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Properties

Label 2-483-7.2-c1-0-12
Degree $2$
Conductor $483$
Sign $-0.594 - 0.804i$
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  + (−0.682 + 1.18i)2-s + (0.5 + 0.866i)3-s + (0.0677 + 0.117i)4-s + (0.663 − 1.14i)5-s − 1.36·6-s + (2.35 + 1.20i)7-s − 2.91·8-s + (−0.499 + 0.866i)9-s + (0.905 + 1.56i)10-s + (0.710 + 1.22i)11-s + (−0.0677 + 0.117i)12-s + 3.16·13-s + (−3.03 + 1.95i)14-s + 1.32·15-s + (1.85 − 3.21i)16-s + (−0.579 − 1.00i)17-s + ⋯
L(s)  = 1  + (−0.482 + 0.836i)2-s + (0.288 + 0.499i)3-s + (0.0338 + 0.0586i)4-s + (0.296 − 0.513i)5-s − 0.557·6-s + (0.889 + 0.456i)7-s − 1.03·8-s + (−0.166 + 0.288i)9-s + (0.286 + 0.495i)10-s + (0.214 + 0.370i)11-s + (−0.0195 + 0.0338i)12-s + 0.878·13-s + (−0.811 + 0.523i)14-s + 0.342·15-s + (0.463 − 0.803i)16-s + (−0.140 − 0.243i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.594 - 0.804i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.594 - 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.627626 + 1.24362i\)
\(L(\frac12)\) \(\approx\) \(0.627626 + 1.24362i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.35 - 1.20i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.682 - 1.18i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.663 + 1.14i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.710 - 1.22i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.16T + 13T^{2} \)
17 \( 1 + (0.579 + 1.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.28 - 7.42i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 7.45T + 29T^{2} \)
31 \( 1 + (0.933 + 1.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.89 + 6.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 + (0.868 - 1.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.07 - 3.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.34 + 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.90 + 11.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.04 - 7.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.18T + 71T^{2} \)
73 \( 1 + (-5.68 - 9.84i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.86 + 10.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.70T + 83T^{2} \)
89 \( 1 + (-3.17 + 5.50i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.408T + 97T^{2} \)
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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

−11.25491044113808862650119515535, −10.17082791006910905134118084674, −9.199607165038104802981777474795, −8.396780681286585308855571955832, −8.094007202884842702892152061939, −6.72424942078608064479164846786, −5.79443457360982472273715337188, −4.78644436327548772216165198592, −3.51072135777687319842508961262, −1.88196913396553043880581768061, 1.04268003487256538182380933392, 2.21720083322135796933028674679, 3.29894816254258549213863630150, 4.82345798852343495605169715191, 6.34080344453475003973703586959, 6.82685070539145687841318194781, 8.421608873236439088216490933128, 8.704316389689105618575817643555, 10.06273230705119156890911094710, 10.71476964122517624637448929065

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