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Properties

Label 2-486-27.13-c1-0-7
Degree $2$
Conductor $486$
Sign $0.973 - 0.230i$
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  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (−1.93 − 1.62i)5-s + (2.20 + 0.802i)7-s + (−0.5 − 0.866i)8-s + (1.26 − 2.19i)10-s + (3.28 − 2.75i)11-s + (0.131 − 0.747i)13-s + (−0.407 + 2.31i)14-s + (0.766 − 0.642i)16-s + (2.46 − 4.26i)17-s + (3.62 + 6.27i)19-s + (2.37 + 0.866i)20-s + (3.28 + 2.75i)22-s + (−0.286 + 0.104i)23-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (−0.469 + 0.171i)4-s + (−0.867 − 0.727i)5-s + (0.833 + 0.303i)7-s + (−0.176 − 0.306i)8-s + (0.400 − 0.693i)10-s + (0.991 − 0.831i)11-s + (0.0365 − 0.207i)13-s + (−0.108 + 0.617i)14-s + (0.191 − 0.160i)16-s + (0.596 − 1.03i)17-s + (0.831 + 1.44i)19-s + (0.532 + 0.193i)20-s + (0.700 + 0.588i)22-s + (−0.0598 + 0.0217i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(486\)    =    \(2 \cdot 3^{5}\)
Sign: $0.973 - 0.230i$
Analytic conductor: \(3.88072\)
Root analytic conductor: \(1.96995\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{486} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 486,\ (\ :1/2),\ 0.973 - 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41441 + 0.165321i\)
\(L(\frac12)\) \(\approx\) \(1.41441 + 0.165321i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 \)
good5 \( 1 + (1.93 + 1.62i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-2.20 - 0.802i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-3.28 + 2.75i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.131 + 0.747i)T + (-12.2 - 4.44i)T^{2} \)
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 \( 1 + (0.286 - 0.104i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.539 - 3.05i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-8.07 + 2.94i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-3.78 + 6.55i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.851 - 4.82i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (1.24 - 1.04i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (1.70 + 0.620i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 0.573T + 53T^{2} \)
59 \( 1 + (4.19 + 3.51i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (10.3 + 3.77i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.00727 + 0.0412i)T + (-62.9 - 22.9i)T^{2} \)
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 \( 1 + (1.28 + 7.26i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.18 + 6.71i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-3.96 - 6.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.51 - 2.11i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.36222823819732931852423966409, −9.885114638692236758835405863085, −8.949699082952500464263454526520, −8.105072002634830894608023349840, −7.67594120067157432674926143399, −6.28863591440108541316716451548, −5.32648147247686655464108805412, −4.41779194293015198387035161191, −3.35733238060527191972412566791, −1.06198434287026828266213618225, 1.39670223015593237239631961439, 2.97513585296223564949660147025, 4.08927291456472979046889078585, 4.82635761575022684295877873420, 6.42292940247342389107357239480, 7.37820099540764010117409236651, 8.208146647863617092298955242276, 9.317747723943019172882547515927, 10.25793487541471408662492945735, 11.10187447748204071591574783773

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