Properties

Label 2-483-23.13-c1-0-3
Degree $2$
Conductor $483$
Sign $0.432 - 0.901i$
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  + (−0.544 − 0.627i)2-s + (0.841 + 0.540i)3-s + (0.186 − 1.29i)4-s + (−1.36 + 2.98i)5-s + (−0.118 − 0.822i)6-s + (−0.959 − 0.281i)7-s + (−2.31 + 1.48i)8-s + (0.415 + 0.909i)9-s + (2.62 − 0.769i)10-s + (−0.745 + 0.860i)11-s + (0.857 − 0.989i)12-s + (4.40 − 1.29i)13-s + (0.345 + 0.755i)14-s + (−2.76 + 1.77i)15-s + (−0.321 − 0.0943i)16-s + (0.756 + 5.26i)17-s + ⋯
L(s)  = 1  + (−0.384 − 0.443i)2-s + (0.485 + 0.312i)3-s + (0.0931 − 0.648i)4-s + (−0.610 + 1.33i)5-s + (−0.0482 − 0.335i)6-s + (−0.362 − 0.106i)7-s + (−0.817 + 0.525i)8-s + (0.138 + 0.303i)9-s + (0.828 − 0.243i)10-s + (−0.224 + 0.259i)11-s + (0.247 − 0.285i)12-s + (1.22 − 0.358i)13-s + (0.0922 + 0.201i)14-s + (−0.713 + 0.458i)15-s + (−0.0803 − 0.0235i)16-s + (0.183 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.432 - 0.901i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (358, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.432 - 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.858350 + 0.540123i\)
\(L(\frac12)\) \(\approx\) \(0.858350 + 0.540123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (0.959 + 0.281i)T \)
23 \( 1 + (-1.20 - 4.64i)T \)
good2 \( 1 + (0.544 + 0.627i)T + (-0.284 + 1.97i)T^{2} \)
5 \( 1 + (1.36 - 2.98i)T + (-3.27 - 3.77i)T^{2} \)
11 \( 1 + (0.745 - 0.860i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-4.40 + 1.29i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.756 - 5.26i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (1.11 - 7.72i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (1.10 + 7.69i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (2.20 - 1.41i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-1.90 - 4.16i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (0.343 - 0.751i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (2.23 + 1.43i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 - 5.46T + 47T^{2} \)
53 \( 1 + (10.0 + 2.94i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (-5.91 + 1.73i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (-0.526 + 0.338i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (1.30 + 1.50i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (-5.90 - 6.81i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-1.55 + 10.8i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (-4.53 + 1.33i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-2.51 - 5.50i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (11.7 + 7.53i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-1.80 + 3.95i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.89177020517382046493271253338, −10.29141095476870992224871606859, −9.726098446024424305060516391084, −8.433293094291204379211799950277, −7.73542682507660204075754085890, −6.43569084040702878504831021405, −5.76344405402304282653993345326, −3.88772951021429901994292136083, −3.21440532522295016008423610663, −1.79660520109270860891510396148, 0.67151439655837106534373602846, 2.79555401029301748140922355139, 3.93126426230636419228962665343, 5.05171057668661951835738466960, 6.52198209474314928315021108868, 7.32363076131191476800840069681, 8.258335084950806562126433027026, 9.048588709826122230622599906488, 9.137933949179806865451622565703, 11.01128860994437782226260666677

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