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Properties

Label 2-486-81.13-c1-0-1
Degree $2$
Conductor $486$
Sign $0.271 - 0.962i$
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  + (−0.893 − 0.448i)2-s + (0.597 + 0.802i)4-s + (1.15 + 3.85i)5-s + (−0.663 − 1.53i)7-s + (−0.173 − 0.984i)8-s + (0.698 − 3.95i)10-s + (−0.0131 + 0.00311i)11-s + (3.84 + 2.52i)13-s + (−0.0974 + 1.67i)14-s + (−0.286 + 0.957i)16-s + (−4.83 + 1.76i)17-s + (2.66 + 0.968i)19-s + (−2.40 + 3.22i)20-s + (0.0131 + 0.00311i)22-s + (0.253 − 0.586i)23-s + ⋯
L(s)  = 1  + (−0.631 − 0.317i)2-s + (0.298 + 0.401i)4-s + (0.515 + 1.72i)5-s + (−0.250 − 0.581i)7-s + (−0.0613 − 0.348i)8-s + (0.220 − 1.25i)10-s + (−0.00396 + 0.000940i)11-s + (1.06 + 0.700i)13-s + (−0.0260 + 0.447i)14-s + (−0.0717 + 0.239i)16-s + (−1.17 + 0.426i)17-s + (0.610 + 0.222i)19-s + (−0.536 + 0.721i)20-s + (0.00280 + 0.000664i)22-s + (0.0527 − 0.122i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(486\)    =    \(2 \cdot 3^{5}\)
Sign: $0.271 - 0.962i$
Analytic conductor: \(3.88072\)
Root analytic conductor: \(1.96995\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{486} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 486,\ (\ :1/2),\ 0.271 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821979 + 0.621849i\)
\(L(\frac12)\) \(\approx\) \(0.821979 + 0.621849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.893 + 0.448i)T \)
3 \( 1 \)
good5 \( 1 + (-1.15 - 3.85i)T + (-4.17 + 2.74i)T^{2} \)
7 \( 1 + (0.663 + 1.53i)T + (-4.80 + 5.09i)T^{2} \)
11 \( 1 + (0.0131 - 0.00311i)T + (9.82 - 4.93i)T^{2} \)
13 \( 1 + (-3.84 - 2.52i)T + (5.14 + 11.9i)T^{2} \)
17 \( 1 + (4.83 - 1.76i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (-2.66 - 0.968i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-0.253 + 0.586i)T + (-15.7 - 16.7i)T^{2} \)
29 \( 1 + (-0.262 - 4.51i)T + (-28.8 + 3.36i)T^{2} \)
31 \( 1 + (9.48 - 1.10i)T + (30.1 - 7.14i)T^{2} \)
37 \( 1 + (-6.03 - 5.06i)T + (6.42 + 36.4i)T^{2} \)
41 \( 1 + (6.30 - 3.16i)T + (24.4 - 32.8i)T^{2} \)
43 \( 1 + (-4.34 - 4.60i)T + (-2.50 + 42.9i)T^{2} \)
47 \( 1 + (-11.3 - 1.32i)T + (45.7 + 10.8i)T^{2} \)
53 \( 1 + (-2.96 + 5.14i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.42 - 0.573i)T + (52.7 + 26.4i)T^{2} \)
61 \( 1 + (-1.39 + 1.87i)T + (-17.4 - 58.4i)T^{2} \)
67 \( 1 + (-0.232 + 3.99i)T + (-66.5 - 7.77i)T^{2} \)
71 \( 1 + (-1.01 + 5.73i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-1.45 - 8.24i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (4.39 + 2.20i)T + (47.1 + 63.3i)T^{2} \)
83 \( 1 + (13.5 + 6.80i)T + (49.5 + 66.5i)T^{2} \)
89 \( 1 + (1.34 + 7.65i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-0.923 + 3.08i)T + (-81.0 - 53.3i)T^{2} \)
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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

−10.99431778164932587597035772888, −10.40098050446893191236032801515, −9.555145834884201949039430529877, −8.618525689421039678134536140348, −7.30442212859426328719622498930, −6.77671847593668490158958157891, −5.93862979036980990193553070572, −3.98933324664932363222716428182, −3.06705429372199901926574693028, −1.80173206079405224014376150658, 0.78713507161248787630530745515, 2.23417154616475099261591731655, 4.14952573299254316481303779603, 5.48434016844488905514586460954, 5.84065289492778798004913128056, 7.29098064189515659401115161426, 8.418699443817626853705709313657, 9.010558643534501570071866402648, 9.466194920156569652492051115020, 10.66746662266842059804448888423

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