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Properties

Label 2-2e9-128.101-c1-0-2
Degree $2$
Conductor $512$
Sign $-0.137 - 0.990i$
Analytic cond. $4.08834$
Root an. cond. $2.02196$
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  + (−2.34 + 1.25i)3-s + (−2.35 − 2.86i)5-s + (1.77 − 1.18i)7-s + (2.27 − 3.40i)9-s + (−3.00 + 0.911i)11-s + (4.40 + 3.61i)13-s + (9.11 + 3.77i)15-s + (−2.12 + 0.878i)17-s + (0.222 + 2.26i)19-s + (−2.68 + 5.01i)21-s + (−4.14 + 0.825i)23-s + (−1.70 + 8.56i)25-s + (−0.285 + 2.89i)27-s + (−1.05 + 3.49i)29-s + (0.733 + 0.733i)31-s + ⋯
L(s)  = 1  + (−1.35 + 0.724i)3-s + (−1.05 − 1.28i)5-s + (0.670 − 0.448i)7-s + (0.757 − 1.13i)9-s + (−0.905 + 0.274i)11-s + (1.22 + 1.00i)13-s + (2.35 + 0.975i)15-s + (−0.514 + 0.213i)17-s + (0.0511 + 0.519i)19-s + (−0.584 + 1.09i)21-s + (−0.864 + 0.172i)23-s + (−0.340 + 1.71i)25-s + (−0.0549 + 0.557i)27-s + (−0.196 + 0.648i)29-s + (0.131 + 0.131i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $-0.137 - 0.990i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ -0.137 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.312536 + 0.358890i\)
\(L(\frac12)\) \(\approx\) \(0.312536 + 0.358890i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (2.34 - 1.25i)T + (1.66 - 2.49i)T^{2} \)
5 \( 1 + (2.35 + 2.86i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (-1.77 + 1.18i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (3.00 - 0.911i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (-4.40 - 3.61i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (2.12 - 0.878i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-0.222 - 2.26i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (4.14 - 0.825i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (1.05 - 3.49i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (-0.733 - 0.733i)T + 31iT^{2} \)
37 \( 1 + (-1.40 - 0.138i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (-1.67 - 8.43i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-2.27 - 1.21i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (-4.22 - 10.1i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (0.810 + 2.67i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (-6.48 + 5.32i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (-4.18 - 7.83i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (1.02 + 1.91i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (-4.85 - 7.27i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (9.35 + 6.24i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-1.50 + 3.63i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-0.375 + 0.0370i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (-5.47 - 1.09i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (4.20 + 4.20i)T + 97iT^{2} \)
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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

−11.21904522930334310182043452466, −10.56212165496652572594313819483, −9.429095884341360088457466364184, −8.410177468258554331934441677194, −7.68456161897740351294429164996, −6.30309041202574520223511482568, −5.29065634173825519614730784736, −4.45275059877369382371277045203, −4.01307137091810914896808852085, −1.24965786084827960736527368960, 0.38237351974156344060890263644, 2.44756967695468000056183419622, 3.84888851656661993969099909287, 5.29556865736065908221401422637, 6.03909291905767057059177480542, 6.97240414254638533861342375187, 7.76170302560461545665986599313, 8.486511988918627481258854525594, 10.39513585837886792630019411452, 10.88030044435153593256252894805

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