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Properties

Label 2-486-27.7-c1-0-8
Degree $2$
Conductor $486$
Sign $0.363 + 0.931i$
Analytic cond. $3.88072$
Root an. cond. $1.96995$
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 + (0.766 − 0.642i)4-s + (−0.550 − 3.11i)5-s + (2.82 + 2.36i)7-s + (0.500 − 0.866i)8-s + (−1.58 − 2.74i)10-s + (0.396 − 2.25i)11-s + (−2.91 − 1.06i)13-s + (3.46 + 1.26i)14-s + (0.173 − 0.984i)16-s + (0.862 + 1.49i)17-s + (1.69 − 2.93i)19-s + (−2.42 − 2.03i)20-s + (−0.396 − 2.25i)22-s + (2.56 − 2.15i)23-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.383 − 0.321i)4-s + (−0.246 − 1.39i)5-s + (1.06 + 0.895i)7-s + (0.176 − 0.306i)8-s + (−0.500 − 0.867i)10-s + (0.119 − 0.678i)11-s + (−0.808 − 0.294i)13-s + (0.925 + 0.336i)14-s + (0.0434 − 0.246i)16-s + (0.209 + 0.362i)17-s + (0.389 − 0.674i)19-s + (−0.542 − 0.455i)20-s + (−0.0845 − 0.479i)22-s + (0.535 − 0.449i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(486\)    =    \(2 \cdot 3^{5}\)
Sign: $0.363 + 0.931i$
Analytic conductor: \(3.88072\)
Root analytic conductor: \(1.96995\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{486} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 486,\ (\ :1/2),\ 0.363 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77735 - 1.21429i\)
\(L(\frac12)\) \(\approx\) \(1.77735 - 1.21429i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (0.550 + 3.11i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-2.82 - 2.36i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.396 + 2.25i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (2.91 + 1.06i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.862 - 1.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.69 + 2.93i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.56 + 2.15i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.550 - 0.200i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (3.55 - 2.98i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-3.65 - 6.32i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.65 + 2.42i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.287 + 1.63i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.95 - 2.47i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 2.58T + 53T^{2} \)
59 \( 1 + (-1.67 - 9.51i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-10.0 - 8.41i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-8.08 - 2.94i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-0.993 - 1.72i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.32 - 9.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (13.2 - 4.82i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (2.91 - 1.05i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (8.67 - 15.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.59 + 9.05i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.18571825955688775195165527369, −9.938133249749779949428221033606, −8.742457061036862639736398522305, −8.404426838387674862950456327287, −7.14240926937206936539216495153, −5.56585793662677475494450519659, −5.16115070794982260477357650273, −4.23118651358445976914132650553, −2.67840074548673625074075050339, −1.21930253720738745890730070554, 2.03447097857481393462412707415, 3.39101053602177878703735985834, 4.36940405429205180770662350124, 5.39510265553497485685229952896, 6.78382673814075714655051828116, 7.34313798489800119872472146020, 7.928213255182268064574004319280, 9.591791276268884604646753650344, 10.46894591133479037011779883502, 11.26924605488972539148301696363

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