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Properties

Label 12-3645e6-1.1-c0e6-0-4
Degree $12$
Conductor $2.345\times 10^{21}$
Sign $1$
Analytic cond. $36.2349$
Root an. cond. $1.34873$
Motivic weight $0$
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  − 2·8-s + 3·17-s + 3·19-s − 6·53-s + 64-s + 12·107-s − 6·109-s − 125-s + 127-s + 131-s − 6·136-s + 137-s + 139-s + 149-s + 151-s − 6·152-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2·8-s + 3·17-s + 3·19-s − 6·53-s + 64-s + 12·107-s − 6·109-s − 125-s + 127-s + 131-s − 6·136-s + 137-s + 139-s + 149-s + 151-s − 6·152-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{36} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(36.2349\)
Root analytic conductor: \(1.34873\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{36} \cdot 5^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.468018813\)
\(L(\frac12)\) \(\approx\) \(1.468018813\)
\(L(1)\) 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 \( 1 + T^{3} + T^{6} \)
good2 \( ( 1 + T^{3} + T^{6} )^{2} \)
7 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
11 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
23 \( ( 1 + T^{3} + T^{6} )^{2} \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
47 \( ( 1 + T^{3} + T^{6} )^{2} \)
53 \( ( 1 + T + T^{2} )^{6} \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 + T^{3} + T^{6} )^{2} \)
67 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
73 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
79 \( ( 1 + T^{3} + T^{6} )^{2} \)
83 \( ( 1 + T^{3} + T^{6} )^{2} \)
89 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
97 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
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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

−4.69484339025276265151916051614, −4.50076985377928656162454540485, −4.43340708462142710077882539718, −4.12012722682835764452730413348, −4.10432322931937498003178646044, −3.57099887046083271141538820372, −3.48699743339343711447598302716, −3.43627472177431147534552245088, −3.41266129674094107368156179787, −3.37762823176824652585141770970, −3.34804170001894437915050768054, −2.98791396666301419606232888925, −2.94774433366296085061769486561, −2.77051637224199330263393336784, −2.64299498112681813761917726903, −2.41455057050641743911004809742, −2.07099682339411435032142427070, −2.06688156982795190444875373392, −1.76840038390502273896940685863, −1.54330409116017070381441533477, −1.27545315398917443475644337942, −1.23515980323002493233666220642, −1.10804703631823423267509270960, −0.74330842508823620032946020676, −0.39039799677121155815876909475, 0.39039799677121155815876909475, 0.74330842508823620032946020676, 1.10804703631823423267509270960, 1.23515980323002493233666220642, 1.27545315398917443475644337942, 1.54330409116017070381441533477, 1.76840038390502273896940685863, 2.06688156982795190444875373392, 2.07099682339411435032142427070, 2.41455057050641743911004809742, 2.64299498112681813761917726903, 2.77051637224199330263393336784, 2.94774433366296085061769486561, 2.98791396666301419606232888925, 3.34804170001894437915050768054, 3.37762823176824652585141770970, 3.41266129674094107368156179787, 3.43627472177431147534552245088, 3.48699743339343711447598302716, 3.57099887046083271141538820372, 4.10432322931937498003178646044, 4.12012722682835764452730413348, 4.43340708462142710077882539718, 4.50076985377928656162454540485, 4.69484339025276265151916051614

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

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