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Properties

Label 2-473-473.343-c1-0-16
Degree $2$
Conductor $473$
Sign $0.903 - 0.429i$
Analytic cond. $3.77692$
Root an. cond. $1.94343$
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  + (1.61 + 1.17i)4-s + (0.927 − 2.85i)9-s + (0.5 + 3.27i)11-s + (0.786 + 0.255i)13-s + (1.23 + 3.80i)16-s + (7.84 − 2.54i)17-s − 4.07·23-s + (−4.04 + 2.93i)25-s + (−1.05 + 3.25i)31-s + (4.85 − 3.52i)36-s + (3.85 + 5.30i)41-s + 6.55i·43-s + (−3.04 + 5.89i)44-s + (8.85 − 6.43i)47-s + (−2.16 − 6.65i)49-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)4-s + (0.309 − 0.951i)9-s + (0.150 + 0.988i)11-s + (0.218 + 0.0708i)13-s + (0.309 + 0.951i)16-s + (1.90 − 0.617i)17-s − 0.849·23-s + (−0.809 + 0.587i)25-s + (−0.190 + 0.585i)31-s + (0.809 − 0.587i)36-s + (0.601 + 0.828i)41-s + 0.999i·43-s + (−0.459 + 0.888i)44-s + (1.29 − 0.938i)47-s + (−0.309 − 0.951i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $0.903 - 0.429i$
Analytic conductor: \(3.77692\)
Root analytic conductor: \(1.94343\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{473} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 473,\ (\ :1/2),\ 0.903 - 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73191 + 0.390479i\)
\(L(\frac12)\) \(\approx\) \(1.73191 + 0.390479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.5 - 3.27i)T \)
43 \( 1 - 6.55iT \)
good2 \( 1 + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.786 - 0.255i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-7.84 + 2.54i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 4.07T + 23T^{2} \)
29 \( 1 + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.05 - 3.25i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.85 - 5.30i)T + (-12.6 + 38.9i)T^{2} \)
47 \( 1 + (-8.85 + 6.43i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.44 + 13.6i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (12.0 + 8.78i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.38 - 1.09i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.45 - 1.44i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (5.87 - 18.0i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.25737099199043621877632467666, −10.03526637476898615810869902397, −9.509013318487268895512391033910, −8.153992845398582974316823525825, −7.39056918556683682964533655248, −6.61140043829916620529090671955, −5.54719766238720288448543253582, −4.03342976494380809147158939154, −3.13833038780399078367135923942, −1.61449630975523071626262183026, 1.34863001345825413328462444604, 2.71300954150793427891747979512, 4.08302712320708524484468207068, 5.72446886973747137046195299768, 5.91912707184602819622488565942, 7.47900036068885220876743161092, 7.959561981315063148784067991874, 9.286880186110562465656364405058, 10.46119542034651246176905629646, 10.63135253716332310905456810400

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