Properties

Label 2-483-7.2-c1-0-20
Degree $2$
Conductor $483$
Sign $0.415 + 0.909i$
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  + (1.03 − 1.78i)2-s + (0.5 + 0.866i)3-s + (−1.13 − 1.96i)4-s + (0.304 − 0.527i)5-s + 2.06·6-s + (2.05 + 1.66i)7-s − 0.546·8-s + (−0.499 + 0.866i)9-s + (−0.628 − 1.08i)10-s + (0.551 + 0.954i)11-s + (1.13 − 1.96i)12-s + 1.84·13-s + (5.10 − 1.95i)14-s + 0.608·15-s + (1.70 − 2.94i)16-s + (−2.94 − 5.09i)17-s + ⋯
L(s)  = 1  + (0.730 − 1.26i)2-s + (0.288 + 0.499i)3-s + (−0.566 − 0.980i)4-s + (0.136 − 0.235i)5-s + 0.843·6-s + (0.776 + 0.630i)7-s − 0.193·8-s + (−0.166 + 0.288i)9-s + (−0.198 − 0.344i)10-s + (0.166 + 0.287i)11-s + (0.326 − 0.566i)12-s + 0.512·13-s + (1.36 − 0.521i)14-s + 0.157·15-s + (0.425 − 0.736i)16-s + (−0.713 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.415 + 0.909i$
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: \((2,\ 483,\ (\ :1/2),\ 0.415 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10367 - 1.35140i\)
\(L(\frac12)\) \(\approx\) \(2.10367 - 1.35140i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.05 - 1.66i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1.03 + 1.78i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.304 + 0.527i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.551 - 0.954i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.84T + 13T^{2} \)
17 \( 1 + (2.94 + 5.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.544 - 0.942i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 0.804T + 29T^{2} \)
31 \( 1 + (4.38 + 7.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.55 - 7.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.79T + 41T^{2} \)
43 \( 1 - 5.23T + 43T^{2} \)
47 \( 1 + (-2.95 + 5.11i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.25 + 3.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.70 - 6.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.51 - 4.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.03 + 1.78i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.30T + 71T^{2} \)
73 \( 1 + (-0.578 - 1.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.58 - 13.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + (-3.96 + 6.86i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.0T + 97T^{2} \)
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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

−11.16946258713913444025656823518, −10.09735152121462702758994041780, −9.273922226104356829240771446094, −8.443199504298590762423532964328, −7.20223582233620849733019920535, −5.54510853617089380536003775565, −4.83750580270006806622825709993, −3.89914401420775694603099958491, −2.72317594257025182974737819363, −1.66535305996224421131580637362, 1.71426181594447319013837080181, 3.62143094720690895229786206940, 4.56573249961311117258118476974, 5.71729874318110765875061541694, 6.60054420097063704738581354193, 7.24224149805426122274493082961, 8.252027052795126634081603691565, 8.778051417656990767116398383208, 10.51548164337198617222406828574, 11.00702389036707274521128491145

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