Properties

Label 2-6e3-1.1-c3-0-7
Degree $2$
Conductor $216$
Sign $-1$
Analytic cond. $12.7444$
Root an. cond. $3.56993$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 15.4·5-s + 23.8·7-s + 14.2·11-s − 13.8·13-s − 80.5·17-s − 144.·19-s − 141.·23-s + 112.·25-s − 251.·29-s − 16.6·31-s − 367.·35-s + 305.·37-s + 429.·41-s − 181.·43-s − 79.4·47-s + 225.·49-s − 663.·53-s − 219.·55-s + 220.·59-s − 473.·61-s + 213.·65-s − 647.·67-s + 14.4·71-s + 776.·73-s + 339.·77-s − 257.·79-s + 1.28e3·83-s + ⋯
L(s)  = 1  − 1.37·5-s + 1.28·7-s + 0.390·11-s − 0.295·13-s − 1.14·17-s − 1.74·19-s − 1.27·23-s + 0.901·25-s − 1.60·29-s − 0.0965·31-s − 1.77·35-s + 1.35·37-s + 1.63·41-s − 0.644·43-s − 0.246·47-s + 0.655·49-s − 1.72·53-s − 0.538·55-s + 0.486·59-s − 0.993·61-s + 0.406·65-s − 1.18·67-s + 0.0242·71-s + 1.24·73-s + 0.502·77-s − 0.367·79-s + 1.69·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(12.7444\)
Root analytic conductor: \(3.56993\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 216,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 15.4T + 125T^{2} \)
7 \( 1 - 23.8T + 343T^{2} \)
11 \( 1 - 14.2T + 1.33e3T^{2} \)
13 \( 1 + 13.8T + 2.19e3T^{2} \)
17 \( 1 + 80.5T + 4.91e3T^{2} \)
19 \( 1 + 144.T + 6.85e3T^{2} \)
23 \( 1 + 141.T + 1.21e4T^{2} \)
29 \( 1 + 251.T + 2.43e4T^{2} \)
31 \( 1 + 16.6T + 2.97e4T^{2} \)
37 \( 1 - 305.T + 5.06e4T^{2} \)
41 \( 1 - 429.T + 6.89e4T^{2} \)
43 \( 1 + 181.T + 7.95e4T^{2} \)
47 \( 1 + 79.4T + 1.03e5T^{2} \)
53 \( 1 + 663.T + 1.48e5T^{2} \)
59 \( 1 - 220.T + 2.05e5T^{2} \)
61 \( 1 + 473.T + 2.26e5T^{2} \)
67 \( 1 + 647.T + 3.00e5T^{2} \)
71 \( 1 - 14.4T + 3.57e5T^{2} \)
73 \( 1 - 776.T + 3.89e5T^{2} \)
79 \( 1 + 257.T + 4.93e5T^{2} \)
83 \( 1 - 1.28e3T + 5.71e5T^{2} \)
89 \( 1 + 156.T + 7.04e5T^{2} \)
97 \( 1 - 1.16e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.28681890177975284825942107145, −10.86272433452038570441576687664, −9.217849848799679557240575756936, −8.163597681983709554458218544042, −7.63205285410961500652380795349, −6.27406080109797370552182757789, −4.60455133020779375551468754608, −4.00875344980959931260724918236, −2.02675066573573264484141549007, 0, 2.02675066573573264484141549007, 4.00875344980959931260724918236, 4.60455133020779375551468754608, 6.27406080109797370552182757789, 7.63205285410961500652380795349, 8.163597681983709554458218544042, 9.217849848799679557240575756936, 10.86272433452038570441576687664, 11.28681890177975284825942107145

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