Properties

Label 12-5225e6-1.1-c1e6-0-1
Degree $12$
Conductor $2.035\times 10^{22}$
Sign $1$
Analytic cond. $5.27448\times 10^{9}$
Root an. cond. $6.45924$
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  + 2·2-s + 3-s − 2·4-s + 2·6-s − 5·7-s − 4·8-s − 8·9-s − 6·11-s − 2·12-s + 5·13-s − 10·14-s + 5·16-s − 17-s − 16·18-s − 6·19-s − 5·21-s − 12·22-s − 4·23-s − 4·24-s + 10·26-s − 4·27-s + 10·28-s − 9·29-s − 21·31-s − 32-s − 6·33-s − 2·34-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s − 4-s + 0.816·6-s − 1.88·7-s − 1.41·8-s − 8/3·9-s − 1.80·11-s − 0.577·12-s + 1.38·13-s − 2.67·14-s + 5/4·16-s − 0.242·17-s − 3.77·18-s − 1.37·19-s − 1.09·21-s − 2.55·22-s − 0.834·23-s − 0.816·24-s + 1.96·26-s − 0.769·27-s + 1.88·28-s − 1.67·29-s − 3.77·31-s − 0.176·32-s − 1.04·33-s − 0.342·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(5^{12} \cdot 11^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(5.27448\times 10^{9}\)
Root analytic conductor: \(6.45924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 5^{12} \cdot 11^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad5 \( 1 \)
11 \( ( 1 + T )^{6} \)
19 \( ( 1 + T )^{6} \)
good2 \( 1 - p T + 3 p T^{2} - 3 p^{2} T^{3} + 23 T^{4} - 35 T^{5} + 59 T^{6} - 35 p T^{7} + 23 p^{2} T^{8} - 3 p^{5} T^{9} + 3 p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
3 \( 1 - T + p^{2} T^{2} - 13 T^{3} + 46 T^{4} - 73 T^{5} + 158 T^{6} - 73 p T^{7} + 46 p^{2} T^{8} - 13 p^{3} T^{9} + p^{6} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 5 T + 5 p T^{2} + 117 T^{3} + 486 T^{4} + 1257 T^{5} + 4062 T^{6} + 1257 p T^{7} + 486 p^{2} T^{8} + 117 p^{3} T^{9} + 5 p^{5} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 5 T + 71 T^{2} - 22 p T^{3} + 2193 T^{4} - 6950 T^{5} + 37416 T^{6} - 6950 p T^{7} + 2193 p^{2} T^{8} - 22 p^{4} T^{9} + 71 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + T + 29 T^{2} + 60 T^{3} + 597 T^{4} + 286 T^{5} + 706 p T^{6} + 286 p T^{7} + 597 p^{2} T^{8} + 60 p^{3} T^{9} + 29 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 4 T + 40 T^{2} + 301 T^{3} + 1789 T^{4} + 8268 T^{5} + 55396 T^{6} + 8268 p T^{7} + 1789 p^{2} T^{8} + 301 p^{3} T^{9} + 40 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 9 T + 133 T^{2} + 958 T^{3} + 7815 T^{4} + 47918 T^{5} + 9704 p T^{6} + 47918 p T^{7} + 7815 p^{2} T^{8} + 958 p^{3} T^{9} + 133 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 21 T + 317 T^{2} + 3422 T^{3} + 30173 T^{4} + 217138 T^{5} + 1321014 T^{6} + 217138 p T^{7} + 30173 p^{2} T^{8} + 3422 p^{3} T^{9} + 317 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T + 140 T^{2} + 67 T^{3} + 6768 T^{4} + 29689 T^{5} + 5898 p T^{6} + 29689 p T^{7} + 6768 p^{2} T^{8} + 67 p^{3} T^{9} + 140 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 23 T + 426 T^{2} + 5180 T^{3} + 54132 T^{4} + 439592 T^{5} + 3140244 T^{6} + 439592 p T^{7} + 54132 p^{2} T^{8} + 5180 p^{3} T^{9} + 426 p^{4} T^{10} + 23 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 7 T + 217 T^{2} + 1283 T^{3} + 20890 T^{4} + 101311 T^{5} + 1152762 T^{6} + 101311 p T^{7} + 20890 p^{2} T^{8} + 1283 p^{3} T^{9} + 217 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 18 T + 332 T^{2} - 3797 T^{3} + 41419 T^{4} - 339760 T^{5} + 2633636 T^{6} - 339760 p T^{7} + 41419 p^{2} T^{8} - 3797 p^{3} T^{9} + 332 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 17 T + 379 T^{2} - 4246 T^{3} + 52843 T^{4} - 431160 T^{5} + 3765806 T^{6} - 431160 p T^{7} + 52843 p^{2} T^{8} - 4246 p^{3} T^{9} + 379 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 