Properties

Label 2-483-161.61-c1-0-25
Degree $2$
Conductor $483$
Sign $-0.634 + 0.772i$
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  + (0.230 + 0.180i)2-s + (−0.814 − 0.580i)3-s + (−0.451 − 1.86i)4-s + (−1.27 + 0.245i)5-s + (−0.0824 − 0.280i)6-s + (2.61 − 0.430i)7-s + (0.475 − 1.04i)8-s + (0.327 + 0.945i)9-s + (−0.337 − 0.173i)10-s + (0.611 + 0.778i)11-s + (−0.711 + 1.77i)12-s + (−2.09 − 3.25i)13-s + (0.678 + 0.373i)14-s + (1.18 + 0.538i)15-s + (−3.10 + 1.60i)16-s + (−4.95 − 4.72i)17-s + ⋯
L(s)  = 1  + (0.162 + 0.127i)2-s + (−0.470 − 0.334i)3-s + (−0.225 − 0.930i)4-s + (−0.569 + 0.109i)5-s + (−0.0336 − 0.114i)6-s + (0.986 − 0.162i)7-s + (0.168 − 0.368i)8-s + (0.109 + 0.315i)9-s + (−0.106 − 0.0550i)10-s + (0.184 + 0.234i)11-s + (−0.205 + 0.513i)12-s + (−0.580 − 0.903i)13-s + (0.181 + 0.0997i)14-s + (0.304 + 0.139i)15-s + (−0.776 + 0.400i)16-s + (−1.20 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.634 + 0.772i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.634 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.384991 - 0.814335i\)
\(L(\frac12)\) \(\approx\) \(0.384991 - 0.814335i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.814 + 0.580i)T \)
7 \( 1 + (-2.61 + 0.430i)T \)
23 \( 1 + (3.75 - 2.97i)T \)
good2 \( 1 + (-0.230 - 0.180i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (1.27 - 0.245i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (-0.611 - 0.778i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (2.09 + 3.25i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (4.95 + 4.72i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (-4.13 + 3.94i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (4.64 - 1.36i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (0.638 + 6.69i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (4.80 - 1.66i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (-5.07 + 4.39i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (-4.50 + 2.05i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-1.16 - 0.671i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-12.3 - 0.587i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (3.40 - 6.61i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (-3.79 - 5.33i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (1.03 + 2.59i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (-1.06 + 7.44i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-11.5 + 2.81i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (0.687 - 0.0327i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (-5.99 + 6.91i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-0.895 - 0.0855i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (9.96 + 11.4i)T + (-13.8 + 96.0i)T^{2} \)
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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

−10.91434180442874675375398996709, −9.828862647898291481651176209753, −8.966161773927497516474570660077, −7.53751511183862103706981379551, −7.20352633507244924952659088825, −5.74399010159317580227204909468, −5.07118882279378484286176557430, −4.13168031066517763409789670086, −2.17295768581094303309655845416, −0.54814562739479007471951097028, 2.06884253999181975749454378745, 3.85640787041558095970822217490, 4.32839661576150941726152756098, 5.45495556764770627060000078458, 6.78847655347414754633518188341, 7.85560513218925977043050230978, 8.482054650213244905812881470507, 9.420890950187493244257342448540, 10.70931566117896716346920094268, 11.46804611766550752304527590844

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