30 T + 441 T^{2} - 5010 T^{3} + 52200 T^{4} - 489990 T^{5} + 4032079 T^{6} - 489990 p T^{7} + 52200 p^{2} T^{8} - 5010 p^{3} T^{9} + 441 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
61 \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \)
67 \( 1 - 162 T^{2} - 176 T^{3} + 15390 T^{4} + 14256 T^{5} - 1149078 T^{6} + 14256 p T^{7} + 15390 p^{2} T^{8} - 176 p^{3} T^{9} - 162 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 12 T + 27 T^{2} + 396 T^{3} + 27 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 3 T - 36 T^{2} + 887 T^{3} - 36 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )( 1 + 21 T + 123 T^{2} + 266 T^{3} + 123 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} ) \)
79 \( 1 - 6 T - 114 T^{2} + 932 T^{3} + 5154 T^{4} - 32646 T^{5} - 177042 T^{6} - 32646 p T^{7} + 5154 p^{2} T^{8} + 932 p^{3} T^{9} - 114 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 6 T + 243 T^{2} + 948 T^{3} + 243 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 177 T^{2} + 584 T^{3} + 15576 T^{4} - 51684 T^{5} - 1293079 T^{6} - 51684 p T^{7} + 15576 p^{2} T^{8} + 584 p^{3} T^{9} - 177 p^{4} T^{10} + p^{6} T^{12} \)
97 \( ( 1 - 8 T + p T^{2} )^{6} \)
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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

−4.83526525511328457342839742249, −4.77012183717626622800290617639, −4.56187965960075036909377890283, −4.50239579310215658796897273663, −4.27723784882652851560720159748, −4.14354541173826357530056680893, −3.95466969700081031597230570870, −3.62555406049829866538716136269, −3.57431420486419470027456187148, −3.45820767401382408886681173336, −3.24415618197963131238473012085, −2.94626401559529630655921305362, −2.87841661121519747938145832927, −2.86462696011883405096737668018, −2.32629750348473789017423777145, −2.27703082264184881162816148602, −2.24151866656048863547949085955, −2.22837775637324665941174664900, −2.20921371405296256982263906230, −1.82545380480142947965529767327, −1.34553656938248704343014882161, −1.11760115790506573613924689159, −1.03368931846337799919101645160, −0.48088743018928932095811419482, −0.40052525254586409025884160256, 0.40052525254586409025884160256, 0.48088743018928932095811419482, 1.03368931846337799919101645160, 1.11760115790506573613924689159, 1.34553656938248704343014882161, 1.82545380480142947965529767327, 2.20921371405296256982263906230, 2.22837775637324665941174664900, 2.24151866656048863547949085955, 2.27703082264184881162816148602, 2.32629750348473789017423777145, 2.86462696011883405096737668018, 2.87841661121519747938145832927, 2.94626401559529630655921305362, 3.24415618197963131238473012085, 3.45820767401382408886681173336, 3.57431420486419470027456187148, 3.62555406049829866538716136269, 3.95466969700081031597230570870, 4.14354541173826357530056680893, 4.27723784882652851560720159748, 4.50239579310215658796897273663, 4.56187965960075036909377890283, 4.77012183717626622800290617639, 4.83526525511328457342839742249

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

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