Properties

Label 2-483-161.2-c1-0-28
Degree $2$
Conductor $483$
Sign $0.270 + 0.962i$
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  + (1.38 − 0.713i)2-s + (0.786 − 0.618i)3-s + (0.247 − 0.347i)4-s + (2.73 − 2.61i)5-s + (0.647 − 1.41i)6-s + (−2.07 − 1.64i)7-s + (−0.348 + 2.42i)8-s + (0.235 − 0.971i)9-s + (1.92 − 5.56i)10-s + (4.47 + 2.30i)11-s + (−0.0202 − 0.425i)12-s + (0.104 + 0.121i)13-s + (−4.04 − 0.799i)14-s + (0.538 − 3.74i)15-s + (1.52 + 4.41i)16-s + (−6.76 + 0.646i)17-s + ⋯
L(s)  = 1  + (0.978 − 0.504i)2-s + (0.453 − 0.356i)3-s + (0.123 − 0.173i)4-s + (1.22 − 1.16i)5-s + (0.264 − 0.578i)6-s + (−0.783 − 0.621i)7-s + (−0.123 + 0.857i)8-s + (0.0785 − 0.323i)9-s + (0.609 − 1.76i)10-s + (1.34 + 0.695i)11-s + (−0.00585 − 0.122i)12-s + (0.0290 + 0.0335i)13-s + (−1.08 − 0.213i)14-s + (0.139 − 0.967i)15-s + (0.381 + 1.10i)16-s + (−1.64 + 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.270 + 0.962i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.270 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28093 - 1.72856i\)
\(L(\frac12)\) \(\approx\) \(2.28093 - 1.72856i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.786 + 0.618i)T \)
7 \( 1 + (2.07 + 1.64i)T \)
23 \( 1 + (-4.57 - 1.42i)T \)
good2 \( 1 + (-1.38 + 0.713i)T + (1.16 - 1.62i)T^{2} \)
5 \( 1 + (-2.73 + 2.61i)T + (0.237 - 4.99i)T^{2} \)
11 \( 1 + (-4.47 - 2.30i)T + (6.38 + 8.96i)T^{2} \)
13 \( 1 + (-0.104 - 0.121i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (6.76 - 0.646i)T + (16.6 - 3.21i)T^{2} \)
19 \( 1 + (6.95 + 0.664i)T + (18.6 + 3.59i)T^{2} \)
29 \( 1 + (-0.219 + 0.479i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-7.79 + 3.12i)T + (22.4 - 21.3i)T^{2} \)
37 \( 1 + (1.56 - 6.45i)T + (-32.8 - 16.9i)T^{2} \)
41 \( 1 + (-4.08 - 1.20i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-0.461 - 3.21i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (2.20 - 3.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.259 + 0.0500i)T + (49.2 + 19.6i)T^{2} \)
59 \( 1 + (-0.182 + 0.527i)T + (-46.3 - 36.4i)T^{2} \)
61 \( 1 + (7.10 + 5.58i)T + (14.3 + 59.2i)T^{2} \)
67 \( 1 + (0.280 - 5.88i)T + (-66.6 - 6.36i)T^{2} \)
71 \( 1 + (-0.857 - 0.551i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (6.30 - 8.85i)T + (-23.8 - 68.9i)T^{2} \)
79 \( 1 + (-4.13 + 0.797i)T + (73.3 - 29.3i)T^{2} \)
83 \( 1 + (-6.36 + 1.86i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (7.52 + 3.01i)T + (64.4 + 61.4i)T^{2} \)
97 \( 1 + (4.63 + 1.36i)T + (81.6 + 52.4i)T^{2} \)
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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

−10.96570181427861176451378629550, −9.747081915671856189138505265117, −9.079326973140598804443341460665, −8.398120676584791099356122370455, −6.69378693248932647482926564467, −6.17476715635276121836483812042, −4.61030069398981607633740310936, −4.21609395038309719563957009455, −2.63517238913123653834865280075, −1.54211288195020719499908407061, 2.32620572035926819167167754488, 3.33701614276983647536422729396, 4.41943956311219393931077887488, 5.76175570602571935053878382623, 6.58046766182760524825150175487, 6.70339544464691903642916581118, 8.878387742738130341453673338054, 9.222984255876158865588130721923, 10.29676202146922653017460710106, 10.97518219524691414810162598801

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