Properties

Label 2-483-161.61-c1-0-7
Degree $2$
Conductor $483$
Sign $0.939 - 0.342i$
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  + (−1.96 − 1.54i)2-s + (0.814 + 0.580i)3-s + (0.997 + 4.11i)4-s + (2.33 − 0.450i)5-s + (−0.703 − 2.39i)6-s + (−0.792 + 2.52i)7-s + (2.31 − 5.06i)8-s + (0.327 + 0.945i)9-s + (−5.27 − 2.72i)10-s + (3.31 + 4.21i)11-s + (−1.57 + 3.92i)12-s + (−1.53 − 2.38i)13-s + (5.44 − 3.72i)14-s + (2.16 + 0.988i)15-s + (−4.82 + 2.48i)16-s + (−2.09 − 1.99i)17-s + ⋯
L(s)  = 1  + (−1.38 − 1.09i)2-s + (0.470 + 0.334i)3-s + (0.498 + 2.05i)4-s + (1.04 − 0.201i)5-s + (−0.287 − 0.977i)6-s + (−0.299 + 0.954i)7-s + (0.817 − 1.79i)8-s + (0.109 + 0.315i)9-s + (−1.66 − 0.860i)10-s + (0.998 + 1.26i)11-s + (−0.453 + 1.13i)12-s + (−0.425 − 0.661i)13-s + (1.45 − 0.996i)14-s + (0.558 + 0.255i)15-s + (−1.20 + 0.622i)16-s + (−0.507 − 0.484i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.939 - 0.342i$
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: \((2,\ 483,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.889376 + 0.156853i\)
\(L(\frac12)\) \(\approx\) \(0.889376 + 0.156853i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.792 - 2.52i)T \)
23 \( 1 + (-2.72 - 3.94i)T \)
good2 \( 1 + (1.96 + 1.54i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (-2.33 + 0.450i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (-3.31 - 4.21i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (1.53 + 2.38i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (2.09 + 1.99i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (5.86 - 5.58i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (2.12 - 0.625i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-0.694 - 7.26i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (-2.18 + 0.754i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (-4.41 + 3.82i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (-6.34 + 2.89i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (6.81 + 3.93i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-12.7 - 0.609i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (-4.16 + 8.08i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (1.35 + 1.90i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (-0.407 - 1.01i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (-0.579 + 4.03i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-2.69 + 0.653i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (-7.83 + 0.373i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (-0.179 + 0.206i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-14.6 - 1.39i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (-5.61 - 6.48i)T + (-13.8 + 96.0i)T^{2} \)
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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

−10.61126629653820262940336377440, −9.975661228532706946452510817603, −9.263123536013318550703221125667, −8.953916139946063672958844202124, −7.84276378483780141021258578756, −6.67968665793711738800741864676, −5.30345987154618964604962287752, −3.74562156898374537401546394567, −2.39542534628584165844693367852, −1.77086890687993219607607116198, 0.832359817499852363371902052769, 2.31616249631407175960143270591, 4.25052335787101034758890912961, 6.11795437320425950318816769144, 6.44462671848368102444359140794, 7.23234211207574331393957982817, 8.360823906848412382702474080321, 9.088106253761932405144489761344, 9.612884872892301825950537522081, 10.62117282423268575808928930875

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