Properties

Label 2-6e3-24.5-c2-0-3
Degree $2$
Conductor $216$
Sign $-0.790 - 0.612i$
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  + (−1.90 + 0.598i)2-s + (3.28 − 2.28i)4-s − 2.62·5-s + 0.432·7-s + (−4.90 + 6.32i)8-s + (5 − 1.56i)10-s − 5.01·11-s + 1.56i·13-s + (−0.824 + 0.258i)14-s + (5.56 − 15i)16-s + 19.8i·17-s + 24.1i·19-s + (−8.60 + 5.98i)20-s + (9.56 − 3i)22-s − 3.07i·23-s + ⋯
L(s)  = 1  + (−0.954 + 0.299i)2-s + (0.820 − 0.570i)4-s − 0.524·5-s + 0.0617·7-s + (−0.612 + 0.790i)8-s + (0.5 − 0.156i)10-s − 0.455·11-s + 0.120i·13-s + (−0.0589 + 0.0184i)14-s + (0.347 − 0.937i)16-s + 1.16i·17-s + 1.27i·19-s + (−0.430 + 0.299i)20-s + (0.434 − 0.136i)22-s − 0.133i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.790 - 0.612i$
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: \((2,\ 216,\ (\ :1),\ -0.790 - 0.612i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.136901 + 0.400164i\)
\(L(\frac12)\) \(\approx\) \(0.136901 + 0.400164i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.90 - 0.598i)T \)
3 \( 1 \)
good5 \( 1 + 2.62T + 25T^{2} \)
7 \( 1 - 0.432T + 49T^{2} \)
11 \( 1 + 5.01T + 121T^{2} \)
13 \( 1 - 1.56iT - 169T^{2} \)
17 \( 1 - 19.8iT - 289T^{2} \)
19 \( 1 - 24.1iT - 361T^{2} \)
23 \( 1 + 3.07iT - 529T^{2} \)
29 \( 1 + 34.4T + 841T^{2} \)
31 \( 1 + 43.4T + 961T^{2} \)
37 \( 1 - 52.9iT - 1.36e3T^{2} \)
41 \( 1 - 56.7iT - 1.68e3T^{2} \)
43 \( 1 - 27.6iT - 1.84e3T^{2} \)
47 \( 1 + 83.7iT - 2.20e3T^{2} \)
53 \( 1 - 41.4T + 2.80e3T^{2} \)
59 \( 1 - 74.2T + 3.48e3T^{2} \)
61 \( 1 - 28.7iT - 3.72e3T^{2} \)
67 \( 1 + 33.8iT - 4.48e3T^{2} \)
71 \( 1 + 104. iT - 5.04e3T^{2} \)
73 \( 1 - 53.2T + 5.32e3T^{2} \)
79 \( 1 - 51.8T + 6.24e3T^{2} \)
83 \( 1 - 76.3T + 6.88e3T^{2} \)
89 \( 1 - 131. iT - 7.92e3T^{2} \)
97 \( 1 - 68.9T + 9.40e3T^{2} \)
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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

−12.21967430480879175357890269961, −11.31719155711757706059864513440, −10.42290924044820817247717266143, −9.545892853867966090590837281507, −8.282474557600761391596650001546, −7.78267437981032314650896519798, −6.51593465774239798999949910561, −5.42918215267009956520977877588, −3.66358026538266707944747699787, −1.79444983368095650296312115114, 0.30496554667058034169939045768, 2.35169300117927240355623629578, 3.76614386662994033572047628249, 5.45790568297386065633495761530, 7.09331562742726761962831125703, 7.64759576562718712132000952651, 8.897409456012420967121711045041, 9.601569913454119223151217122580, 10.90063537888643994622450820872, 11.38662402974338052696511502296

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