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Properties

Label 2-483-161.10-c1-0-23
Degree $2$
Conductor $483$
Sign $-0.269 + 0.962i$
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  + (0.761 − 2.20i)2-s + (0.458 + 0.888i)3-s + (−2.69 − 2.11i)4-s + (0.293 − 0.0279i)5-s + (2.30 − 0.331i)6-s + (2.64 + 0.132i)7-s + (−2.79 + 1.79i)8-s + (−0.580 + 0.814i)9-s + (0.161 − 0.666i)10-s + (3.62 − 1.25i)11-s + (0.648 − 3.36i)12-s + (0.269 + 0.918i)13-s + (2.30 − 5.71i)14-s + (0.159 + 0.247i)15-s + (0.209 + 0.863i)16-s + (−5.89 − 2.35i)17-s + ⋯
L(s)  = 1  + (0.538 − 1.55i)2-s + (0.264 + 0.513i)3-s + (−1.34 − 1.05i)4-s + (0.131 − 0.0125i)5-s + (0.941 − 0.135i)6-s + (0.998 + 0.0499i)7-s + (−0.988 + 0.635i)8-s + (−0.193 + 0.271i)9-s + (0.0511 − 0.210i)10-s + (1.09 − 0.378i)11-s + (0.187 − 0.971i)12-s + (0.0748 + 0.254i)13-s + (0.615 − 1.52i)14-s + (0.0411 + 0.0639i)15-s + (0.0523 + 0.215i)16-s + (−1.42 − 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.269 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.269 + 0.962i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.269 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32193 - 1.74338i\)
\(L(\frac12)\) \(\approx\) \(1.32193 - 1.74338i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.458 - 0.888i)T \)
7 \( 1 + (-2.64 - 0.132i)T \)
23 \( 1 + (4.65 - 1.15i)T \)
good2 \( 1 + (-0.761 + 2.20i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (-0.293 + 0.0279i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-3.62 + 1.25i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.269 - 0.918i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (5.89 + 2.35i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-6.87 + 2.75i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.806 - 5.60i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-7.80 + 0.371i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (5.92 + 4.21i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-1.00 - 0.457i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (4.84 - 7.54i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-4.82 + 2.78i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.35 + 3.51i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (6.04 + 1.46i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-12.8 - 6.64i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-2.31 - 12.0i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (5.34 + 6.16i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (5.18 - 6.59i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (6.46 - 6.78i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-5.68 - 12.4i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.0342 - 0.718i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (3.88 - 8.50i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.05735235392589221654284165527, −9.947812796058446227208621343509, −9.284589583329868540429607098143, −8.463241768315159959367965742011, −7.03105673618577516870646098455, −5.47529629456918719357719524509, −4.56959445236495232581859755957, −3.79335780784847221419141086938, −2.58660296729664876839643022300, −1.38593609645721434860777592849, 1.82637605344638631719328429003, 3.86697178316635323068724394957, 4.72661309848423560788753733793, 5.90511715150647930933977841741, 6.57029687616373887128331829858, 7.56780165560051379632748604311, 8.175475753511331271116640864526, 8.968819957978103624164099346292, 10.17323605223913655320723653429, 11.62631960285130276528133724941

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