Properties

Label 12-370e6-1.1-c1e6-0-1
Degree $12$
Conductor $2.566\times 10^{15}$
Sign $1$
Analytic cond. $665.078$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 3·4-s + 10·7-s + 6·9-s − 2·11-s + 12·12-s + 6·16-s − 40·21-s − 3·25-s − 4·27-s − 30·28-s + 8·33-s − 18·36-s − 10·37-s + 2·41-s + 6·44-s − 4·47-s − 24·48-s + 25·49-s + 2·53-s + 60·63-s − 10·64-s + 8·67-s − 20·71-s − 12·73-s + 12·75-s − 20·77-s + ⋯
L(s)  = 1  − 2.30·3-s − 3/2·4-s + 3.77·7-s + 2·9-s − 0.603·11-s + 3.46·12-s + 3/2·16-s − 8.72·21-s − 3/5·25-s − 0.769·27-s − 5.66·28-s + 1.39·33-s − 3·36-s − 1.64·37-s + 0.312·41-s + 0.904·44-s − 0.583·47-s − 3.46·48-s + 25/7·49-s + 0.274·53-s + 7.55·63-s − 5/4·64-s + 0.977·67-s − 2.37·71-s − 1.40·73-s + 1.38·75-s − 2.27·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(665.078\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 37^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7838104848\)
\(L(\frac12)\) \(\approx\) \(0.7838104848\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{3} \)
5 \( ( 1 + T^{2} )^{3} \)
37 \( 1 + 10 T + 79 T^{2} + 388 T^{3} + 79 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
good3 \( ( 1 + 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
7 \( ( 1 - 5 T + 25 T^{2} - 68 T^{3} + 25 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 + T + 17 T^{2} + 38 T^{3} + 17 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 42 T^{2} + 983 T^{4} - 15500 T^{6} + 983 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 21 T^{2} + 875 T^{4} - 11054 T^{6} + 875 p^{2} T^{8} - 21 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 - 78 T^{2} + 2939 T^{4} - 69068 T^{6} + 2939 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 - 102 T^{2} + 4943 T^{4} - 143540 T^{6} + 4943 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 - 125 T^{2} + 6987 T^{4} - 243758 T^{6} + 6987 p^{2} T^{8} - 125 p^{4} T^{10} + p^{6} T^{12} \)
31 \( 1 - 129 T^{2} + 8423 T^{4} - 327146 T^{6} + 8423 p^{2} T^{8} - 129 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 - T + 67 T^{2} + 90 T^{3} + 67 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 225 T^{2} + 22283 T^{4} - 1243046 T^{6} + 22283 p^{2} T^{8} - 225 p^{4} T^{10} + p^{6} T^{12} \)
47 \( ( 1 + 2 T + 127 T^{2} + 156 T^{3} + 127 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 - T + 55 T^{2} + 246 T^{3} + 55 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 258 T^{2} + 31835 T^{4} - 2371028 T^{6} + 31835 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 - 245 T^{2} + 30091 T^{4} - 2286398 T^{6} + 30091 p^{2} T^{8} - 245 p^{4} T^{10} + p^{6} T^{12} \)
67 \( ( 1 - 4 T + 95 T^{2} - 660 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 + 10 T + 229 T^{2} + 1404 T^{3} + 229 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 6 T + 119 T^{2} + 532 T^{3} + 119 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 394 T^{2} + 69907 T^{4} - 7109380 T^{6} + 69907 p^{2} T^{8} - 394 p^{4} T^{10} + p^{6} T^{12} \)
83 \( ( 1 + 10 T + 187 T^{2} + 1668 T^{3} + 187 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 278 T^{2} + 45087 T^{4} - 4769204 T^{6} + 45087 p^{2} T^{8} - 278 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 - 293 T^{2} + 35491 T^{4} - 3167342 T^{6} + 35491 p^{2} T^{8} - 293 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.80410064147084335077337122192, −5.80038710907015613689055800045, −5.73125675866419033039564940374, −5.72554692809842502880262909603, −5.43886068796977401790629280949, −5.25489651537270955427242124003, −5.09980658806681565630354670580, −5.00624569012031963966616369741, −4.68472604067084461177339499677, −4.44689520430763720914960289870, −4.44076266777878734100764604922, −4.43998740984593056475423863170, −4.34378041327204859113369277379, −3.84498652546583190383149282653, −3.46476932458775479750475337363, −3.33995885709405898999953572755, −3.03382614885322601685958042059, −2.92521037072475036209692270690, −2.38149182456606463498627883928, −1.76169980122040947232616136882, −1.75434877437549426617228037769, −1.75099906028526191248994240465, −1.44721676661157891359357735599, −0.57253874518889279046480561353, −0.52537967101425068786978737934, 0.52537967101425068786978737934, 0.57253874518889279046480561353, 1.44721676661157891359357735599, 1.75099906028526191248994240465, 1.75434877437549426617228037769, 1.76169980122040947232616136882, 2.38149182456606463498627883928, 2.92521037072475036209692270690, 3.03382614885322601685958042059, 3.33995885709405898999953572755, 3.46476932458775479750475337363, 3.84498652546583190383149282653, 4.34378041327204859113369277379, 4.43998740984593056475423863170, 4.44076266777878734100764604922, 4.44689520430763720914960289870, 4.68472604067084461177339499677, 5.00624569012031963966616369741, 5.09980658806681565630354670580, 5.25489651537270955427242124003, 5.43886068796977401790629280949, 5.72554692809842502880262909603, 5.73125675866419033039564940374, 5.80038710907015613689055800045, 5.80410064147084335077337122192

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

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