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Properties

Label 2-2e9-128.101-c1-0-5
Degree $2$
Conductor $512$
Sign $0.841 + 0.540i$
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  + (−1.68 + 0.898i)3-s + (−1.56 − 1.91i)5-s + (−3.63 + 2.43i)7-s + (0.353 − 0.529i)9-s + (4.39 − 1.33i)11-s + (1.74 + 1.42i)13-s + (4.35 + 1.80i)15-s + (3.75 − 1.55i)17-s + (−0.804 − 8.16i)19-s + (3.93 − 7.36i)21-s + (1.74 − 0.347i)23-s + (−0.217 + 1.09i)25-s + (0.441 − 4.48i)27-s + (0.598 − 1.97i)29-s + (3.81 + 3.81i)31-s + ⋯
L(s)  = 1  + (−0.970 + 0.518i)3-s + (−0.701 − 0.854i)5-s + (−1.37 + 0.919i)7-s + (0.117 − 0.176i)9-s + (1.32 − 0.401i)11-s + (0.483 + 0.396i)13-s + (1.12 + 0.466i)15-s + (0.909 − 0.376i)17-s + (−0.184 − 1.87i)19-s + (0.858 − 1.60i)21-s + (0.364 − 0.0724i)23-s + (−0.0435 + 0.219i)25-s + (0.0850 − 0.863i)27-s + (0.111 − 0.366i)29-s + (0.685 + 0.685i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.841 + 0.540i$
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: \((2,\ 512,\ (\ :1/2),\ 0.841 + 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.684549 - 0.201042i\)
\(L(\frac12)\) \(\approx\) \(0.684549 - 0.201042i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 - 0.898i)T + (1.66 - 2.49i)T^{2} \)
5 \( 1 + (1.56 + 1.91i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (3.63 - 2.43i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-4.39 + 1.33i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (-1.74 - 1.42i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (-3.75 + 1.55i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (0.804 + 8.16i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (-1.74 + 0.347i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-0.598 + 1.97i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (-3.81 - 3.81i)T + 31iT^{2} \)
37 \( 1 + (1.04 + 0.102i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (-0.711 - 3.57i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-0.793 - 0.423i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (1.43 + 3.46i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (-1.79 - 5.91i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (-4.26 + 3.49i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (2.89 + 5.41i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (-2.61 - 4.89i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (0.916 + 1.37i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-6.52 - 4.36i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-4.05 + 9.80i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-0.630 + 0.0621i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (-3.26 - 0.649i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (12.6 + 12.6i)T + 97iT^{2} \)
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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

−11.04831075343197827555854759232, −9.801964285382529327753302163504, −9.098487976199701925101348913956, −8.456605036816878708961885270183, −6.82564299358751361402471601685, −6.15408340962519964843617957955, −5.14698205092377941232737732577, −4.22934874553344973229122403914, −3.02405596839892287322875326357, −0.62316588215023938811021263016, 1.07370969730662865065299498785, 3.43949877408571173461378848345, 3.86704020679096681011239846697, 5.78288029323312779481067321593, 6.49998095514358663737825090063, 7.04126924663490432384620156251, 7.981835599521462834018177550296, 9.458447138341027235951692867215, 10.29921445191837873568892476501, 10.95532892068163756858770949027

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