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Properties

Label 2-6e3-24.5-c2-0-1
Degree $2$
Conductor $216$
Sign $-0.804 + 0.594i$
Analytic cond. $5.88557$
Root an. cond. $2.42602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.420 + 1.95i)2-s + (−3.64 + 1.64i)4-s + 2.22·5-s − 8.64·7-s + (−4.75 − 6.43i)8-s + (0.937 + 4.35i)10-s − 19.8·11-s + 14.5i·13-s + (−3.63 − 16.9i)14-s + (10.5 − 12i)16-s − 5.29i·17-s + 24.2i·19-s + (−8.11 + 3.66i)20-s + (−8.35 − 38.8i)22-s − 29.6i·23-s + ⋯
L(s)  = 1  + (0.210 + 0.977i)2-s + (−0.911 + 0.411i)4-s + 0.445·5-s − 1.23·7-s + (−0.594 − 0.804i)8-s + (0.0937 + 0.435i)10-s − 1.80·11-s + 1.11i·13-s + (−0.259 − 1.20i)14-s + (0.661 − 0.750i)16-s − 0.311i·17-s + 1.27i·19-s + (−0.405 + 0.183i)20-s + (−0.379 − 1.76i)22-s − 1.28i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.594i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.804 + 0.594i$
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: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.804 + 0.594i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.145493 - 0.441963i\)
\(L(\frac12)\) \(\approx\) \(0.145493 - 0.441963i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.420 - 1.95i)T \)
3 \( 1 \)
good5 \( 1 - 2.22T + 25T^{2} \)
7 \( 1 + 8.64T + 49T^{2} \)
11 \( 1 + 19.8T + 121T^{2} \)
13 \( 1 - 14.5iT - 169T^{2} \)
17 \( 1 + 5.29iT - 289T^{2} \)
19 \( 1 - 24.2iT - 361T^{2} \)
23 \( 1 + 29.6iT - 529T^{2} \)
29 \( 1 - 6.13T + 841T^{2} \)
31 \( 1 - 10.1T + 961T^{2} \)
37 \( 1 - 16.6iT - 1.36e3T^{2} \)
41 \( 1 - 7.82iT - 1.68e3T^{2} \)
43 \( 1 + 1.54iT - 1.84e3T^{2} \)
47 \( 1 - 63.7iT - 2.20e3T^{2} \)
53 \( 1 - 76.5T + 2.80e3T^{2} \)
59 \( 1 + 41.0T + 3.48e3T^{2} \)
61 \( 1 - 52.0iT - 3.72e3T^{2} \)
67 \( 1 - 110. iT - 4.48e3T^{2} \)
71 \( 1 + 100. iT - 5.04e3T^{2} \)
73 \( 1 + 93.9T + 5.32e3T^{2} \)
79 \( 1 + 10.7T + 6.24e3T^{2} \)
83 \( 1 + 118.T + 6.88e3T^{2} \)
89 \( 1 - 146. iT - 7.92e3T^{2} \)
97 \( 1 + 3.70T + 9.40e3T^{2} \)
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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

−12.95179893277547261102562186774, −12.08833719794720673879887463766, −10.32868901972206522146646597145, −9.723563410983025787693277053879, −8.592478898201329852968229911715, −7.53096300793785276193340013922, −6.44099349888877889706148924840, −5.65280643322952554126339626074, −4.33009830852515080798813120186, −2.81067849036606813900657988214, 0.22511907150119761399687066159, 2.47265017319130347722259665676, 3.40256915511145817299184563436, 5.12646651294273380199777142434, 5.92483369134512524143494734460, 7.57121550118099081331227757369, 8.834521051593439305028716047872, 9.977878549077725118530836542024, 10.32867105622198022409109114526, 11.46725423344344403847910674216

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