Properties

Label 2-483-161.61-c1-0-15
Degree $2$
Conductor $483$
Sign $0.307 + 0.951i$
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  + (−1.93 − 1.52i)2-s + (0.814 + 0.580i)3-s + (0.955 + 3.93i)4-s + (2.26 − 0.437i)5-s + (−0.693 − 2.36i)6-s + (2.27 − 1.35i)7-s + (2.09 − 4.59i)8-s + (0.327 + 0.945i)9-s + (−5.05 − 2.60i)10-s + (−3.37 − 4.29i)11-s + (−1.50 + 3.76i)12-s + (1.63 + 2.53i)13-s + (−6.45 − 0.847i)14-s + (2.10 + 0.959i)15-s + (−3.84 + 1.98i)16-s + (2.54 + 2.42i)17-s + ⋯
L(s)  = 1  + (−1.36 − 1.07i)2-s + (0.470 + 0.334i)3-s + (0.477 + 1.96i)4-s + (1.01 − 0.195i)5-s + (−0.282 − 0.963i)6-s + (0.859 − 0.510i)7-s + (0.742 − 1.62i)8-s + (0.109 + 0.315i)9-s + (−1.59 − 0.823i)10-s + (−1.01 − 1.29i)11-s + (−0.434 + 1.08i)12-s + (0.452 + 0.704i)13-s + (−1.72 − 0.226i)14-s + (0.542 + 0.247i)15-s + (−0.961 + 0.495i)16-s + (0.617 + 0.589i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.307 + 0.951i$
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: \((2,\ 483,\ (\ :1/2),\ 0.307 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869256 - 0.632856i\)
\(L(\frac12)\) \(\approx\) \(0.869256 - 0.632856i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.27 + 1.35i)T \)
23 \( 1 + (-1.29 + 4.61i)T \)
good2 \( 1 + (1.93 + 1.52i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (-2.26 + 0.437i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (3.37 + 4.29i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (-1.63 - 2.53i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-2.54 - 2.42i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (0.307 - 0.293i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (-8.46 + 2.48i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-0.321 - 3.36i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (-4.10 + 1.41i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (6.43 - 5.57i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (0.852 - 0.389i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (3.54 + 2.04i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.08 - 0.242i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (2.72 - 5.28i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (3.60 + 5.05i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (1.94 + 4.85i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (-1.96 + 13.6i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (7.99 - 1.94i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (-14.1 + 0.671i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (-1.19 + 1.38i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-7.77 - 0.742i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (-0.0813 - 0.0938i)T + (-13.8 + 96.0i)T^{2} \)
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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

−10.51113244258827462737681514543, −10.13330081254776754272746866768, −9.047086023506057076914779224563, −8.380593352762056934469735302264, −7.85369028588824419972521531238, −6.27542164820214927559644024751, −4.88340533369430346767249814408, −3.41745206710287475733879186443, −2.30899139050632980394229866031, −1.16448568908613193386004912739, 1.44778432520541135811499947150, 2.57363545819081804414186968751, 5.06808973518322293835467829467, 5.77254983200121814234305983240, 6.88958682540236596383277006411, 7.71905437202145058570665175220, 8.291566036871709708153631818252, 9.277019195013340241756482279828, 9.984706097653547847041184991319, 10.56921159064703563612022878906

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