Properties

Label 2-483-23.4-c1-0-13
Degree $2$
Conductor $483$
Sign $0.998 + 0.0486i$
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  + (0.698 + 1.53i)2-s + (−0.959 + 0.281i)3-s + (−0.544 + 0.627i)4-s + (−2.50 − 1.60i)5-s + (−1.10 − 1.27i)6-s + (−0.142 − 0.989i)7-s + (1.88 + 0.554i)8-s + (0.841 − 0.540i)9-s + (0.712 − 4.95i)10-s + (2.05 − 4.49i)11-s + (0.345 − 0.755i)12-s + (−0.165 + 1.14i)13-s + (1.41 − 0.909i)14-s + (2.85 + 0.837i)15-s + (0.707 + 4.92i)16-s + (−1.62 − 1.87i)17-s + ⋯
L(s)  = 1  + (0.494 + 1.08i)2-s + (−0.553 + 0.162i)3-s + (−0.272 + 0.313i)4-s + (−1.11 − 0.719i)5-s + (−0.449 − 0.519i)6-s + (−0.0537 − 0.374i)7-s + (0.667 + 0.195i)8-s + (0.280 − 0.180i)9-s + (0.225 − 1.56i)10-s + (0.618 − 1.35i)11-s + (0.0996 − 0.218i)12-s + (−0.0458 + 0.318i)13-s + (0.378 − 0.243i)14-s + (0.736 + 0.216i)15-s + (0.176 + 1.23i)16-s + (−0.394 − 0.455i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.998 + 0.0486i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.998 + 0.0486i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31142 - 0.0319163i\)
\(L(\frac12)\) \(\approx\) \(1.31142 - 0.0319163i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (-0.794 + 4.72i)T \)
good2 \( 1 + (-0.698 - 1.53i)T + (-1.30 + 1.51i)T^{2} \)
5 \( 1 + (2.50 + 1.60i)T + (2.07 + 4.54i)T^{2} \)
11 \( 1 + (-2.05 + 4.49i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.165 - 1.14i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (1.62 + 1.87i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-3.78 + 4.37i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-3.45 - 3.98i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-6.16 - 1.80i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-0.716 + 0.460i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (7.78 + 5.00i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (6.05 - 1.77i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 7.03T + 47T^{2} \)
53 \( 1 + (1.92 + 13.4i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-0.0785 + 0.546i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-6.57 - 1.93i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-2.36 - 5.17i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-1.31 - 2.87i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-3.77 + 4.35i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (0.688 - 4.79i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (7.71 - 4.96i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-0.622 + 0.182i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-2.40 - 1.54i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.36962351899486004941877219849, −10.17311253975153138384063080474, −8.752732584754142040751584800245, −8.216318724579481285929240809672, −6.96621290884623674950814193487, −6.51775263884784573480952780918, −5.15523963875860426072747615684, −4.62341106752473707755441506103, −3.51116669771881290043776314502, −0.798576087453934138304224208700, 1.64383173779387555748568323849, 3.07249822170323075906857240829, 4.00169221106649798082423464686, 4.91044261044398893314565431706, 6.40676234200576766407474276126, 7.34435861764791043124096737680, 8.041041230340233324783866636504, 9.756493031665629212217041646868, 10.28534846418018160070916889147, 11.43401849568738773109340706681

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