Properties

Label 2-483-7.2-c1-0-7
Degree $2$
Conductor $483$
Sign $-0.876 - 0.482i$
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  + (−1.39 + 2.42i)2-s + (0.5 + 0.866i)3-s + (−2.91 − 5.05i)4-s + (0.973 − 1.68i)5-s − 2.79·6-s + (−2.64 − 0.113i)7-s + 10.7·8-s + (−0.499 + 0.866i)9-s + (2.72 + 4.72i)10-s + (1.34 + 2.32i)11-s + (2.91 − 5.05i)12-s + 4.19·13-s + (3.97 − 6.25i)14-s + 1.94·15-s + (−9.20 + 15.9i)16-s + (0.316 + 0.547i)17-s + ⋯
L(s)  = 1  + (−0.989 + 1.71i)2-s + (0.288 + 0.499i)3-s + (−1.45 − 2.52i)4-s + (0.435 − 0.754i)5-s − 1.14·6-s + (−0.999 − 0.0428i)7-s + 3.79·8-s + (−0.166 + 0.288i)9-s + (0.861 + 1.49i)10-s + (0.405 + 0.701i)11-s + (0.842 − 1.45i)12-s + 1.16·13-s + (1.06 − 1.67i)14-s + 0.502·15-s + (−2.30 + 3.98i)16-s + (0.0766 + 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.876 - 0.482i$
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: \((2,\ 483,\ (\ :1/2),\ -0.876 - 0.482i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.204503 + 0.795855i\)
\(L(\frac12)\) \(\approx\) \(0.204503 + 0.795855i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.113i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1.39 - 2.42i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.973 + 1.68i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.34 - 2.32i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.19T + 13T^{2} \)
17 \( 1 + (-0.316 - 0.547i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.77 - 4.81i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 5.57T + 29T^{2} \)
31 \( 1 + (-2.79 - 4.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.05 - 3.56i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.19T + 41T^{2} \)
43 \( 1 + 1.98T + 43T^{2} \)
47 \( 1 + (-0.146 + 0.253i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.08 - 8.80i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.65 + 6.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.14 + 5.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.57 + 2.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.00T + 71T^{2} \)
73 \( 1 + (6.93 + 12.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.89 - 13.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + (0.905 - 1.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.95T + 97T^{2} \)
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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

−10.63839595342634303051287216914, −10.03711515719921838975677665118, −9.272788678221357908129749286811, −8.720085971405114217695193537385, −7.921273898739320191534357603333, −6.66660492662197826720015349205, −6.08803343479461035651497522733, −5.06889785717764751210706576270, −3.99307796820410006619373144420, −1.35615717072271873587125412889, 0.78013281836934378093277937930, 2.39041908772093576831996885402, 3.09699381638539473364556951801, 4.08657725438873804623198555580, 6.23756570658600618080864172489, 7.14388761812123831013221037991, 8.464120185220750324795929637419, 8.914970113221708955546348219266, 9.877608534653952696282776709440, 10.59826446399147114851350582442

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