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Properties

Label 2-6e3-72.67-c2-0-5
Degree $2$
Conductor $216$
Sign $0.510 - 0.860i$
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  + (−0.827 + 1.82i)2-s + (−2.63 − 3.01i)4-s + (−3.84 − 2.22i)5-s + (0.704 − 0.406i)7-s + (7.66 − 2.30i)8-s + (7.22 − 5.16i)10-s + (3.72 + 6.44i)11-s + (18.0 + 10.4i)13-s + (0.157 + 1.61i)14-s + (−2.14 + 15.8i)16-s − 1.74·17-s + 31.7·19-s + (3.43 + 17.4i)20-s + (−14.8 + 1.44i)22-s + (6.44 + 3.72i)23-s + ⋯
L(s)  = 1  + (−0.413 + 0.910i)2-s + (−0.658 − 0.753i)4-s + (−0.769 − 0.444i)5-s + (0.100 − 0.0580i)7-s + (0.957 − 0.287i)8-s + (0.722 − 0.516i)10-s + (0.338 + 0.585i)11-s + (1.39 + 0.803i)13-s + (0.0112 + 0.115i)14-s + (−0.134 + 0.990i)16-s − 0.102·17-s + 1.67·19-s + (0.171 + 0.871i)20-s + (−0.673 + 0.0657i)22-s + (0.280 + 0.161i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.510 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.510 - 0.860i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.510 - 0.860i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.963049 + 0.548464i\)
\(L(\frac12)\) \(\approx\) \(0.963049 + 0.548464i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.827 - 1.82i)T \)
3 \( 1 \)
good5 \( 1 + (3.84 + 2.22i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-0.704 + 0.406i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-3.72 - 6.44i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-18.0 - 10.4i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 1.74T + 289T^{2} \)
19 \( 1 - 31.7T + 361T^{2} \)
23 \( 1 + (-6.44 - 3.72i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-26.9 + 15.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (4.91 + 2.83i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 62.0iT - 1.36e3T^{2} \)
41 \( 1 + (2.74 - 4.74i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (22.3 + 38.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-71.1 + 41.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 85.0iT - 2.80e3T^{2} \)
59 \( 1 + (-21.8 + 37.8i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-61.1 + 35.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-9.91 + 17.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 69.3iT - 5.04e3T^{2} \)
73 \( 1 + 21.7T + 5.32e3T^{2} \)
79 \( 1 + (37.1 - 21.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-3.53 - 6.11i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 37.1T + 7.92e3T^{2} \)
97 \( 1 + (-9.38 - 16.2i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.14119407462000271824191024656, −11.32352043206072732906865826109, −10.04370454234458608300232782891, −9.053287389826050037739261360752, −8.245523670709656844903617811236, −7.27794908736755621369166559811, −6.25735615584058714136019914526, −4.91084264356003342655020922408, −3.88093313866410155445745996768, −1.15850702098050869983194866713, 0.984975971216918852437915876225, 3.07345614954998887410283909298, 3.85546392234552083372259500932, 5.50268957890525651003048561567, 7.15558977265059917271895862718, 8.159963441108807408093478075829, 8.964405759370119601845598141043, 10.18774619233102588441968192791, 11.17012716000177440332475415604, 11.55923465176783506426182375592

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