Properties

Label 12-9680e6-1.1-c1e6-0-7
Degree $12$
Conductor $8.227\times 10^{23}$
Sign $1$
Analytic cond. $2.13262\times 10^{11}$
Root an. cond. $8.79176$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 6·5-s + 15·9-s − 36·15-s − 12·23-s + 21·25-s − 22·27-s + 90·45-s − 42·47-s − 33·49-s + 24·53-s − 30·67-s + 72·69-s − 126·75-s + 27·81-s − 30·89-s + 24·97-s − 36·103-s − 36·113-s − 72·115-s + 56·125-s + 127-s + 131-s − 132·135-s + 137-s + 139-s + 252·141-s + ⋯
L(s)  = 1  − 3.46·3-s + 2.68·5-s + 5·9-s − 9.29·15-s − 2.50·23-s + 21/5·25-s − 4.23·27-s + 13.4·45-s − 6.12·47-s − 4.71·49-s + 3.29·53-s − 3.66·67-s + 8.66·69-s − 14.5·75-s + 3·81-s − 3.17·89-s + 2.43·97-s − 3.54·103-s − 3.38·113-s − 6.71·115-s + 5.00·125-s + 0.0887·127-s + 0.0873·131-s − 11.3·135-s + 0.0854·137-s + 0.0848·139-s + 21.2·141-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 5^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(2.13262\times 10^{11}\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{24} \cdot 5^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T )^{6} \)
11 \( 1 \)
good3 \( ( 1 + p T + 2 p T^{2} + 11 T^{3} + 2 p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} )^{2} \)
7 \( 1 + 33 T^{2} + 498 T^{4} + 4421 T^{6} + 498 p^{2} T^{8} + 33 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 6 T^{2} + 87 T^{4} + 740 T^{6} + 87 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 54 T^{2} + 1455 T^{4} + 27316 T^{6} + 1455 p^{2} T^{8} + 54 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 + 78 T^{2} + 2775 T^{4} + 62804 T^{6} + 2775 p^{2} T^{8} + 78 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 + 6 T + 57 T^{2} + 192 T^{3} + 57 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 + 30 T^{2} + 1095 T^{4} + 54916 T^{6} + 1095 p^{2} T^{8} + 30 p^{4} T^{10} + p^{6} T^{12} \)
31 \( ( 1 + 45 T^{2} + 4 p T^{3} + 45 p T^{4} + p^{3} T^{6} )^{2} \)
37 \( ( 1 + 87 T^{2} - 16 T^{3} + 87 p T^{4} + p^{3} T^{6} )^{2} \)
41 \( 1 + 141 T^{2} + 10494 T^{4} + 517105 T^{6} + 10494 p^{2} T^{8} + 141 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 + 3 p T^{2} + 10242 T^{4} + 518861 T^{6} + 10242 p^{2} T^{8} + 3 p^{5} T^{10} + p^{6} T^{12} \)
47 \( ( 1 + 21 T + 6 p T^{2} + 2277 T^{3} + 6 p^{2} T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 - 12 T + 75 T^{2} - 540 T^{3} + 75 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( ( 1 + 165 T^{2} - 12 T^{3} + 165 p T^{4} + p^{3} T^{6} )^{2} \)
61 \( 1 + 333 T^{2} + 48102 T^{4} + 3843137 T^{6} + 48102 p^{2} T^{8} + 333 p^{4} T^{10} + p^{6} T^{12} \)
67 \( ( 1 + 15 T + 270 T^{2} + 2107 T^{3} + 270 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 + 201 T^{2} + 12 T^{3} + 201 p T^{4} + p^{3} T^{6} )^{2} \)
73 \( 1 + 246 T^{2} + 30015 T^{4} + 2489204 T^{6} + 30015 p^{2} T^{8} + 246 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 + 198 T^{2} + 17583 T^{4} + 1191188 T^{6} + 17583 p^{2} T^{8} + 198 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 + 186 T^{2} + 14391 T^{4} + 929404 T^{6} + 14391 p^{2} T^{8} + 186 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 + 15 T + 246 T^{2} + 2523 T^{3} + 246 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( ( 1 - 12 T + 255 T^{2} - 2252 T^{3} + 255 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−4.19138693475961006561153994029, −4.12181835363155525537829104018, −4.07189988402498180315282524767, −4.00506151924960621103519799465, −3.92547838835551180918145779714, −3.77607698088082297546524089098, −3.48899970533025603316507659978, −3.11661323787712253842128996497, −3.07704868381115422623052429474, −3.06670541933424248517123972505, −3.05826498489662531367607682491, −2.94821067728314056834174655503, −2.78833228768212124806687225520, −2.29132354648124925549451661848, −2.21005821148544698171767593448, −2.14335290582608573538349729594, −2.13871788240719464606428448281, −1.88473678397618844296316000580, −1.75604265231901161488080157701, −1.57485129763552112585278128696, −1.45593625052150332507840899975, −1.14812006596463241414543051839, −1.14399003837289476860163283854, −1.05966893290093877574874795402, −0.993133882555519771496658443771, 0, 0, 0, 0, 0, 0, 0.993133882555519771496658443771, 1.05966893290093877574874795402, 1.14399003837289476860163283854, 1.14812006596463241414543051839, 1.45593625052150332507840899975, 1.57485129763552112585278128696, 1.75604265231901161488080157701, 1.88473678397618844296316000580, 2.13871788240719464606428448281, 2.14335290582608573538349729594, 2.21005821148544698171767593448, 2.29132354648124925549451661848, 2.78833228768212124806687225520, 2.94821067728314056834174655503, 3.05826498489662531367607682491, 3.06670541933424248517123972505, 3.07704868381115422623052429474, 3.11661323787712253842128996497, 3.48899970533025603316507659978, 3.77607698088082297546524089098, 3.92547838835551180918145779714, 4.00506151924960621103519799465, 4.07189988402498180315282524767, 4.12181835363155525537829104018, 4.19138693475961006561153994029

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

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