Properties

Label 2-483-161.20-c1-0-10
Degree $2$
Conductor $483$
Sign $-0.206 - 0.978i$
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  + (0.0752 + 0.523i)2-s + (0.909 + 0.415i)3-s + (1.65 − 0.484i)4-s + (−1.76 + 2.03i)5-s + (−0.148 + 0.507i)6-s + (0.0392 + 2.64i)7-s + (0.817 + 1.78i)8-s + (0.654 + 0.755i)9-s + (−1.19 − 0.771i)10-s + (−2.70 − 0.388i)11-s + (1.70 + 0.244i)12-s + (−0.411 + 0.640i)13-s + (−1.38 + 0.219i)14-s + (−2.45 + 1.12i)15-s + (2.01 − 1.29i)16-s + (−0.483 − 0.141i)17-s + ⋯
L(s)  = 1  + (0.0532 + 0.370i)2-s + (0.525 + 0.239i)3-s + (0.825 − 0.242i)4-s + (−0.789 + 0.911i)5-s + (−0.0608 + 0.207i)6-s + (0.0148 + 0.999i)7-s + (0.288 + 0.632i)8-s + (0.218 + 0.251i)9-s + (−0.379 − 0.243i)10-s + (−0.814 − 0.117i)11-s + (0.491 + 0.0706i)12-s + (−0.114 + 0.177i)13-s + (−0.369 + 0.0586i)14-s + (−0.633 + 0.289i)15-s + (0.504 − 0.324i)16-s + (−0.117 − 0.0344i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.206 - 0.978i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09825 + 1.35477i\)
\(L(\frac12)\) \(\approx\) \(1.09825 + 1.35477i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.909 - 0.415i)T \)
7 \( 1 + (-0.0392 - 2.64i)T \)
23 \( 1 + (3.73 + 3.00i)T \)
good2 \( 1 + (-0.0752 - 0.523i)T + (-1.91 + 0.563i)T^{2} \)
5 \( 1 + (1.76 - 2.03i)T + (-0.711 - 4.94i)T^{2} \)
11 \( 1 + (2.70 + 0.388i)T + (10.5 + 3.09i)T^{2} \)
13 \( 1 + (0.411 - 0.640i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (0.483 + 0.141i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-0.0399 + 0.0117i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-3.79 - 1.11i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-5.71 + 2.60i)T + (20.3 - 23.4i)T^{2} \)
37 \( 1 + (-4.10 + 3.55i)T + (5.26 - 36.6i)T^{2} \)
41 \( 1 + (-5.63 - 4.88i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (-8.94 - 4.08i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 - 0.346iT - 47T^{2} \)
53 \( 1 + (1.27 + 1.98i)T + (-22.0 + 48.2i)T^{2} \)
59 \( 1 + (-3.76 + 5.86i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (4.33 + 9.49i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (12.8 - 1.84i)T + (64.2 - 18.8i)T^{2} \)
71 \( 1 + (0.383 + 2.66i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-2.43 - 8.27i)T + (-61.4 + 39.4i)T^{2} \)
79 \( 1 + (-0.221 + 0.345i)T + (-32.8 - 71.8i)T^{2} \)
83 \( 1 + (-8.16 - 9.42i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (-2.18 + 4.78i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (-0.138 + 0.160i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.17133963102048544690717575876, −10.49153924234612534188865587062, −9.475131186130585745679357402106, −8.169424088105930975714542964210, −7.75228554965322154216301498474, −6.65027599777040166708767546392, −5.82300079368664807053687470395, −4.53366428295403582117490707629, −3.00323764807153946305425329800, −2.36652315177318159487891727211, 1.00978194422326288520276379079, 2.57438184331482997661895034911, 3.78564829146704942885900357014, 4.62974940832096789916811849941, 6.21416705724817742654642388774, 7.58097173210796742060262857355, 7.68056584086175936742238407671, 8.784306947530588783728074762329, 10.08538576182363967937930898363, 10.68506558580525798036284427023

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