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Properties

Label 2-469-469.356-c0-0-0
Degree $2$
Conductor $469$
Sign $0.0909 - 0.995i$
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  + (−1.88 + 0.363i)2-s + (2.49 − 0.996i)4-s + (−0.327 + 0.945i)7-s + (−2.71 + 1.74i)8-s + (−0.654 − 0.755i)9-s + (0.462 + 1.90i)11-s + (0.273 − 1.89i)14-s + (2.54 − 2.42i)16-s + (1.50 + 1.18i)18-s + (−1.56 − 3.42i)22-s + (0.0552 + 0.0775i)23-s + (0.841 + 0.540i)25-s + (0.127 + 2.67i)28-s + (0.327 + 0.566i)29-s + (−2.03 + 2.86i)32-s + ⋯
L(s)  = 1  + (−1.88 + 0.363i)2-s + (2.49 − 0.996i)4-s + (−0.327 + 0.945i)7-s + (−2.71 + 1.74i)8-s + (−0.654 − 0.755i)9-s + (0.462 + 1.90i)11-s + (0.273 − 1.89i)14-s + (2.54 − 2.42i)16-s + (1.50 + 1.18i)18-s + (−1.56 − 3.42i)22-s + (0.0552 + 0.0775i)23-s + (0.841 + 0.540i)25-s + (0.127 + 2.67i)28-s + (0.327 + 0.566i)29-s + (−2.03 + 2.86i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0909 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0909 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.0909 - 0.995i$
Analytic conductor: \(0.234061\)
Root analytic conductor: \(0.483799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (356, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 469,\ (\ :0),\ 0.0909 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3415918683\)
\(L(\frac12)\) \(\approx\) \(0.3415918683\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.327 - 0.945i)T \)
67 \( 1 + (-0.723 + 0.690i)T \)
good2 \( 1 + (1.88 - 0.363i)T + (0.928 - 0.371i)T^{2} \)
3 \( 1 + (0.654 + 0.755i)T^{2} \)
5 \( 1 + (-0.841 - 0.540i)T^{2} \)
11 \( 1 + (-0.462 - 1.90i)T + (-0.888 + 0.458i)T^{2} \)
13 \( 1 + (-0.580 + 0.814i)T^{2} \)
17 \( 1 + (-0.723 - 0.690i)T^{2} \)
19 \( 1 + (0.786 - 0.618i)T^{2} \)
23 \( 1 + (-0.0552 - 0.0775i)T + (-0.327 + 0.945i)T^{2} \)
29 \( 1 + (-0.327 - 0.566i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.580 - 0.814i)T^{2} \)
37 \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.235 + 0.971i)T^{2} \)
43 \( 1 + (-0.223 - 1.55i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + (-0.981 + 0.189i)T^{2} \)
53 \( 1 + (-0.283 + 1.97i)T + (-0.959 - 0.281i)T^{2} \)
59 \( 1 + (-0.415 + 0.909i)T^{2} \)
61 \( 1 + (0.888 + 0.458i)T^{2} \)
71 \( 1 + (-0.437 + 0.175i)T + (0.723 - 0.690i)T^{2} \)
73 \( 1 + (0.888 + 0.458i)T^{2} \)
79 \( 1 + (-0.0395 + 0.829i)T + (-0.995 - 0.0950i)T^{2} \)
83 \( 1 + (-0.0475 + 0.998i)T^{2} \)
89 \( 1 + (0.654 - 0.755i)T^{2} \)
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

−11.29474181804329589189455671337, −10.10868885678023989115156477481, −9.499687132561553559993780187676, −8.927667498437508866395187039954, −8.071058945252824973634878074165, −6.89332920308136809182455103144, −6.46757823745712666266732635436, −5.18103082086648426679038894738, −2.96917983999307920847891502511, −1.71372483739206753914299019160, 0.818933133218357113705426334412, 2.58061037285104260090156658034, 3.65698632295369828403359970676, 5.81906951401099669048674319768, 6.77692635287451856627943951858, 7.72113906161112555994078529215, 8.544547429342873952753404413979, 9.063382577924161845792563645209, 10.29877025515328284091879358286, 10.79980373546239703330473233157

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