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Properties

Label 2-1027-1027.394-c0-0-2
Degree $2$
Conductor $1027$
Sign $-0.394 + 0.918i$
Analytic cond. $0.512539$
Root an. cond. $0.715918$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.72 − 0.994i)2-s + (1.47 + 2.56i)4-s + 0.415i·5-s − 3.89i·8-s + (−0.5 − 0.866i)9-s + (0.413 − 0.716i)10-s + (−0.704 − 0.406i)11-s + (0.809 − 0.587i)13-s + (−2.39 + 4.14i)16-s + 1.98i·18-s + (−1.28 + 0.743i)19-s + (−1.06 + 0.614i)20-s + (0.809 + 1.40i)22-s + (0.669 − 1.15i)23-s + 0.827·25-s + (−1.97 + 0.207i)26-s + ⋯
L(s)  = 1  + (−1.72 − 0.994i)2-s + (1.47 + 2.56i)4-s + 0.415i·5-s − 3.89i·8-s + (−0.5 − 0.866i)9-s + (0.413 − 0.716i)10-s + (−0.704 − 0.406i)11-s + (0.809 − 0.587i)13-s + (−2.39 + 4.14i)16-s + 1.98i·18-s + (−1.28 + 0.743i)19-s + (−1.06 + 0.614i)20-s + (0.809 + 1.40i)22-s + (0.669 − 1.15i)23-s + 0.827·25-s + (−1.97 + 0.207i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $-0.394 + 0.918i$
Analytic conductor: \(0.512539\)
Root analytic conductor: \(0.715918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1027} (394, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1027,\ (\ :0),\ -0.394 + 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3828751038\)
\(L(\frac12)\) \(\approx\) \(0.3828751038\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + T \)
good2 \( 1 + (1.72 + 0.994i)T + (0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 - 0.415iT - T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.704 + 0.406i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.28 - 0.743i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.669 + 1.15i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.98iT - T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.64 - 0.951i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + 1.48iT - T^{2} \)
83 \( 1 + 1.73iT - T^{2} \)
89 \( 1 + (-1.01 - 0.587i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.360 - 0.207i)T + (0.5 - 0.866i)T^{2} \)
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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

−10.02777834266870999953117677088, −8.987494014083089143414328396461, −8.481162292925088710725482819715, −7.83354099186302306084258329855, −6.73329236335828902404552504946, −6.05520236626729053497898493684, −3.99818539146744067301266855726, −3.11862474283628699611305453460, −2.29121326069633076521606718783, −0.65302194485941688397919091799, 1.42285135650887549677651461213, 2.56654867647127186284322873496, 4.84908082650445957824452321829, 5.44567259094174741904979702843, 6.54819611145079492591593334553, 7.18499517602484460601867161437, 8.124521357486085493861496350985, 8.695985546337516193754265892166, 9.215260889671188400034613728874, 10.28669845262325820451637222770

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