29 T + 615 T^{2} + 8836 T^{3} + 106273 T^{4} + 1021136 T^{5} + 8614374 T^{6} + 1021136 p T^{7} + 106273 p^{2} T^{8} + 8836 p^{3} T^{9} + 615 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 17 T + 329 T^{2} - 4296 T^{3} + 48027 T^{4} - 474378 T^{5} + 3867132 T^{6} - 474378 p T^{7} + 48027 p^{2} T^{8} - 4296 p^{3} T^{9} + 329 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 8 T + 205 T^{2} + 1428 T^{3} + 24143 T^{4} + 153535 T^{5} + 1923192 T^{6} + 153535 p T^{7} + 24143 p^{2} T^{8} + 1428 p^{3} T^{9} + 205 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 12 T + 370 T^{2} + 2983 T^{3} + 53939 T^{4} + 322960 T^{5} + 4638908 T^{6} + 322960 p T^{7} + 53939 p^{2} T^{8} + 2983 p^{3} T^{9} + 370 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 2 T + 367 T^{2} + 726 T^{3} + 60373 T^{4} + 105223 T^{5} + 5677006 T^{6} + 105223 p T^{7} + 60373 p^{2} T^{8} + 726 p^{3} T^{9} + 367 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 3 T + 240 T^{2} - 947 T^{3} + 27448 T^{4} - 159411 T^{5} + 2355446 T^{6} - 159411 p T^{7} + 27448 p^{2} T^{8} - 947 p^{3} T^{9} + 240 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 11 T + 253 T^{2} - 2614 T^{3} + 28281 T^{4} - 304912 T^{5} + 2269804 T^{6} - 304912 p T^{7} + 28281 p^{2} T^{8} - 2614 p^{3} T^{9} + 253 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 22 T + 599 T^{2} + 8914 T^{3} + 137429 T^{4} + 1507023 T^{5} + 16386228 T^{6} + 1507023 p T^{7} + 137429 p^{2} T^{8} + 8914 p^{3} T^{9} + 599 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 2 T + 213 T^{2} - 416 T^{3} + 34651 T^{4} - 51915 T^{5} + 3572138 T^{6} - 51915 p T^{7} + 34651 p^{2} T^{8} - 416 p^{3} T^{9} + 213 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.53523628040975980854970776496, −4.47238633013134364037590610893, −4.28704067477380519447870042436, −4.26044184100335858799576816521, −3.95002777299341442359056854829, −3.88679971840672550853059722991, −3.70961997032124795321536934594, −3.50259288105516162046199928074, −3.48347513890674881295383752611, −3.42405242702127160038685865990, −3.40950683382318976703046372339, −3.28770619104718461646965726090, −3.11325271803697874964856763875, −2.83153788009659822346453478972, −2.62454211503532306998599327189, −2.48324365131056809851542005958, −2.46117008658273751582501103606, −2.27893937685491490671310919376, −1.97316608872787410540804979640, −1.89773313807452550415470649086, −1.86778494508366795015283438128, −1.58074256803024119440923740029, −1.11900554281075984487679112186, −1.10110698003327743606222080187, −0.997055640129493686873125897135, 0, 0, 0, 0, 0, 0, 0.997055640129493686873125897135, 1.10110698003327743606222080187, 1.11900554281075984487679112186, 1.58074256803024119440923740029, 1.86778494508366795015283438128, 1.89773313807452550415470649086, 1.97316608872787410540804979640, 2.27893937685491490671310919376, 2.46117008658273751582501103606, 2.48324365131056809851542005958, 2.62454211503532306998599327189, 2.83153788009659822346453478972, 3.11325271803697874964856763875, 3.28770619104718461646965726090, 3.40950683382318976703046372339, 3.42405242702127160038685865990, 3.48347513890674881295383752611, 3.50259288105516162046199928074, 3.70961997032124795321536934594, 3.88679971840672550853059722991, 3.95002777299341442359056854829, 4.26044184100335858799576816521, 4.28704067477380519447870042436, 4.47238633013134364037590610893, 4.53523628040975980854970776496

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

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