Properties

Label 2-483-161.153-c1-0-27
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: $\chi_{483} (475, \cdot )$
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

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

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