Properties

Label 2-483-161.34-c1-0-24
Degree $2$
Conductor $483$
Sign $0.981 + 0.193i$
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  + (−0.818 + 0.525i)2-s + (0.989 + 0.142i)3-s + (−0.437 + 0.958i)4-s + (0.929 − 0.272i)5-s + (−0.884 + 0.404i)6-s + (−0.0253 − 2.64i)7-s + (−0.422 − 2.94i)8-s + (0.959 + 0.281i)9-s + (−0.617 + 0.712i)10-s + (2.79 − 4.34i)11-s + (−0.569 + 0.886i)12-s + (−0.940 − 0.815i)13-s + (1.41 + 2.15i)14-s + (0.958 − 0.137i)15-s + (0.512 + 0.591i)16-s + (−0.140 − 0.307i)17-s + ⋯
L(s)  = 1  + (−0.578 + 0.371i)2-s + (0.571 + 0.0821i)3-s + (−0.218 + 0.479i)4-s + (0.415 − 0.122i)5-s + (−0.361 + 0.164i)6-s + (−0.00959 − 0.999i)7-s + (−0.149 − 1.03i)8-s + (0.319 + 0.0939i)9-s + (−0.195 + 0.225i)10-s + (0.842 − 1.31i)11-s + (−0.164 + 0.255i)12-s + (−0.260 − 0.226i)13-s + (0.377 + 0.575i)14-s + (0.247 − 0.0355i)15-s + (0.128 + 0.147i)16-s + (−0.0341 − 0.0746i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.981 + 0.193i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.981 + 0.193i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26558 - 0.123675i\)
\(L(\frac12)\) \(\approx\) \(1.26558 - 0.123675i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (0.0253 + 2.64i)T \)
23 \( 1 + (-4.07 - 2.52i)T \)
good2 \( 1 + (0.818 - 0.525i)T + (0.830 - 1.81i)T^{2} \)
5 \( 1 + (-0.929 + 0.272i)T + (4.20 - 2.70i)T^{2} \)
11 \( 1 + (-2.79 + 4.34i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (0.940 + 0.815i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (0.140 + 0.307i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-2.62 + 5.74i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-2.59 - 5.68i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (6.95 - 0.999i)T + (29.7 - 8.73i)T^{2} \)
37 \( 1 + (0.359 - 1.22i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (-1.06 - 3.62i)T + (-34.4 + 22.1i)T^{2} \)
43 \( 1 + (-2.84 - 0.409i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + 8.50iT - 47T^{2} \)
53 \( 1 + (-6.02 + 5.21i)T + (7.54 - 52.4i)T^{2} \)
59 \( 1 + (-6.94 - 6.01i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (0.508 + 3.53i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-3.35 - 5.21i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (5.04 - 3.24i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-1.57 - 0.721i)T + (47.8 + 55.1i)T^{2} \)
79 \( 1 + (-2.46 - 2.13i)T + (11.2 + 78.1i)T^{2} \)
83 \( 1 + (-5.97 - 1.75i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-1.32 + 9.22i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (14.8 - 4.35i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.81772902319387687433945123476, −9.704859600456280356314790561759, −9.088446569058366652968807484683, −8.403666219178127550157818414758, −7.30471474570457808603470665492, −6.79950922626795292801933914577, −5.27947265791627088637720304847, −3.89897819577959178531689700554, −3.13304771516866148124112578626, −0.984030941054028132146993393905, 1.66797347166594847990497820260, 2.46381365035374015670502352703, 4.18273070708357329225266983361, 5.40921533249012683835873004500, 6.34528116549174422068681187087, 7.59025438034588307557376523357, 8.607498633674989382323992571630, 9.544117901909710130345503201780, 9.666607675176324340778702275548, 10.79642871959696858354007569054

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