Properties

Label 12-585e6-1.1-c1e6-0-2
Degree $12$
Conductor $4.008\times 10^{16}$
Sign $1$
Analytic cond. $10389.5$
Root an. cond. $2.16130$
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  + 4-s + 12·11-s − 2·16-s + 25-s + 12·29-s − 20·31-s + 8·41-s + 12·44-s + 30·49-s − 16·59-s + 12·61-s − 10·64-s + 24·71-s + 32·79-s + 20·89-s + 100-s + 4·101-s + 48·109-s + 12·116-s + 26·121-s − 20·124-s + 16·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  + 1/2·4-s + 3.61·11-s − 1/2·16-s + 1/5·25-s + 2.22·29-s − 3.59·31-s + 1.24·41-s + 1.80·44-s + 30/7·49-s − 2.08·59-s + 1.53·61-s − 5/4·64-s + 2.84·71-s + 3.60·79-s + 2.11·89-s + 1/10·100-s + 0.398·101-s + 4.59·109-s + 1.11·116-s + 2.36·121-s − 1.79·124-s + 1.43·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 5^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(10389.5\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{585} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 5^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.443919240\)
\(L(\frac12)\) \(\approx\) \(7.443919240\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T^{2} - 16 T^{3} - p T^{4} + p^{3} T^{6} \)
13 \( ( 1 + T^{2} )^{3} \)
good2 \( 1 - T^{2} + 3 T^{4} + 5 T^{6} + 3 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \)
7 \( 1 - 30 T^{2} + 431 T^{4} - 3764 T^{6} + 431 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \)
11 \( ( 1 - 6 T + 41 T^{2} - 130 T^{3} + 41 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 58 T^{2} + 1455 T^{4} - 25708 T^{6} + 1455 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 + 53 T^{2} + 2 T^{3} + 53 p T^{4} + p^{3} T^{6} )^{2} \)
23 \( 1 - 66 T^{2} + 2747 T^{4} - 73472 T^{6} + 2747 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 6 T + 51 T^{2} - 240 T^{3} + 51 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 + 10 T + 113 T^{2} + 594 T^{3} + 113 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 166 T^{2} + 13031 T^{4} - 608388 T^{6} + 13031 p^{2} T^{8} - 166 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 - 4 T + 91 T^{2} - 360 T^{3} + 91 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 130 T^{2} + 11171 T^{4} - 561696 T^{6} + 11171 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 238 T^{2} + 25247 T^{4} - 1528980 T^{6} + 25247 p^{2} T^{8} - 238 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 174 T^{2} + 18071 T^{4} - 1143332 T^{6} + 18071 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 + 8 T + 137 T^{2} + 682 T^{3} + 137 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 6 T + 167 T^{2} - 736 T^{3} + 167 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 182 T^{2} + 24055 T^{4} - 1826084 T^{6} + 24055 p^{2} T^{8} - 182 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 12 T + 125 T^{2} - 950 T^{3} + 125 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 190 T^{2} + 23087 T^{4} - 2068020 T^{6} + 23087 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 - 16 T + 261 T^{2} - 2512 T^{3} + 261 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 318 T^{2} + 52775 T^{4} - 5422964 T^{6} + 52775 p^{2} T^{8} - 318 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 - 10 T + 215 T^{2} - 1580 T^{3} + 215 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 218 T^{2} + 12559 T^{4} - 119468 T^{6} + 12559 p^{2} T^{8} - 218 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.72469226326141289095688224520, −5.51557856025886068045755947593, −5.48330927181671786166445385368, −5.41931515407399546264626998436, −4.96214973605473106653766755151, −4.85239923400168241173715497957, −4.58219796518417560285471589218, −4.48921942771823225494957860959, −4.33846474712727425608845723228, −4.19878837782279341927976134552, −3.75331338795445326740238017521, −3.67585058022720638016301710304, −3.62283103810032928372709524319, −3.52238445067189867829646812452, −3.42719481109665123836523182031, −2.88363842836354740083778552999, −2.70272030596729739694972576361, −2.33820813853914405151011278517, −2.20532411060666277455594590042, −1.98745775113197713419034813024, −1.95419050851494371909103058201, −1.33175594654473272691716627954, −1.19725618245584263065815061063, −0.835257741726428431749845635275, −0.65263460360565691165992837320, 0.65263460360565691165992837320, 0.835257741726428431749845635275, 1.19725618245584263065815061063, 1.33175594654473272691716627954, 1.95419050851494371909103058201, 1.98745775113197713419034813024, 2.20532411060666277455594590042, 2.33820813853914405151011278517, 2.70272030596729739694972576361, 2.88363842836354740083778552999, 3.42719481109665123836523182031, 3.52238445067189867829646812452, 3.62283103810032928372709524319, 3.67585058022720638016301710304, 3.75331338795445326740238017521, 4.19878837782279341927976134552, 4.33846474712727425608845723228, 4.48921942771823225494957860959, 4.58219796518417560285471589218, 4.85239923400168241173715497957, 4.96214973605473106653766755151, 5.41931515407399546264626998436, 5.48330927181671786166445385368, 5.51557856025886068045755947593, 5.72469226326141289095688224520

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

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