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Properties

Label 2-483-161.10-c1-0-26
Degree $2$
Conductor $483$
Sign $-0.657 + 0.753i$
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  + (−0.311 + 0.898i)2-s + (−0.458 − 0.888i)3-s + (0.861 + 0.677i)4-s + (−1.14 + 0.109i)5-s + (0.941 − 0.135i)6-s + (−2.64 − 0.149i)7-s + (−2.47 + 1.59i)8-s + (−0.580 + 0.814i)9-s + (0.258 − 1.06i)10-s + (−0.332 + 0.115i)11-s + (0.207 − 1.07i)12-s + (−1.47 − 5.00i)13-s + (0.955 − 2.32i)14-s + (0.623 + 0.969i)15-s + (−0.143 − 0.590i)16-s + (−4.33 − 1.73i)17-s + ⋯
L(s)  = 1  + (−0.219 + 0.635i)2-s + (−0.264 − 0.513i)3-s + (0.430 + 0.338i)4-s + (−0.513 + 0.0489i)5-s + (0.384 − 0.0552i)6-s + (−0.998 − 0.0563i)7-s + (−0.875 + 0.562i)8-s + (−0.193 + 0.271i)9-s + (0.0817 − 0.336i)10-s + (−0.100 + 0.0347i)11-s + (0.0598 − 0.310i)12-s + (−0.407 − 1.38i)13-s + (0.255 − 0.622i)14-s + (0.160 + 0.250i)15-s + (−0.0358 − 0.147i)16-s + (−1.05 − 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.657 + 0.753i$
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: \((2,\ 483,\ (\ :1/2),\ -0.657 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0489597 - 0.107778i\)
\(L(\frac12)\) \(\approx\) \(0.0489597 - 0.107778i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.149i)T \)
23 \( 1 + (4.70 + 0.927i)T \)
good2 \( 1 + (0.311 - 0.898i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (1.14 - 0.109i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (0.332 - 0.115i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (1.47 + 5.00i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (4.33 + 1.73i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-4.87 + 1.95i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.605 - 4.21i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (6.47 - 0.308i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (0.503 + 0.358i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-1.88 - 0.861i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-4.99 + 7.76i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (6.00 - 3.46i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.51 + 2.63i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (4.75 + 1.15i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (0.315 + 0.162i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-2.40 - 12.4i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (3.13 + 3.61i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-1.30 + 1.65i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (6.08 - 6.37i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-6.77 - 14.8i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.398 + 8.36i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-2.69 + 5.89i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.78366190562993743113065414072, −9.650845323463555434363052562876, −8.627121899328288885915355248751, −7.58708914065848400162697920648, −7.19552654657395541111299185230, −6.18726338354295678918321803139, −5.31027030995590528104523467324, −3.55449136954638459592399852172, −2.55392018270824597487614767958, −0.07179585196656573793457846372, 2.02944619438418644587684681060, 3.40159884895102937810864000489, 4.32429450813569488991200159079, 5.85289587774175983649614875718, 6.54931643795131814507425461356, 7.63571817107794438176047883929, 9.150000666913245372018825578562, 9.593951468744783682658315482575, 10.40238208815306576059196593041, 11.40282235777543895010938125338

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