Properties

Label 2-483-7.2-c1-0-5
Degree $2$
Conductor $483$
Sign $0.109 + 0.994i$
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  + (−1.25 + 2.17i)2-s + (0.5 + 0.866i)3-s + (−2.16 − 3.74i)4-s + (−2.15 + 3.73i)5-s − 2.51·6-s + (1.42 + 2.22i)7-s + 5.85·8-s + (−0.499 + 0.866i)9-s + (−5.42 − 9.40i)10-s + (−0.864 − 1.49i)11-s + (2.16 − 3.74i)12-s − 2.88·13-s + (−6.64 + 0.306i)14-s − 4.31·15-s + (−3.03 + 5.25i)16-s + (2.46 + 4.27i)17-s + ⋯
L(s)  = 1  + (−0.889 + 1.54i)2-s + (0.288 + 0.499i)3-s + (−1.08 − 1.87i)4-s + (−0.965 + 1.67i)5-s − 1.02·6-s + (0.539 + 0.842i)7-s + 2.06·8-s + (−0.166 + 0.288i)9-s + (−1.71 − 2.97i)10-s + (−0.260 − 0.451i)11-s + (0.624 − 1.08i)12-s − 0.801·13-s + (−1.77 + 0.0818i)14-s − 1.11·15-s + (−0.757 + 1.31i)16-s + (0.598 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.109 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.109 + 0.994i$
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: \((2,\ 483,\ (\ :1/2),\ 0.109 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.412941 - 0.370060i\)
\(L(\frac12)\) \(\approx\) \(0.412941 - 0.370060i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 - 2.22i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1.25 - 2.17i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (2.15 - 3.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.864 + 1.49i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.88T + 13T^{2} \)
17 \( 1 + (-2.46 - 4.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.22 - 2.11i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 8.05T + 29T^{2} \)
31 \( 1 + (0.457 + 0.793i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.57 + 7.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.60T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + (-0.199 + 0.346i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.27 + 5.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.917 - 1.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.650 + 1.12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.10 - 8.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.00T + 71T^{2} \)
73 \( 1 + (-1.30 - 2.26i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.09 - 8.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + (7.49 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.07T + 97T^{2} \)
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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

−11.27752930333724454763148185898, −10.45415955927797990984855748834, −9.819165805263082626560846447424, −8.552160056077467560900164965607, −8.031249217391877301365620630881, −7.34027656784809240481684266745, −6.32673060117596687740826745553, −5.50250125500398025538565398791, −4.10326465661217841929067795252, −2.67268366338395328094695585087, 0.47379162985919674995987383752, 1.41461360840507551873904449931, 2.91775488674725555285131155745, 4.31496485695622351621984208505, 4.85355418735337844085594291372, 7.31913364558359978795307794347, 7.928674909324020828784319672635, 8.600825157631603425423167916697, 9.437537632847720633580778680197, 10.23995245316993649429289056912

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