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Properties

Label 2-6e3-72.67-c2-0-8
Degree $2$
Conductor $216$
Sign $0.920 - 0.391i$
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  + (−1.94 − 0.470i)2-s + (3.55 + 1.82i)4-s + (5.15 + 2.97i)5-s + (4.09 − 2.36i)7-s + (−6.05 − 5.23i)8-s + (−8.62 − 8.21i)10-s + (6.94 + 12.0i)11-s + (−4.03 − 2.32i)13-s + (−9.06 + 2.66i)14-s + (9.30 + 13.0i)16-s − 21.5·17-s + 3.83·19-s + (12.8 + 20.0i)20-s + (−7.84 − 26.6i)22-s + (30.0 + 17.3i)23-s + ⋯
L(s)  = 1  + (−0.971 − 0.235i)2-s + (0.889 + 0.457i)4-s + (1.03 + 0.595i)5-s + (0.584 − 0.337i)7-s + (−0.756 − 0.653i)8-s + (−0.862 − 0.821i)10-s + (0.631 + 1.09i)11-s + (−0.310 − 0.179i)13-s + (−0.647 + 0.190i)14-s + (0.581 + 0.813i)16-s − 1.26·17-s + 0.201·19-s + (0.644 + 1.00i)20-s + (−0.356 − 1.21i)22-s + (1.30 + 0.753i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.920 - 0.391i$
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: \((2,\ 216,\ (\ :1),\ 0.920 - 0.391i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.25112 + 0.255261i\)
\(L(\frac12)\) \(\approx\) \(1.25112 + 0.255261i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.94 + 0.470i)T \)
3 \( 1 \)
good5 \( 1 + (-5.15 - 2.97i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-4.09 + 2.36i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-6.94 - 12.0i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (4.03 + 2.32i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 21.5T + 289T^{2} \)
19 \( 1 - 3.83T + 361T^{2} \)
23 \( 1 + (-30.0 - 17.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-39.3 + 22.7i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-31.8 - 18.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 36.0iT - 1.36e3T^{2} \)
41 \( 1 + (-10.2 + 17.7i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-3.50 - 6.07i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (53.1 - 30.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 58.7iT - 2.80e3T^{2} \)
59 \( 1 + (11.0 - 19.1i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-47.1 + 27.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-56.9 + 98.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 8.30iT - 5.04e3T^{2} \)
73 \( 1 + 114.T + 5.32e3T^{2} \)
79 \( 1 + (47.5 - 27.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (15.6 + 27.0i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 47.7T + 7.92e3T^{2} \)
97 \( 1 + (28.7 + 49.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.84847795780954827601541987422, −10.99809447705013464653159878239, −10.05253841657062570170913111898, −9.468823550573533587986086728260, −8.289428269793184867934285360670, −7.04880186591990388592994819680, −6.42564240395827958223009046625, −4.67576904590698090190196010126, −2.75790150557272631376632380581, −1.54307472146719442230621101758, 1.12467962462640640669255902612, 2.54039503126094571130831681085, 4.89116635870216218586813461177, 5.99595650859845514264965555485, 6.90723341842053267021732329523, 8.548906104068522027062016927021, 8.830229791621549845438228014678, 9.875630602928336076082527415091, 10.97713348022698110415223713599, 11.71575291376239820928071709535

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