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Properties

Label 2-6e3-9.2-c2-0-0
Degree $2$
Conductor $216$
Sign $-0.999 - 0.0415i$
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  + (−3.44 + 1.98i)5-s + (−1.80 + 3.12i)7-s + (−11.8 − 6.83i)11-s + (−8.96 − 15.5i)13-s + 21.6i·17-s − 23.4·19-s + (−13.0 + 7.54i)23-s + (−4.59 + 7.96i)25-s + (20.4 + 11.8i)29-s + (−23.5 − 40.7i)31-s − 14.3i·35-s + 54.6·37-s + (−24.6 + 14.2i)41-s + (−23.8 + 41.3i)43-s + (−30.5 − 17.6i)47-s + ⋯
L(s)  = 1  + (−0.688 + 0.397i)5-s + (−0.257 + 0.446i)7-s + (−1.07 − 0.621i)11-s + (−0.689 − 1.19i)13-s + 1.27i·17-s − 1.23·19-s + (−0.568 + 0.327i)23-s + (−0.183 + 0.318i)25-s + (0.706 + 0.407i)29-s + (−0.758 − 1.31i)31-s − 0.409i·35-s + 1.47·37-s + (−0.601 + 0.347i)41-s + (−0.555 + 0.961i)43-s + (−0.649 − 0.375i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0415i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.999 - 0.0415i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.999 - 0.0415i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00284135 + 0.136649i\)
\(L(\frac12)\) \(\approx\) \(0.00284135 + 0.136649i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (3.44 - 1.98i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (1.80 - 3.12i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (11.8 + 6.83i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (8.96 + 15.5i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 21.6iT - 289T^{2} \)
19 \( 1 + 23.4T + 361T^{2} \)
23 \( 1 + (13.0 - 7.54i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-20.4 - 11.8i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (23.5 + 40.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 54.6T + 1.36e3T^{2} \)
41 \( 1 + (24.6 - 14.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (23.8 - 41.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (30.5 + 17.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 65.0iT - 2.80e3T^{2} \)
59 \( 1 + (-76.0 + 43.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (6.46 - 11.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-1.55 - 2.69i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 49.5iT - 5.04e3T^{2} \)
73 \( 1 - 102.T + 5.32e3T^{2} \)
79 \( 1 + (18.7 - 32.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (14.6 + 8.48i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 14.1iT - 7.92e3T^{2} \)
97 \( 1 + (24.1 - 41.8i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.78001120958443268416405521655, −11.50174005401180719501062434835, −10.67889985686841288406395580594, −9.820472687680235315683546703873, −8.262311035242833120967873161814, −7.86039134819743415734723359618, −6.36213865553359768688186574687, −5.34076172212214225075655792566, −3.78287520451304394684977145696, −2.55650683163885217225789943043, 0.07022404959542682344813204512, 2.39292226703255033597477585824, 4.16103808126711847490594391816, 4.96975423254316763118185470601, 6.68151370742530516952276250476, 7.52947701651394920153812779906, 8.575301700155686234481007705999, 9.717169840896285555621939220868, 10.59444074204400292305681027073, 11.80030208698239262590336938478

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