Properties

Label 12-825e6-1.1-c1e6-0-1
Degree $12$
Conductor $3.153\times 10^{17}$
Sign $1$
Analytic cond. $81730.9$
Root an. cond. $2.56664$
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  − 2·4-s − 3·9-s + 6·11-s − 16-s + 2·19-s − 4·29-s + 34·31-s + 6·36-s + 8·41-s − 12·44-s + 19·49-s + 12·59-s − 6·61-s + 52·71-s − 4·76-s − 12·79-s + 6·81-s − 4·89-s − 18·99-s − 4·101-s − 6·109-s + 8·116-s + 21·121-s − 68·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 4-s − 9-s + 1.80·11-s − 1/4·16-s + 0.458·19-s − 0.742·29-s + 6.10·31-s + 36-s + 1.24·41-s − 1.80·44-s + 19/7·49-s + 1.56·59-s − 0.768·61-s + 6.17·71-s − 0.458·76-s − 1.35·79-s + 2/3·81-s − 0.423·89-s − 1.80·99-s − 0.398·101-s − 0.574·109-s + 0.742·116-s + 1.90·121-s − 6.10·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{12} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(81730.9\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{12} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.256208539\)
\(L(\frac12)\) \(\approx\) \(5.256208539\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T^{2} )^{3} \)
5 \( 1 \)
11 \( ( 1 - T )^{6} \)
good2 \( 1 + p T^{2} + 5 T^{4} + 3 p^{2} T^{6} + 5 p^{2} T^{8} + p^{5} T^{10} + p^{6} T^{12} \)
7 \( 1 - 19 T^{2} + 242 T^{4} - 1923 T^{6} + 242 p^{2} T^{8} - 19 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 - 53 T^{2} + 1315 T^{4} - 20606 T^{6} + 1315 p^{2} T^{8} - 53 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 36 T^{2} + 296 T^{4} + 1402 T^{6} + 296 p^{2} T^{8} - 36 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 - T + 2 p T^{2} - 63 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 52 T^{2} + 1872 T^{4} - 52066 T^{6} + 1872 p^{2} T^{8} - 52 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 + 2 T + 63 T^{2} + 132 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 - 17 T + 181 T^{2} - 1190 T^{3} + 181 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 72 T^{2} + 3960 T^{4} - 196054 T^{6} + 3960 p^{2} T^{8} - 72 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 - 4 T + 122 T^{2} - 326 T^{3} + 122 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 + 95 T^{2} + 5443 T^{4} + 231578 T^{6} + 5443 p^{2} T^{8} + 95 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 232 T^{2} + 24200 T^{4} - 1457406 T^{6} + 24200 p^{2} T^{8} - 232 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 + 14 T^{2} + 3543 T^{4} - 72988 T^{6} + 3543 p^{2} T^{8} + 14 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 - 6 T + 146 T^{2} - 572 T^{3} + 146 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 3 T + 95 T^{2} + 10 p T^{3} + 95 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 + 31 T^{2} + 10387 T^{4} + 250042 T^{6} + 10387 p^{2} T^{8} + 31 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 26 T + 430 T^{2} - 4272 T^{3} + 430 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 250 T^{2} + 36415 T^{4} - 3207340 T^{6} + 36415 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 6 T + 200 T^{2} + 736 T^{3} + 200 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 30 T^{2} + 10551 T^{4} - 8652 p T^{6} + 10551 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 + 2 T + 167 T^{2} + 28 T^{3} + 167 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 + 109 T^{2} + 3526 T^{4} + 74857 T^{6} + 3526 p^{2} T^{8} + 109 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.40650040803966792130461646539, −5.21840213223943142923554578482, −5.09527266976550820687975804697, −4.98381156796781296434293067236, −4.63145756213600638102151179305, −4.58765595344754486774577922356, −4.40477711544612496636891062686, −4.23043069662593306223666013062, −4.21327859517911766577734547427, −3.97935231045098896335378031425, −3.77309870844691475126191373343, −3.52121541720560906173723089509, −3.42886479257348325944268052734, −3.15688940903444291727880181408, −2.85306763793579772075483243932, −2.83294617430316222245350672914, −2.48770765736175355205274650945, −2.32648292819594252770638668375, −2.32285850768391606188067586596, −1.76338113636392119566882099693, −1.68729420040418093409655607667, −1.04798408301246304867224940283, −0.870957704152696615837526597776, −0.789668403743869964662877748110, −0.56744977828891505152320765071, 0.56744977828891505152320765071, 0.789668403743869964662877748110, 0.870957704152696615837526597776, 1.04798408301246304867224940283, 1.68729420040418093409655607667, 1.76338113636392119566882099693, 2.32285850768391606188067586596, 2.32648292819594252770638668375, 2.48770765736175355205274650945, 2.83294617430316222245350672914, 2.85306763793579772075483243932, 3.15688940903444291727880181408, 3.42886479257348325944268052734, 3.52121541720560906173723089509, 3.77309870844691475126191373343, 3.97935231045098896335378031425, 4.21327859517911766577734547427, 4.23043069662593306223666013062, 4.40477711544612496636891062686, 4.58765595344754486774577922356, 4.63145756213600638102151179305, 4.98381156796781296434293067236, 5.09527266976550820687975804697, 5.21840213223943142923554578482, 5.40650040803966792130461646539

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

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