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Properties

Label 2-54-27.4-c1-0-1
Degree $2$
Conductor $54$
Sign $0.835 - 0.549i$
Analytic cond. $0.431192$
Root an. cond. $0.656652$
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  + (0.939 + 0.342i)2-s + (−1.11 + 1.32i)3-s + (0.766 + 0.642i)4-s + (0.439 − 2.49i)5-s + (−1.5 + 0.866i)6-s + (−1.79 + 1.50i)7-s + (0.500 + 0.866i)8-s + (−0.520 − 2.95i)9-s + (1.26 − 2.19i)10-s + (−0.745 − 4.22i)11-s + (−1.70 + 0.300i)12-s + (−0.713 + 0.259i)13-s + (−2.20 + 0.802i)14-s + (2.81 + 3.35i)15-s + (0.173 + 0.984i)16-s + (−2.46 + 4.26i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (−0.642 + 0.766i)3-s + (0.383 + 0.321i)4-s + (0.196 − 1.11i)5-s + (−0.612 + 0.353i)6-s + (−0.679 + 0.570i)7-s + (0.176 + 0.306i)8-s + (−0.173 − 0.984i)9-s + (0.400 − 0.693i)10-s + (−0.224 − 1.27i)11-s + (−0.492 + 0.0868i)12-s + (−0.197 + 0.0719i)13-s + (−0.589 + 0.214i)14-s + (0.727 + 0.867i)15-s + (0.0434 + 0.246i)16-s + (−0.596 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $0.835 - 0.549i$
Analytic conductor: \(0.431192\)
Root analytic conductor: \(0.656652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{54} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 54,\ (\ :1/2),\ 0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.929566 + 0.278293i\)
\(L(\frac12)\) \(\approx\) \(0.929566 + 0.278293i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (1.11 - 1.32i)T \)
good5 \( 1 + (-0.439 + 2.49i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (1.79 - 1.50i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.745 + 4.22i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (0.713 - 0.259i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.46 - 4.26i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.62 - 6.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.233 + 0.196i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-2.91 - 1.06i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (6.58 + 5.52i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-3.78 + 6.55i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.60 - 1.67i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.283 + 1.60i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (1.39 - 1.16i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 0.573T + 53T^{2} \)
59 \( 1 + (-0.950 + 5.39i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-8.46 + 7.10i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.0393 - 0.0143i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-2.10 + 3.64i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.54 - 9.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.92 - 2.52i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (6.41 + 2.33i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (3.96 + 6.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.570 + 3.23i)T + (-91.1 + 33.1i)T^{2} \)
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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

−15.77621274435088217946463173599, −14.47011777915254830561903046478, −13.03477811383327586193709992224, −12.28217799209658423776321063989, −11.05447164511094643243591016677, −9.584600478947968357259233567496, −8.421337929443840754432665237503, −6.11644928847088868153130300080, −5.36002748445467669611286149486, −3.73362244416975614718113193614, 2.70182334989983768221891677530, 4.98056117730359741131778298575, 6.81778934042816611592621963328, 7.11390196016222587747188680258, 9.842787852491263120741460873091, 10.87797341539091563079830117386, 11.91714563610639375103515204358, 13.12558336336656766506595488042, 13.84702960009547673667086079729, 15.11869521607941332312603868635

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