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Properties

Label 2-459-153.50-c2-0-14
Degree $2$
Conductor $459$
Sign $0.971 - 0.237i$
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  + (−2.65 + 1.53i)2-s + (2.68 − 4.65i)4-s + (−4.03 + 6.99i)5-s + (−8.17 + 4.72i)7-s + 4.20i·8-s − 24.7i·10-s + (−5.27 − 9.13i)11-s + (10.9 − 18.9i)13-s + (14.4 − 25.0i)14-s + (4.31 + 7.46i)16-s + (−15.3 + 7.32i)17-s − 14.6·19-s + (21.7 + 37.5i)20-s + (27.9 + 16.1i)22-s + (−3.86 + 6.69i)23-s + ⋯
L(s)  = 1  + (−1.32 + 0.765i)2-s + (0.671 − 1.16i)4-s + (−0.807 + 1.39i)5-s + (−1.16 + 0.674i)7-s + 0.525i·8-s − 2.47i·10-s + (−0.479 − 0.830i)11-s + (0.839 − 1.45i)13-s + (1.03 − 1.78i)14-s + (0.269 + 0.466i)16-s + (−0.902 + 0.430i)17-s − 0.771·19-s + (1.08 + 1.87i)20-s + (1.27 + 0.733i)22-s + (−0.168 + 0.291i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $0.971 - 0.237i$
Analytic conductor: \(12.5068\)
Root analytic conductor: \(3.53650\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 459,\ (\ :1),\ 0.971 - 0.237i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2963992896\)
\(L(\frac12)\) \(\approx\) \(0.2963992896\)
\(L(2)\) 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 \( 1 + (15.3 - 7.32i)T \)
good2 \( 1 + (2.65 - 1.53i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (4.03 - 6.99i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (8.17 - 4.72i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (5.27 + 9.13i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-10.9 + 18.9i)T + (-84.5 - 146. i)T^{2} \)
19 \( 1 + 14.6T + 361T^{2} \)
23 \( 1 + (3.86 - 6.69i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-3.81 - 6.60i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-15.0 - 8.66i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 13.8iT - 1.36e3T^{2} \)
41 \( 1 + (-23.1 + 40.0i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-27.7 - 48.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (18.6 - 10.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 61.3iT - 2.80e3T^{2} \)
59 \( 1 + (-76.3 - 44.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (34.9 - 20.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (36.3 - 62.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 10.1T + 5.04e3T^{2} \)
73 \( 1 + 34.6iT - 5.32e3T^{2} \)
79 \( 1 + (-96.7 + 55.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-3.77 + 2.17i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 67.1iT - 7.92e3T^{2} \)
97 \( 1 + (89.7 - 51.8i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.63144903275051130589677795301, −10.03592660089001438802333976193, −8.787531475568583878048459776203, −8.232257830740851790899480079956, −7.32565504939440785654068952024, −6.35191921435320839135267656944, −5.95404514853067390136722274363, −3.64015671041830231475188940312, −2.78330548014685359614973587388, −0.29815818877858528426596810102, 0.74728358448889850776841696691, 2.16303434873374981689846505340, 3.86706875934172190697578788103, 4.67130114574868177782688806932, 6.49425894311860350765099228705, 7.48519325740654075335723560870, 8.457264894998945659472718597386, 9.121706302457498480319031918968, 9.712507839267696632463042824833, 10.72444783924962490723025934341

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