Properties

Label 2-483-161.34-c1-0-17
Degree $2$
Conductor $483$
Sign $0.888 - 0.459i$
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  + (−2.18 + 1.40i)2-s + (0.989 + 0.142i)3-s + (1.96 − 4.30i)4-s + (1.99 − 0.584i)5-s + (−2.36 + 1.07i)6-s + (2.37 − 1.16i)7-s + (1.01 + 7.02i)8-s + (0.959 + 0.281i)9-s + (−3.52 + 4.06i)10-s + (−0.546 + 0.849i)11-s + (2.56 − 3.98i)12-s + (−2.07 − 1.80i)13-s + (−3.55 + 5.87i)14-s + (2.05 − 0.295i)15-s + (−5.86 − 6.77i)16-s + (1.29 + 2.84i)17-s + ⋯
L(s)  = 1  + (−1.54 + 0.992i)2-s + (0.571 + 0.0821i)3-s + (0.983 − 2.15i)4-s + (0.889 − 0.261i)5-s + (−0.963 + 0.440i)6-s + (0.897 − 0.440i)7-s + (0.357 + 2.48i)8-s + (0.319 + 0.0939i)9-s + (−1.11 + 1.28i)10-s + (−0.164 + 0.256i)11-s + (0.739 − 1.15i)12-s + (−0.576 − 0.499i)13-s + (−0.948 + 1.57i)14-s + (0.530 − 0.0762i)15-s + (−1.46 − 1.69i)16-s + (0.315 + 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.888 - 0.459i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.888 - 0.459i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01469 + 0.246723i\)
\(L(\frac12)\) \(\approx\) \(1.01469 + 0.246723i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (-2.37 + 1.16i)T \)
23 \( 1 + (-1.55 + 4.53i)T \)
good2 \( 1 + (2.18 - 1.40i)T + (0.830 - 1.81i)T^{2} \)
5 \( 1 + (-1.99 + 0.584i)T + (4.20 - 2.70i)T^{2} \)
11 \( 1 + (0.546 - 0.849i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (2.07 + 1.80i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (-1.29 - 2.84i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-2.30 + 5.05i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (0.204 + 0.447i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-6.70 + 0.963i)T + (29.7 - 8.73i)T^{2} \)
37 \( 1 + (-0.106 + 0.362i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (1.45 + 4.95i)T + (-34.4 + 22.1i)T^{2} \)
43 \( 1 + (8.56 + 1.23i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 - 12.1iT - 47T^{2} \)
53 \( 1 + (0.165 - 0.143i)T + (7.54 - 52.4i)T^{2} \)
59 \( 1 + (-3.18 - 2.76i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (-1.30 - 9.06i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-5.08 - 7.90i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (8.80 - 5.65i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-10.4 - 4.79i)T + (47.8 + 55.1i)T^{2} \)
79 \( 1 + (5.51 + 4.77i)T + (11.2 + 78.1i)T^{2} \)
83 \( 1 + (-4.57 - 1.34i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-2.41 + 16.8i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-0.968 + 0.284i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.39119099910606110165232127209, −10.03266218812070070176163392947, −9.087979135176999013719590080073, −8.380890399677456328623787315259, −7.62288917686188491343807987253, −6.82786294036011495081987158068, −5.66047499702997702295182977602, −4.72486470970711447733747509242, −2.40535553313772978203505856502, −1.15584904414138431327096925284, 1.49831871303425657349761786549, 2.32333074531181019489132091882, 3.39630427760291425976155954452, 5.18671208026037329289820435765, 6.71787800677721373532044630911, 7.83292299881697524192299063817, 8.317969471150226527894421265285, 9.457829683558311822779827523422, 9.767522051202734308209889596730, 10.66015903519764351857420857087

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