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Properties

Label 2-483-21.20-c1-0-11
Degree $2$
Conductor $483$
Sign $0.498 + 0.867i$
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  − 2.69i·2-s + (−0.983 + 1.42i)3-s − 5.24·4-s − 3.85·5-s + (3.83 + 2.64i)6-s + (2.38 + 1.14i)7-s + 8.74i·8-s + (−1.06 − 2.80i)9-s + 10.3i·10-s − 0.326i·11-s + (5.16 − 7.48i)12-s − 1.38i·13-s + (3.06 − 6.42i)14-s + (3.79 − 5.49i)15-s + 13.0·16-s + 5.56·17-s + ⋯
L(s)  = 1  − 1.90i·2-s + (−0.567 + 0.823i)3-s − 2.62·4-s − 1.72·5-s + (1.56 + 1.08i)6-s + (0.902 + 0.430i)7-s + 3.09i·8-s + (−0.355 − 0.934i)9-s + 3.28i·10-s − 0.0984i·11-s + (1.49 − 2.15i)12-s − 0.383i·13-s + (0.820 − 1.71i)14-s + (0.979 − 1.41i)15-s + 3.26·16-s + 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.498 + 0.867i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.498 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591444 - 0.342376i\)
\(L(\frac12)\) \(\approx\) \(0.591444 - 0.342376i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.983 - 1.42i)T \)
7 \( 1 + (-2.38 - 1.14i)T \)
23 \( 1 - iT \)
good2 \( 1 + 2.69iT - 2T^{2} \)
5 \( 1 + 3.85T + 5T^{2} \)
11 \( 1 + 0.326iT - 11T^{2} \)
13 \( 1 + 1.38iT - 13T^{2} \)
17 \( 1 - 5.56T + 17T^{2} \)
19 \( 1 + 1.06iT - 19T^{2} \)
29 \( 1 - 9.99iT - 29T^{2} \)
31 \( 1 + 0.901iT - 31T^{2} \)
37 \( 1 - 3.32T + 37T^{2} \)
41 \( 1 + 2.51T + 41T^{2} \)
43 \( 1 + 1.16T + 43T^{2} \)
47 \( 1 - 9.48T + 47T^{2} \)
53 \( 1 - 9.04iT - 53T^{2} \)
59 \( 1 + 0.627T + 59T^{2} \)
61 \( 1 + 2.29iT - 61T^{2} \)
67 \( 1 - 8.95T + 67T^{2} \)
71 \( 1 + 7.51iT - 71T^{2} \)
73 \( 1 - 0.951iT - 73T^{2} \)
79 \( 1 + 6.01T + 79T^{2} \)
83 \( 1 + 4.55T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 - 6.39iT - 97T^{2} \)
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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

−10.90104179961250821719614512109, −10.48403599800795296552341149537, −9.260472870717722090043935134949, −8.527954962672908668938207646526, −7.64338099364810492978494532952, −5.41006472988469761489757180872, −4.69281663277529310642408236514, −3.78179922275475960294070752110, −3.04724829726615230560504862580, −0.958700891856675090192433190477, 0.67359799172708364408133889710, 3.91319829877536958920958746188, 4.71309455350566040518437869382, 5.68454080575969753016727939048, 6.80480534886171954298680790771, 7.57544964141047233486275241616, 7.919026014661386845364546590964, 8.581349552472105608916608193149, 10.12427579476516273507615145915, 11.41871977548773790221587511028

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