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Properties

Label 2-459-51.50-c2-0-10
Degree $2$
Conductor $459$
Sign $0.878 + 0.478i$
Analytic cond. $12.5068$
Root an. cond. $3.53650$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.92i·2-s − 4.57·4-s − 4.30·5-s + 7.91i·7-s + 1.67i·8-s + 12.6i·10-s − 13.3·11-s + 15.9·13-s + 23.1·14-s − 13.3·16-s + (−8.12 + 14.9i)17-s + 36.2·19-s + 19.6·20-s + 38.9i·22-s + 27.6·23-s + ⋯
L(s)  = 1  − 1.46i·2-s − 1.14·4-s − 0.861·5-s + 1.13i·7-s + 0.209i·8-s + 1.26i·10-s − 1.20·11-s + 1.22·13-s + 1.65·14-s − 0.836·16-s + (−0.478 + 0.878i)17-s + 1.90·19-s + 0.984·20-s + 1.77i·22-s + 1.20·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.478i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $0.878 + 0.478i$
Analytic conductor: \(12.5068\)
Root analytic conductor: \(3.53650\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (458, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 459,\ (\ :1),\ 0.878 + 0.478i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.195131181\)
\(L(\frac12)\) \(\approx\) \(1.195131181\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 + (8.12 - 14.9i)T \)
good2 \( 1 + 2.92iT - 4T^{2} \)
5 \( 1 + 4.30T + 25T^{2} \)
7 \( 1 - 7.91iT - 49T^{2} \)
11 \( 1 + 13.3T + 121T^{2} \)
13 \( 1 - 15.9T + 169T^{2} \)
19 \( 1 - 36.2T + 361T^{2} \)
23 \( 1 - 27.6T + 529T^{2} \)
29 \( 1 - 33.4T + 841T^{2} \)
31 \( 1 - 41.1iT - 961T^{2} \)
37 \( 1 - 4.06iT - 1.36e3T^{2} \)
41 \( 1 - 37.0T + 1.68e3T^{2} \)
43 \( 1 + 36.7T + 1.84e3T^{2} \)
47 \( 1 + 40.4iT - 2.20e3T^{2} \)
53 \( 1 - 46.8iT - 2.80e3T^{2} \)
59 \( 1 - 71.2iT - 3.48e3T^{2} \)
61 \( 1 - 71.0iT - 3.72e3T^{2} \)
67 \( 1 + 48.0T + 4.48e3T^{2} \)
71 \( 1 + 116.T + 5.04e3T^{2} \)
73 \( 1 - 30.5iT - 5.32e3T^{2} \)
79 \( 1 + 125. iT - 6.24e3T^{2} \)
83 \( 1 - 57.9iT - 6.88e3T^{2} \)
89 \( 1 - 92.1iT - 7.92e3T^{2} \)
97 \( 1 - 27.5iT - 9.40e3T^{2} \)
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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

−10.87194281821616741717806978539, −10.24645826874677559434622131823, −9.003028021220973790107396542497, −8.462177189852900626201155708793, −7.26411501172069901659864479910, −5.81299498766782865552346061852, −4.71235603914603044691171697229, −3.42563905065448190257814806793, −2.73545197470986065523954751492, −1.20733306631988421019412286459, 0.58432361944985940962473754216, 3.16478643080095611976092217518, 4.43743748706336266715000790055, 5.29134758045334688839891541910, 6.45963509360781663253720896045, 7.53916276832779266549342417567, 7.66107744673441762867379485267, 8.734629066512646445309438150236, 9.848906996755254797630484344532, 11.06905402628776067827170506578

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