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Properties

Label 2-6e3-27.7-c1-0-8
Degree $2$
Conductor $216$
Sign $-0.149 + 0.988i$
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  + (0.824 − 1.52i)3-s + (−0.265 − 1.50i)5-s + (−3.27 − 2.75i)7-s + (−1.64 − 2.51i)9-s + (−0.469 + 2.66i)11-s + (3.23 + 1.17i)13-s + (−2.51 − 0.836i)15-s + (2.63 + 4.55i)17-s + (3.51 − 6.09i)19-s + (−6.89 + 2.72i)21-s + (0.888 − 0.745i)23-s + (2.50 − 0.910i)25-s + (−5.17 + 0.429i)27-s + (−0.981 + 0.357i)29-s + (4.26 − 3.57i)31-s + ⋯
L(s)  = 1  + (0.475 − 0.879i)3-s + (−0.118 − 0.673i)5-s + (−1.23 − 1.03i)7-s + (−0.547 − 0.837i)9-s + (−0.141 + 0.802i)11-s + (0.896 + 0.326i)13-s + (−0.648 − 0.216i)15-s + (0.638 + 1.10i)17-s + (0.806 − 1.39i)19-s + (−1.50 + 0.595i)21-s + (0.185 − 0.155i)23-s + (0.500 − 0.182i)25-s + (−0.996 + 0.0826i)27-s + (−0.182 + 0.0663i)29-s + (0.765 − 0.642i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.149 + 0.988i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.149 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.788485 - 0.916545i\)
\(L(\frac12)\) \(\approx\) \(0.788485 - 0.916545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.824 + 1.52i)T \)
good5 \( 1 + (0.265 + 1.50i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (3.27 + 2.75i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.469 - 2.66i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-3.23 - 1.17i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.63 - 4.55i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.51 + 6.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.888 + 0.745i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.981 - 0.357i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-4.26 + 3.57i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (0.0292 + 0.0506i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.59 - 3.49i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.26 - 7.18i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (4.71 + 3.95i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 8.72T + 53T^{2} \)
59 \( 1 + (-1.90 - 10.8i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (4.25 + 3.57i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (9.30 + 3.38i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-6.09 - 10.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.371 - 0.643i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.26 - 0.460i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (3.87 - 1.41i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-2.28 + 3.95i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.745 + 4.22i)T + (-91.1 - 33.1i)T^{2} \)
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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

−12.43580826920616254045270459582, −11.19159731965428641684788147069, −9.903186421998714310204818227612, −9.087274300323407221940208995218, −7.959849247097273788110511131945, −7.00430313694474105474008087282, −6.14864142372598245805480209034, −4.34370483827792289984883819929, −3.07733440819484659583596706662, −1.06898952719931238338633012286, 2.98180530265377504235077218787, 3.42604354302727521933205352254, 5.38180501466206063512468749560, 6.22182168574393883805630899759, 7.74805616676822418355748861024, 8.853220601067429564194133389769, 9.634369336846036292632675360926, 10.51246315320689724140884924033, 11.48692692129205239714380978093, 12.55349935294587442548109091775

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