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Properties

Label 2-483-161.17-c1-0-11
Degree $2$
Conductor $483$
Sign $0.238 - 0.971i$
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  + (1.11 − 0.106i)2-s + (0.690 + 0.723i)3-s + (−0.723 + 0.139i)4-s + (−0.742 − 0.382i)5-s + (0.849 + 0.736i)6-s + (1.48 + 2.18i)7-s + (−2.95 + 0.866i)8-s + (−0.0475 + 0.998i)9-s + (−0.871 − 0.348i)10-s + (−0.449 + 4.71i)11-s + (−0.600 − 0.427i)12-s + (2.59 − 0.373i)13-s + (1.89 + 2.28i)14-s + (−0.235 − 0.801i)15-s + (−1.84 + 0.737i)16-s + (1.73 + 5.00i)17-s + ⋯
L(s)  = 1  + (0.791 − 0.0755i)2-s + (0.398 + 0.417i)3-s + (−0.361 + 0.0697i)4-s + (−0.331 − 0.171i)5-s + (0.346 + 0.300i)6-s + (0.562 + 0.826i)7-s + (−1.04 + 0.306i)8-s + (−0.0158 + 0.332i)9-s + (−0.275 − 0.110i)10-s + (−0.135 + 1.42i)11-s + (−0.173 − 0.123i)12-s + (0.720 − 0.103i)13-s + (0.507 + 0.611i)14-s + (−0.0607 − 0.206i)15-s + (−0.460 + 0.184i)16-s + (0.420 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.238 - 0.971i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.238 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50919 + 1.18375i\)
\(L(\frac12)\) \(\approx\) \(1.50919 + 1.18375i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.690 - 0.723i)T \)
7 \( 1 + (-1.48 - 2.18i)T \)
23 \( 1 + (-3.76 + 2.96i)T \)
good2 \( 1 + (-1.11 + 0.106i)T + (1.96 - 0.378i)T^{2} \)
5 \( 1 + (0.742 + 0.382i)T + (2.90 + 4.07i)T^{2} \)
11 \( 1 + (0.449 - 4.71i)T + (-10.8 - 2.08i)T^{2} \)
13 \( 1 + (-2.59 + 0.373i)T + (12.4 - 3.66i)T^{2} \)
17 \( 1 + (-1.73 - 5.00i)T + (-13.3 + 10.5i)T^{2} \)
19 \( 1 + (-0.477 + 1.38i)T + (-14.9 - 11.7i)T^{2} \)
29 \( 1 + (0.680 - 0.784i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (0.978 + 0.237i)T + (27.5 + 14.2i)T^{2} \)
37 \( 1 + (0.294 + 0.0140i)T + (36.8 + 3.51i)T^{2} \)
41 \( 1 + (0.542 + 0.843i)T + (-17.0 + 37.2i)T^{2} \)
43 \( 1 + (-1.86 + 6.36i)T + (-36.1 - 23.2i)T^{2} \)
47 \( 1 + (2.80 - 1.62i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.804 + 1.02i)T + (-12.4 - 51.5i)T^{2} \)
59 \( 1 + (2.39 - 5.98i)T + (-42.7 - 40.7i)T^{2} \)
61 \( 1 + (4.70 + 4.48i)T + (2.90 + 60.9i)T^{2} \)
67 \( 1 + (-12.3 + 8.78i)T + (21.9 - 63.3i)T^{2} \)
71 \( 1 + (-4.39 + 9.62i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-0.0232 - 0.120i)T + (-67.7 + 27.1i)T^{2} \)
79 \( 1 + (3.01 + 3.83i)T + (-18.6 + 76.7i)T^{2} \)
83 \( 1 + (-8.65 - 5.55i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (0.141 + 0.585i)T + (-79.1 + 40.7i)T^{2} \)
97 \( 1 + (5.91 - 3.80i)T + (40.2 - 88.2i)T^{2} \)
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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

−11.32163235064129962709920347710, −10.29086719583367885647621487691, −9.234420155230253813116498362570, −8.560400789547896001120298047466, −7.76616395482898811262387781403, −6.25347003397380054002797732122, −5.16600973738291870003487027647, −4.46853069572021131881648395964, −3.51022759607121134917336217596, −2.15015987553142341259200618871, 0.954188351201468669456469629588, 3.15440064722377356906425547007, 3.79284293849319756110469181336, 5.05389030064128836399240532502, 5.97167845329135446570649204109, 7.13058823477089663790580808937, 8.028688452016743363753637087021, 8.871056515308506719316625894199, 9.830365912058792976154520661176, 11.21354059829485478554538126795

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