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Properties

Label 2-459-153.50-c2-0-16
Degree $2$
Conductor $459$
Sign $0.567 - 0.823i$
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  + (−0.670 + 0.387i)2-s + (−1.70 + 2.94i)4-s + (−1.35 + 2.34i)5-s + (8.30 − 4.79i)7-s − 5.73i·8-s − 2.09i·10-s + (5.14 + 8.91i)11-s + (8.91 − 15.4i)13-s + (−3.71 + 6.42i)14-s + (−4.58 − 7.93i)16-s + (−14.9 + 8.07i)17-s + 25.6·19-s + (−4.59 − 7.96i)20-s + (−6.90 − 3.98i)22-s + (12.5 − 21.6i)23-s + ⋯
L(s)  = 1  + (−0.335 + 0.193i)2-s + (−0.425 + 0.736i)4-s + (−0.270 + 0.468i)5-s + (1.18 − 0.684i)7-s − 0.716i·8-s − 0.209i·10-s + (0.467 + 0.810i)11-s + (0.685 − 1.18i)13-s + (−0.265 + 0.459i)14-s + (−0.286 − 0.496i)16-s + (−0.880 + 0.474i)17-s + 1.35·19-s + (−0.229 − 0.398i)20-s + (−0.313 − 0.181i)22-s + (0.543 − 0.941i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $0.567 - 0.823i$
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: \((2,\ 459,\ (\ :1),\ 0.567 - 0.823i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.469306251\)
\(L(\frac12)\) \(\approx\) \(1.469306251\)
\(L(2)\) 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 \( 1 + (14.9 - 8.07i)T \)
good2 \( 1 + (0.670 - 0.387i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (1.35 - 2.34i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-8.30 + 4.79i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-5.14 - 8.91i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-8.91 + 15.4i)T + (-84.5 - 146. i)T^{2} \)
19 \( 1 - 25.6T + 361T^{2} \)
23 \( 1 + (-12.5 + 21.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-8.47 - 14.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (20.6 + 11.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 50.0iT - 1.36e3T^{2} \)
41 \( 1 + (19.5 - 33.8i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-18.1 - 31.4i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-25.6 + 14.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 39.2iT - 2.80e3T^{2} \)
59 \( 1 + (-64.6 - 37.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-23.7 + 13.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-53.3 + 92.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 25.8T + 5.04e3T^{2} \)
73 \( 1 - 58.8iT - 5.32e3T^{2} \)
79 \( 1 + (69.1 - 39.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-59.2 + 34.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 177. iT - 7.92e3T^{2} \)
97 \( 1 + (-24.6 + 14.2i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.97911694681439465388614361961, −10.13568225941570289442286257332, −8.994664240658622659280704947726, −8.149596587170034886813009089217, −7.48429841975062240946545589615, −6.70417994852350493279686209427, −5.05320058671135204499633295705, −4.15429308784012801143265136743, −3.07978117002495421702051000444, −1.13571373533816609123532071978, 0.922365465374380136083572699757, 2.05161411433311468648339873432, 3.97547893479585164916985678039, 5.05143936737025139871380158676, 5.73025547439254483460102811137, 7.07366620452728842243879194780, 8.465355393012069470187372414471, 8.845911707604195379402788053355, 9.555951792120699770139739197507, 11.02746947116527281620085916462

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