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Properties

Label 12-1218e6-1.1-c1e6-0-3
Degree $12$
Conductor $3.265\times 10^{18}$
Sign $1$
Analytic cond. $846342.$
Root an. cond. $3.11861$
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  − 3·2-s + 3·3-s + 3·4-s − 5-s − 9·6-s + 2·7-s + 2·8-s + 3·9-s + 3·10-s + 11·11-s + 9·12-s + 10·13-s − 6·14-s − 3·15-s − 9·16-s − 6·17-s − 9·18-s + 4·19-s − 3·20-s + 6·21-s − 33·22-s + 14·23-s + 6·24-s + 7·25-s − 30·26-s − 2·27-s + 6·28-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.73·3-s + 3/2·4-s − 0.447·5-s − 3.67·6-s + 0.755·7-s + 0.707·8-s + 9-s + 0.948·10-s + 3.31·11-s + 2.59·12-s + 2.77·13-s − 1.60·14-s − 0.774·15-s − 9/4·16-s − 1.45·17-s − 2.12·18-s + 0.917·19-s − 0.670·20-s + 1.30·21-s − 7.03·22-s + 2.91·23-s + 1.22·24-s + 7/5·25-s − 5.88·26-s − 0.384·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 29^{6}\)
Sign: $1$
Analytic conductor: \(846342.\)
Root analytic conductor: \(3.11861\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 29^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.914853475\)
\(L(\frac12)\) \(\approx\) \(3.914853475\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{3} \)
3 \( ( 1 - T + T^{2} )^{3} \)
7 \( 1 - 2 T - 4 T^{2} + 13 T^{3} - 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
29 \( ( 1 - T )^{6} \)
good5 \( 1 + T - 6 T^{2} - 7 T^{3} + 7 T^{4} + 4 T^{5} + 29 T^{6} + 4 p T^{7} + 7 p^{2} T^{8} - 7 p^{3} T^{9} - 6 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - p T + 52 T^{2} - 205 T^{3} + 955 T^{4} - 3746 T^{5} + 12379 T^{6} - 3746 p T^{7} + 955 p^{2} T^{8} - 205 p^{3} T^{9} + 52 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - 5 T + 37 T^{2} - 105 T^{3} + 37 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 6 T + 18 T^{2} + 26 T^{3} - 228 T^{4} - 1200 T^{5} - 4405 T^{6} - 1200 p T^{7} - 228 p^{2} T^{8} + 26 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 4 T - 2 p T^{2} + 70 T^{3} + 1318 T^{4} - 958 T^{5} - 27013 T^{6} - 958 p T^{7} + 1318 p^{2} T^{8} + 70 p^{3} T^{9} - 2 p^{5} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 14 T + 70 T^{2} - 350 T^{3} + 3080 T^{4} - 15344 T^{5} + 55351 T^{6} - 15344 p T^{7} + 3080 p^{2} T^{8} - 350 p^{3} T^{9} + 70 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 4 T - 26 T^{2} + 6 T^{3} + 138 T^{4} - 3170 T^{5} - 3917 T^{6} - 3170 p T^{7} + 138 p^{2} T^{8} + 6 p^{3} T^{9} - 26 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T - 64 T^{2} - 57 T^{3} + 2385 T^{4} + 6186 T^{5} - 101627 T^{6} + 6186 p T^{7} + 2385 p^{2} T^{8} - 57 p^{3} T^{9} - 64 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 9 T + 145 T^{2} + 753 T^{3} + 145 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 13 T + 104 T^{2} + 593 T^{3} + 104 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 13 T + 42 T^{2} - 89 T^{3} + 7 T^{4} + 27290 T^{5} - 313777 T^{6} + 27290 p T^{7} + 7 p^{2} T^{8} - 89 p^{3} T^{9} + 42 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 14 T + 69 T^{2} - 2 p T^{3} - 2306 T^{4} + 42970 T^{5} - 408751 T^{6} + 42970 p T^{7} - 2306 p^{2} T^{8} - 2 p^{4} T^{9} + 69 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 6 T + 3 T^{2} + 78 T^{3} - 1218 T^{4} - 498 T^{5} + 224791 T^{6} - 498 p T^{7} - 1218 p^{2} T^{8} + 78 p^{3} T^{9} + 3 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 4 T - 4 T^{2} + 1078 T^{3} + 1176 T^{4} - 4856 T^{5} + 619027 T^{6} - 4856 p T^{7} + 1176 p^{2} T^{8} + 1078 p^{3} T^{9} - 4 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 13 T - 36 T^{2} + 557 T^{3} + 7347 T^{4} - 31502 T^{5} - 377469 T^{6} - 31502 p T^{7} + 7347 p^{2} T^{8} + 557 p^{3} T^{9} - 36 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 3 T - 35 T^{2} - 323 T^{3} - 35 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 28 T + 312 T^{2} + 3570 T^{3} + 53284 T^{4} + 496132 T^{5} + 3598195 T^{6} + 496132 p T^{7} + 53284 p^{2} T^{8} + 3570 p^{3} T^{9} + 312 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 19 T + 24 T^{2} - 85 T^{3} + 26925 T^{4} - 171488 T^{5} - 259593 T^{6} - 171488 p T^{7} + 26925 p^{2} T^{8} - 85 p^{3} T^{9} + 24 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 17 T + 341 T^{2} + 2981 T^{3} + 341 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - T - 128 T^{2} - 659 T^{3} + 5425 T^{4} + 49226 T^{5} - 8807 T^{6} + 49226 p T^{7} + 5425 p^{2} T^{8} - 659 p^{3} T^{9} - 128 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 19 T + 199 T^{2} - 2049 T^{3} + 199 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−4.95505271956347972845393064839, −4.90086264516558908516979465442, −4.80265928679068098277960439556, −4.59811593578704311069882198333, −4.44649628232464020801643687182, −4.25077564734728017661453446600, −4.03686977618381163545694369023, −4.02315593965714505970778289080, −3.64198581011193149341041135395, −3.48592430801842060732649323437, −3.47168821715820584541162485322, −3.41344023431757280052978817755, −3.12762003374910502542735978413, −3.10447387148629445679929437133, −2.57608268545088390021470729975, −2.46465825541917563362020814509, −2.38400391737216381801728315661, −1.99821810542839439763531229379, −1.59719285236951988228246975986, −1.46632351456051027209104541108, −1.37783500852319575769723788935, −1.34321052407580886418858018590, −0.990010509318177717414941178795, −0.865745794438448072171032053012, −0.33546266417842007548064355139, 0.33546266417842007548064355139, 0.865745794438448072171032053012, 0.990010509318177717414941178795, 1.34321052407580886418858018590, 1.37783500852319575769723788935, 1.46632351456051027209104541108, 1.59719285236951988228246975986, 1.99821810542839439763531229379, 2.38400391737216381801728315661, 2.46465825541917563362020814509, 2.57608268545088390021470729975, 3.10447387148629445679929437133, 3.12762003374910502542735978413, 3.41344023431757280052978817755, 3.47168821715820584541162485322, 3.48592430801842060732649323437, 3.64198581011193149341041135395, 4.02315593965714505970778289080, 4.03686977618381163545694369023, 4.25077564734728017661453446600, 4.44649628232464020801643687182, 4.59811593578704311069882198333, 4.80265928679068098277960439556, 4.90086264516558908516979465442, 4.95505271956347972845393064839

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

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