Properties

Label 2-483-23.18-c1-0-7
Degree $2$
Conductor $483$
Sign $0.982 - 0.186i$
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  + (0.273 + 0.0801i)2-s + (−0.654 + 0.755i)3-s + (−1.61 − 1.03i)4-s + (−0.0390 + 0.271i)5-s + (−0.239 + 0.153i)6-s + (0.415 + 0.909i)7-s + (−0.730 − 0.843i)8-s + (−0.142 − 0.989i)9-s + (−0.0324 + 0.0711i)10-s + (4.25 − 1.24i)11-s + (1.84 − 0.540i)12-s + (0.917 − 2.01i)13-s + (0.0405 + 0.281i)14-s + (−0.179 − 0.207i)15-s + (1.46 + 3.20i)16-s + (−1.97 + 1.27i)17-s + ⋯
L(s)  = 1  + (0.193 + 0.0567i)2-s + (−0.378 + 0.436i)3-s + (−0.807 − 0.518i)4-s + (−0.0174 + 0.121i)5-s + (−0.0977 + 0.0628i)6-s + (0.157 + 0.343i)7-s + (−0.258 − 0.298i)8-s + (−0.0474 − 0.329i)9-s + (−0.0102 + 0.0224i)10-s + (1.28 − 0.376i)11-s + (0.531 − 0.156i)12-s + (0.254 − 0.557i)13-s + (0.0108 + 0.0752i)14-s + (−0.0464 − 0.0536i)15-s + (0.365 + 0.800i)16-s + (−0.480 + 0.308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.982 - 0.186i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.982 - 0.186i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24047 + 0.116742i\)
\(L(\frac12)\) \(\approx\) \(1.24047 + 0.116742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (-4.62 - 1.25i)T \)
good2 \( 1 + (-0.273 - 0.0801i)T + (1.68 + 1.08i)T^{2} \)
5 \( 1 + (0.0390 - 0.271i)T + (-4.79 - 1.40i)T^{2} \)
11 \( 1 + (-4.25 + 1.24i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-0.917 + 2.01i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (1.97 - 1.27i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-4.49 - 2.88i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-0.958 + 0.615i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-2.03 - 2.34i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (1.32 + 9.22i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (-0.524 + 3.64i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-6.99 + 8.07i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + 9.89T + 47T^{2} \)
53 \( 1 + (-5.06 - 11.0i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (-0.248 + 0.544i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (-9.18 - 10.5i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (5.98 + 1.75i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (14.1 + 4.16i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (6.05 + 3.89i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (0.242 - 0.530i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (0.753 + 5.23i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (9.17 - 10.5i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-1.84 + 12.8i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.90294021126344812454655773425, −10.18336796836572494528344479824, −9.045089583720645645592768337689, −8.798240845050285623129053128128, −7.21562353990155451281241609949, −6.03295649056325234001017076737, −5.41004023278938464885679848974, −4.30305438865877409704602561039, −3.34545538320521751310711432502, −1.12021282532468117129740297205, 1.11108141621508131676494904182, 3.04118957532070162181280669792, 4.37529730156825693743670420784, 4.97778615475446628185847759033, 6.49290540151605427512926222449, 7.16170123882450134570835284021, 8.342858913715737582232766220238, 9.110168734120501276181709391136, 9.913361140235041678264221325869, 11.48529766510085999722394744712

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