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Properties

Label 2-483-161.10-c1-0-1
Degree $2$
Conductor $483$
Sign $-0.156 + 0.987i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
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  + (−0.850 + 2.45i)2-s + (−0.458 − 0.888i)3-s + (−3.74 − 2.94i)4-s + (1.41 − 0.135i)5-s + (2.57 − 0.370i)6-s + (−1.58 + 2.12i)7-s + (6.05 − 3.88i)8-s + (−0.580 + 0.814i)9-s + (−0.871 + 3.59i)10-s + (−1.78 + 0.616i)11-s + (−0.901 + 4.67i)12-s + (0.406 + 1.38i)13-s + (−3.87 − 5.69i)14-s + (−0.768 − 1.19i)15-s + (2.16 + 8.91i)16-s + (−2.52 − 1.00i)17-s + ⋯
L(s)  = 1  + (−0.601 + 1.73i)2-s + (−0.264 − 0.513i)3-s + (−1.87 − 1.47i)4-s + (0.632 − 0.0604i)5-s + (1.05 − 0.151i)6-s + (−0.597 + 0.801i)7-s + (2.13 − 1.37i)8-s + (−0.193 + 0.271i)9-s + (−0.275 + 1.13i)10-s + (−0.536 + 0.185i)11-s + (−0.260 + 1.35i)12-s + (0.112 + 0.383i)13-s + (−1.03 − 1.52i)14-s + (−0.198 − 0.308i)15-s + (0.540 + 2.22i)16-s + (−0.611 − 0.244i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0795861 - 0.0932146i\)
\(L(\frac12)\) \(\approx\) \(0.0795861 - 0.0932146i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.458 + 0.888i)T \)
7 \( 1 + (1.58 - 2.12i)T \)
23 \( 1 + (3.90 + 2.79i)T \)
good2 \( 1 + (0.850 - 2.45i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (-1.41 + 0.135i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (1.78 - 0.616i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.406 - 1.38i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (2.52 + 1.00i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (3.47 - 1.39i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (1.23 + 8.61i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-0.102 + 0.00487i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-0.511 - 0.364i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (6.66 + 3.04i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (1.58 - 2.46i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (8.83 - 5.09i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.74 - 9.17i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (5.18 + 1.25i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-3.25 - 1.67i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-0.262 - 1.36i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-5.92 - 6.83i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-2.10 + 2.68i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (10.1 - 10.6i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (4.62 + 10.1i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.551 + 11.5i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (5.87 - 12.8i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.65284414054317474675440243086, −10.21689906941093760332723655141, −9.603531839126376685917801430089, −8.678111257998076497242359324767, −7.997776225422200471201162612018, −6.91023764403894380985467593575, −6.14597791681833222287424786937, −5.68997033842782797169839673615, −4.48068425641224010688528004875, −2.15635551979173153497041563195, 0.089195528335726773735295693964, 1.85634629913304562580111951890, 3.18387037182709700643358008231, 4.01885304694082935457957393155, 5.22504786480345470043277497900, 6.63561102832509991814348125725, 8.086560803933366314022726667106, 8.973815977570641854108851951825, 9.899342954105437326472255118573, 10.31373126453961514167536932212

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