Properties

Label 2-6e3-216.133-c1-0-32
Degree $2$
Conductor $216$
Sign $-0.776 + 0.630i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.495 − 1.32i)2-s + (0.919 − 1.46i)3-s + (−1.50 − 1.31i)4-s + (−0.185 − 0.221i)5-s + (−1.48 − 1.94i)6-s + (−2.62 + 0.956i)7-s + (−2.48 + 1.35i)8-s + (−1.31 − 2.69i)9-s + (−0.384 + 0.136i)10-s + (3.08 − 3.67i)11-s + (−3.31 + 1.01i)12-s + (3.17 − 0.559i)13-s + (−0.0337 + 3.95i)14-s + (−0.495 + 0.0691i)15-s + (0.559 + 3.96i)16-s + (2.81 + 4.87i)17-s + ⋯
L(s)  = 1  + (0.350 − 0.936i)2-s + (0.530 − 0.847i)3-s + (−0.754 − 0.655i)4-s + (−0.0829 − 0.0988i)5-s + (−0.608 − 0.793i)6-s + (−0.993 + 0.361i)7-s + (−0.878 + 0.477i)8-s + (−0.436 − 0.899i)9-s + (−0.121 + 0.0431i)10-s + (0.928 − 1.10i)11-s + (−0.956 + 0.291i)12-s + (0.880 − 0.155i)13-s + (−0.00902 + 1.05i)14-s + (−0.127 + 0.0178i)15-s + (0.139 + 0.990i)16-s + (0.682 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.776 + 0.630i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.776 + 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.477233 - 1.34562i\)
\(L(\frac12)\) \(\approx\) \(0.477233 - 1.34562i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.495 + 1.32i)T \)
3 \( 1 + (-0.919 + 1.46i)T \)
good5 \( 1 + (0.185 + 0.221i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (2.62 - 0.956i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-3.08 + 3.67i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-3.17 + 0.559i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-2.81 - 4.87i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.00428 - 0.00247i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.84 - 0.670i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-7.88 - 1.38i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (5.45 + 1.98i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (8.20 - 4.73i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.69 - 9.63i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-3.57 + 4.26i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-3.48 + 1.26i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 0.865iT - 53T^{2} \)
59 \( 1 + (3.57 + 4.25i)T + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.43 - 6.68i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.224 - 0.0396i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.429 - 0.744i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.02 - 3.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.797 - 4.52i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (4.00 + 0.705i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (1.77 - 3.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.04 - 5.07i)T + (16.8 + 95.5i)T^{2} \)
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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

−12.20743456572758092254472024635, −11.19413192397274080610829288747, −10.04539542092457021204045588339, −8.873340401926588523140065025098, −8.396984206660400395197597396137, −6.48756751701630802867252285455, −5.87167089763375461222837451427, −3.78572001992104229528768570526, −2.99714663777096794783713650378, −1.19717242985219204005068597174, 3.23302017099548387380781409853, 4.10822710095548295491555609808, 5.31076182074660824812314045747, 6.70784625689854960368259905523, 7.49065421790106468634419977357, 8.969812321127414059241891230446, 9.403718845668240561603242060394, 10.49094838076683228891514552363, 11.94281648984450777208171469750, 12.94367052461307639607613453584

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