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Properties

Label 2-483-161.19-c1-0-12
Degree $2$
Conductor $483$
Sign $0.600 + 0.799i$
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  + (−2.39 − 0.229i)2-s + (0.690 − 0.723i)3-s + (3.74 + 0.721i)4-s + (1.30 − 0.671i)5-s + (−1.82 + 1.57i)6-s + (−2.17 + 1.50i)7-s + (−4.18 − 1.22i)8-s + (−0.0475 − 0.998i)9-s + (−3.28 + 1.31i)10-s + (−0.0780 − 0.817i)11-s + (3.10 − 2.21i)12-s + (5.79 + 0.832i)13-s + (5.57 − 3.10i)14-s + (0.413 − 1.40i)15-s + (2.68 + 1.07i)16-s + (2.49 − 7.21i)17-s + ⋯
L(s)  = 1  + (−1.69 − 0.162i)2-s + (0.398 − 0.417i)3-s + (1.87 + 0.360i)4-s + (0.582 − 0.300i)5-s + (−0.743 + 0.644i)6-s + (−0.823 + 0.567i)7-s + (−1.48 − 0.434i)8-s + (−0.0158 − 0.332i)9-s + (−1.03 + 0.415i)10-s + (−0.0235 − 0.246i)11-s + (0.895 − 0.638i)12-s + (1.60 + 0.230i)13-s + (1.48 − 0.829i)14-s + (0.106 − 0.363i)15-s + (0.672 + 0.269i)16-s + (0.606 − 1.75i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.600 + 0.799i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.600 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.704885 - 0.352012i\)
\(L(\frac12)\) \(\approx\) \(0.704885 - 0.352012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.690 + 0.723i)T \)
7 \( 1 + (2.17 - 1.50i)T \)
23 \( 1 + (4.76 - 0.510i)T \)
good2 \( 1 + (2.39 + 0.229i)T + (1.96 + 0.378i)T^{2} \)
5 \( 1 + (-1.30 + 0.671i)T + (2.90 - 4.07i)T^{2} \)
11 \( 1 + (0.0780 + 0.817i)T + (-10.8 + 2.08i)T^{2} \)
13 \( 1 + (-5.79 - 0.832i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (-2.49 + 7.21i)T + (-13.3 - 10.5i)T^{2} \)
19 \( 1 + (-1.23 - 3.56i)T + (-14.9 + 11.7i)T^{2} \)
29 \( 1 + (-3.34 - 3.85i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-8.23 + 1.99i)T + (27.5 - 14.2i)T^{2} \)
37 \( 1 + (-2.58 + 0.123i)T + (36.8 - 3.51i)T^{2} \)
41 \( 1 + (-6.49 + 10.1i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (2.74 + 9.33i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (7.78 + 4.49i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.23 - 4.11i)T + (-12.4 + 51.5i)T^{2} \)
59 \( 1 + (-3.62 - 9.04i)T + (-42.7 + 40.7i)T^{2} \)
61 \( 1 + (-0.0304 + 0.0290i)T + (2.90 - 60.9i)T^{2} \)
67 \( 1 + (-2.50 - 1.78i)T + (21.9 + 63.3i)T^{2} \)
71 \( 1 + (1.17 + 2.57i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-2.54 + 13.1i)T + (-67.7 - 27.1i)T^{2} \)
79 \( 1 + (-0.619 + 0.788i)T + (-18.6 - 76.7i)T^{2} \)
83 \( 1 + (-8.62 + 5.54i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (1.40 - 5.79i)T + (-79.1 - 40.7i)T^{2} \)
97 \( 1 + (3.43 + 2.20i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.45376520770353119528618007688, −9.719403848960500953337527064527, −9.064107611619612824960505763564, −8.452042695341004420191649980632, −7.49944057670983764299392242948, −6.49146815719750360018698966575, −5.66021256514899072253960471924, −3.42408730532112087779488274164, −2.24296971649444476442254975630, −0.955658898718863265189842042207, 1.26621473575949095326624831912, 2.78418848663683865606263216348, 4.09411094963322562522464402533, 6.26353628415159829548786902974, 6.41126601276111700054086733764, 8.098764465488406183762633744225, 8.228971629365950328173840033822, 9.629913266188258050226725347219, 9.918765683710631827732165192322, 10.62579890430923340054486245458

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