Properties

Label 2-483-23.16-c1-0-16
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: $\chi_{483} (85, \cdot )$
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

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

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