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Properties

Label 2-469-469.412-c0-0-0
Degree $2$
Conductor $469$
Sign $0.991 - 0.133i$
Analytic cond. $0.234061$
Root an. cond. $0.483799$
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  + (0.601 + 0.573i)2-s + (−0.0147 − 0.309i)4-s + (0.235 − 0.971i)7-s + (0.712 − 0.822i)8-s + (−0.959 + 0.281i)9-s + (0.839 + 1.17i)11-s + (0.698 − 0.449i)14-s + (0.591 − 0.0564i)16-s + (−0.738 − 0.380i)18-s + (−0.171 + 1.19i)22-s + (−1.54 + 1.21i)23-s + (−0.654 − 0.755i)25-s + (−0.304 − 0.0586i)28-s + (−0.235 + 0.408i)29-s + (−0.467 − 0.367i)32-s + ⋯
L(s)  = 1  + (0.601 + 0.573i)2-s + (−0.0147 − 0.309i)4-s + (0.235 − 0.971i)7-s + (0.712 − 0.822i)8-s + (−0.959 + 0.281i)9-s + (0.839 + 1.17i)11-s + (0.698 − 0.449i)14-s + (0.591 − 0.0564i)16-s + (−0.738 − 0.380i)18-s + (−0.171 + 1.19i)22-s + (−1.54 + 1.21i)23-s + (−0.654 − 0.755i)25-s + (−0.304 − 0.0586i)28-s + (−0.235 + 0.408i)29-s + (−0.467 − 0.367i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.991 - 0.133i$
Analytic conductor: \(0.234061\)
Root analytic conductor: \(0.483799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (412, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 469,\ (\ :0),\ 0.991 - 0.133i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.154000104\)
\(L(\frac12)\) \(\approx\) \(1.154000104\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.235 + 0.971i)T \)
67 \( 1 + (0.995 - 0.0950i)T \)
good2 \( 1 + (-0.601 - 0.573i)T + (0.0475 + 0.998i)T^{2} \)
3 \( 1 + (0.959 - 0.281i)T^{2} \)
5 \( 1 + (0.654 + 0.755i)T^{2} \)
11 \( 1 + (-0.839 - 1.17i)T + (-0.327 + 0.945i)T^{2} \)
13 \( 1 + (0.786 + 0.618i)T^{2} \)
17 \( 1 + (0.995 + 0.0950i)T^{2} \)
19 \( 1 + (0.888 - 0.458i)T^{2} \)
23 \( 1 + (1.54 - 1.21i)T + (0.235 - 0.971i)T^{2} \)
29 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.786 - 0.618i)T^{2} \)
37 \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.580 + 0.814i)T^{2} \)
43 \( 1 + (1.49 + 0.961i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 + (-0.723 - 0.690i)T^{2} \)
53 \( 1 + (-1.56 + 1.00i)T + (0.415 - 0.909i)T^{2} \)
59 \( 1 + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.327 + 0.945i)T^{2} \)
71 \( 1 + (-0.0552 - 1.15i)T + (-0.995 + 0.0950i)T^{2} \)
73 \( 1 + (0.327 + 0.945i)T^{2} \)
79 \( 1 + (0.279 - 0.0538i)T + (0.928 - 0.371i)T^{2} \)
83 \( 1 + (-0.981 + 0.189i)T^{2} \)
89 \( 1 + (0.959 + 0.281i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−11.39657339358862320314522834364, −10.13931312899715512325206394959, −9.796977467522524207958921763864, −8.335423228986523885459055611883, −7.38968959957527476503277095788, −6.58370320472570887412285613814, −5.61975580407820207276097285724, −4.58707854820947426147429108928, −3.76359856007305651369530779351, −1.74405819130659089119328917482, 2.19395926388987145533005302223, 3.24213539208999837288510726343, 4.22062157233810351768440088206, 5.60119507903777712689846378850, 6.19415354004797674062295759305, 7.85356669730853126808871477794, 8.561646198050393937524448830341, 9.271916291901708822350470296698, 10.71489716607468370186032851723, 11.71074437833429849684656836164

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