Properties

Label 2-483-161.61-c1-0-26
Degree $2$
Conductor $483$
Sign $0.0873 + 0.996i$
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  + (0.0565 + 0.0444i)2-s + (−0.814 − 0.580i)3-s + (−0.470 − 1.93i)4-s + (4.36 − 0.841i)5-s + (−0.0202 − 0.0689i)6-s + (1.52 − 2.16i)7-s + (0.119 − 0.261i)8-s + (0.327 + 0.945i)9-s + (0.284 + 0.146i)10-s + (−2.94 − 3.74i)11-s + (−0.741 + 1.85i)12-s + (2.81 + 4.37i)13-s + (0.182 − 0.0546i)14-s + (−4.04 − 1.84i)15-s + (−3.52 + 1.81i)16-s + (2.37 + 2.26i)17-s + ⋯
L(s)  = 1  + (0.0399 + 0.0314i)2-s + (−0.470 − 0.334i)3-s + (−0.235 − 0.969i)4-s + (1.95 − 0.376i)5-s + (−0.00826 − 0.0281i)6-s + (0.575 − 0.817i)7-s + (0.0421 − 0.0923i)8-s + (0.109 + 0.315i)9-s + (0.0898 + 0.0463i)10-s + (−0.887 − 1.12i)11-s + (−0.214 + 0.534i)12-s + (0.780 + 1.21i)13-s + (0.0486 − 0.0146i)14-s + (−1.04 − 0.476i)15-s + (−0.881 + 0.454i)16-s + (0.575 + 0.548i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0873 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0873 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.0873 + 0.996i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.0873 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22095 - 1.11855i\)
\(L(\frac12)\) \(\approx\) \(1.22095 - 1.11855i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.814 + 0.580i)T \)
7 \( 1 + (-1.52 + 2.16i)T \)
23 \( 1 + (3.81 - 2.90i)T \)
good2 \( 1 + (-0.0565 - 0.0444i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (-4.36 + 0.841i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (2.94 + 3.74i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (-2.81 - 4.37i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-2.37 - 2.26i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (2.63 - 2.51i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (1.25 - 0.368i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (0.457 + 4.78i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (3.22 - 1.11i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (1.13 - 0.986i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (-6.41 + 2.92i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-3.38 - 1.95i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.69 + 0.176i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (2.25 - 4.38i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (0.636 + 0.893i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (-1.84 - 4.61i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (1.79 - 12.4i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-3.87 + 0.939i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (-4.30 + 0.205i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (7.78 - 8.98i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (3.35 + 0.320i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (5.86 + 6.77i)T + (-13.8 + 96.0i)T^{2} \)
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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

−10.59850919603778863989468698019, −10.09579820327338894838085269388, −9.139626703549586998178350209434, −8.202780079648051867326078677487, −6.68803507846992233048157231787, −5.82069136977342730663204225105, −5.54050889919645040900178551612, −4.24880584953234507755506058075, −1.99534306751682057079686340744, −1.19305602340965974416096549483, 2.09408480095035527739846292974, 2.99312045784056002705531763507, 4.81530393614187920941148745743, 5.44203404042247539592617461141, 6.37617785046402036277102078624, 7.57732529286520106581333024487, 8.661159598084352437759425972204, 9.473583614139271913235247921895, 10.35156074840365851990695671181, 10.95052664912508679180520170168

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