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Properties

Label 2-483-7.4-c1-0-21
Degree $2$
Conductor $483$
Sign $0.842 + 0.538i$
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  + (0.526 + 0.912i)2-s + (0.5 − 0.866i)3-s + (0.445 − 0.770i)4-s + (−1.08 − 1.88i)5-s + 1.05·6-s + (1.36 + 2.26i)7-s + 3.04·8-s + (−0.499 − 0.866i)9-s + (1.14 − 1.98i)10-s + (1.31 − 2.27i)11-s + (−0.445 − 0.770i)12-s − 3.30·13-s + (−1.34 + 2.43i)14-s − 2.17·15-s + (0.713 + 1.23i)16-s + (1.48 − 2.57i)17-s + ⋯
L(s)  = 1  + (0.372 + 0.645i)2-s + (0.288 − 0.499i)3-s + (0.222 − 0.385i)4-s + (−0.486 − 0.842i)5-s + 0.430·6-s + (0.515 + 0.856i)7-s + 1.07·8-s + (−0.166 − 0.288i)9-s + (0.362 − 0.627i)10-s + (0.395 − 0.684i)11-s + (−0.128 − 0.222i)12-s − 0.917·13-s + (−0.360 + 0.651i)14-s − 0.561·15-s + (0.178 + 0.309i)16-s + (0.360 − 0.623i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.842 + 0.538i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.842 + 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93214 - 0.564768i\)
\(L(\frac12)\) \(\approx\) \(1.93214 - 0.564768i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 - 2.26i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.526 - 0.912i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.08 + 1.88i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.31 + 2.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.30T + 13T^{2} \)
17 \( 1 + (-1.48 + 2.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.348 + 0.604i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 7.31T + 29T^{2} \)
31 \( 1 + (2.80 - 4.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.26 - 3.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.28T + 41T^{2} \)
43 \( 1 + 7.79T + 43T^{2} \)
47 \( 1 + (-1.90 - 3.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.56 - 2.71i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.08 - 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.742 + 1.28i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.63 - 9.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + (0.352 - 0.611i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.30 - 7.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + (-0.166 - 0.289i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.124T + 97T^{2} \)
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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

−11.11735445061438925286579418538, −9.876483737450486983714264072165, −8.790211166896439743931869695567, −8.154748775753666537264806238418, −7.21457875691199795846076096854, −6.22958479130867323838529265867, −5.23109100752831122265282786123, −4.52852218594354204263313487282, −2.72023167085463195102021878927, −1.20640667234527928484342250152, 1.98407105637847481472499540306, 3.24680072136653892287154921343, 4.04646638964194263271267773978, 4.86969169169238726336312091503, 6.70377104407246597751959680038, 7.51806301796932594657630436510, 8.107394772490787013301132592830, 9.637502536181673902789609723775, 10.42930103081964893922601806429, 11.02354214794406839901942108860

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