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Properties

Label 2-486-9.7-c1-0-4
Degree $2$
Conductor $486$
Sign $0.939 - 0.342i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.5 + 2.59i)5-s + (−1 + 1.73i)7-s − 0.999·8-s + 3·10-s + (2 + 3.46i)13-s + (0.999 + 1.73i)14-s + (−0.5 + 0.866i)16-s + 6·17-s − 7·19-s + (1.50 − 2.59i)20-s + (4.5 + 7.79i)23-s + (−2 + 3.46i)25-s + 3.99·26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.670 + 1.16i)5-s + (−0.377 + 0.654i)7-s − 0.353·8-s + 0.948·10-s + (0.554 + 0.960i)13-s + (0.267 + 0.462i)14-s + (−0.125 + 0.216i)16-s + 1.45·17-s − 1.60·19-s + (0.335 − 0.580i)20-s + (0.938 + 1.62i)23-s + (−0.400 + 0.692i)25-s + 0.784·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(486\)    =    \(2 \cdot 3^{5}\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(3.88072\)
Root analytic conductor: \(1.96995\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{486} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 486,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72919 + 0.304903i\)
\(L(\frac12)\) \(\approx\) \(1.72919 + 0.304903i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.03457261891200222356840903600, −10.20026464587774252493112762676, −9.530771136615603992571230523222, −8.563981486422708538670718926232, −7.12040862029975227921916369845, −6.21896888550664762617002940327, −5.52511068690168711259613773189, −3.96350899921876026263251702378, −2.93010306679192687358797931440, −1.89102821128581011153876899378, 1.05188513745075441533185082570, 3.09445757754292285123753254422, 4.40894079072760182883170848310, 5.28175130366550267513058917458, 6.17423520285278835618122467941, 7.14210723258201388684837341199, 8.438662557282255484760827983418, 8.766045867232726927570207969390, 10.10754235561458609037512342213, 10.64557171912236157911321979084

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