Properties

Label 2-6e3-24.5-c2-0-15
Degree $2$
Conductor $216$
Sign $1$
Analytic cond. $5.88557$
Root an. cond. $2.42602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 7.48·5-s + 13.4·7-s − 8·8-s − 14.9·10-s − 11.9·11-s − 26.9·14-s + 16·16-s + 29.9·20-s + 23.9·22-s + 31.0·25-s + 53.9·28-s + 50·29-s − 61.4·31-s − 32·32-s + 100.·35-s − 59.8·40-s − 47.8·44-s + 132.·49-s − 62.0·50-s + 4.57·53-s − 89.6·55-s − 107.·56-s − 100·58-s − 10·59-s + 122.·62-s + ⋯
L(s)  = 1  − 2-s + 4-s + 1.49·5-s + 1.92·7-s − 8-s − 1.49·10-s − 1.08·11-s − 1.92·14-s + 16-s + 1.49·20-s + 1.08·22-s + 1.24·25-s + 1.92·28-s + 1.72·29-s − 1.98·31-s − 32-s + 2.88·35-s − 1.49·40-s − 1.08·44-s + 2.71·49-s − 1.24·50-s + 0.0862·53-s − 1.62·55-s − 1.92·56-s − 1.72·58-s − 0.169·59-s + 1.98·62-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (53, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.472184695\)
\(L(\frac12)\) \(\approx\) \(1.472184695\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 \)
good5 \( 1 - 7.48T + 25T^{2} \)
7 \( 1 - 13.4T + 49T^{2} \)
11 \( 1 + 11.9T + 121T^{2} \)
13 \( 1 - 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 - 50T + 841T^{2} \)
31 \( 1 + 61.4T + 961T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 - 4.57T + 2.80e3T^{2} \)
59 \( 1 + 10T + 3.48e3T^{2} \)
61 \( 1 - 3.72e3T^{2} \)
67 \( 1 - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 143.T + 5.32e3T^{2} \)
79 \( 1 + 58T + 6.24e3T^{2} \)
83 \( 1 - 17.8T + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 128.T + 9.40e3T^{2} \)
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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

−11.78715110271217362737285844752, −10.76037388878610014864115787410, −10.27978769367116423858490834813, −9.079673366848945864617047251226, −8.241060442128074863351176404074, −7.29507221141366825213814538117, −5.85841523828381449001317060469, −5.01327199838095338496976859364, −2.45819352788878694786123102535, −1.49416466079572913012717171047, 1.49416466079572913012717171047, 2.45819352788878694786123102535, 5.01327199838095338496976859364, 5.85841523828381449001317060469, 7.29507221141366825213814538117, 8.241060442128074863351176404074, 9.079673366848945864617047251226, 10.27978769367116423858490834813, 10.76037388878610014864115787410, 11.78715110271217362737285844752

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