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Properties

Label 2-2e9-128.101-c1-0-11
Degree $2$
Conductor $512$
Sign $0.926 + 0.376i$
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  + (2.04 − 1.09i)3-s + (1.40 + 1.71i)5-s + (3.76 − 2.51i)7-s + (1.32 − 1.97i)9-s + (−2.34 + 0.711i)11-s + (0.439 + 0.360i)13-s + (4.75 + 1.96i)15-s + (−7.29 + 3.01i)17-s + (−0.155 − 1.58i)19-s + (4.95 − 9.26i)21-s + (−1.54 + 0.307i)23-s + (0.0147 − 0.0742i)25-s + (−0.141 + 1.43i)27-s + (−1.22 + 4.04i)29-s + (0.936 + 0.936i)31-s + ⋯
L(s)  = 1  + (1.18 − 0.631i)3-s + (0.629 + 0.767i)5-s + (1.42 − 0.951i)7-s + (0.440 − 0.659i)9-s + (−0.707 + 0.214i)11-s + (0.121 + 0.0999i)13-s + (1.22 + 0.508i)15-s + (−1.76 + 0.732i)17-s + (−0.0357 − 0.363i)19-s + (1.08 − 2.02i)21-s + (−0.321 + 0.0640i)23-s + (0.00295 − 0.0148i)25-s + (−0.0272 + 0.276i)27-s + (−0.227 + 0.750i)29-s + (0.168 + 0.168i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.926 + 0.376i$
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: \((2,\ 512,\ (\ :1/2),\ 0.926 + 0.376i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37427 - 0.463720i\)
\(L(\frac12)\) \(\approx\) \(2.37427 - 0.463720i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 + 1.09i)T + (1.66 - 2.49i)T^{2} \)
5 \( 1 + (-1.40 - 1.71i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (-3.76 + 2.51i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (2.34 - 0.711i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (-0.439 - 0.360i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (7.29 - 3.01i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (0.155 + 1.58i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (1.54 - 0.307i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (1.22 - 4.04i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (-0.936 - 0.936i)T + 31iT^{2} \)
37 \( 1 + (-7.54 - 0.742i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (0.504 + 2.53i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (8.44 + 4.51i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (4.89 + 11.8i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (0.556 + 1.83i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (-0.239 + 0.196i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (-4.32 - 8.09i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (-3.98 - 7.45i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (3.92 + 5.87i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-2.97 - 1.98i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (4.13 - 9.98i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-12.8 + 1.26i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (8.22 + 1.63i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (6.77 + 6.77i)T + 97iT^{2} \)
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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

−10.76932734287290464184629916574, −10.08644138206875357744337409293, −8.719290833448191037144240282461, −8.214421054586484754954312531407, −7.26233614959355298191437878019, −6.63905998824984171443251425167, −5.07937053645655319173653315119, −3.93057396647064222638977799540, −2.47687983170183659209494235463, −1.76945141936669501532063054124, 1.90405251283957520173985658474, 2.75684129244724864086693690190, 4.42232839372385344178675055952, 5.02672506770876034984268246357, 6.13177744725407109026702764371, 7.933338127616915961905557095257, 8.324699531387453921476946164841, 9.174175660729206667586919980592, 9.658579640571055387282540893451, 10.98184439776784692801594639458

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