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Properties

Label 1-473-473.59-r0-0-0
Degree $1$
Conductor $473$
Sign $-0.789 + 0.613i$
Analytic cond. $2.19660$
Root an. cond. $2.19660$
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  + (−0.691 − 0.722i)2-s + (0.473 + 0.880i)3-s + (−0.0448 + 0.998i)4-s + (0.858 + 0.512i)5-s + (0.309 − 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.753 − 0.657i)8-s + (−0.550 + 0.834i)9-s + (−0.222 − 0.974i)10-s + (−0.900 + 0.433i)12-s + (−0.995 + 0.0896i)13-s + (0.983 + 0.178i)14-s + (−0.0448 + 0.998i)15-s + (−0.995 − 0.0896i)16-s + (−0.995 − 0.0896i)17-s + (0.983 − 0.178i)18-s + ⋯
L(s)  = 1  + (−0.691 − 0.722i)2-s + (0.473 + 0.880i)3-s + (−0.0448 + 0.998i)4-s + (0.858 + 0.512i)5-s + (0.309 − 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.753 − 0.657i)8-s + (−0.550 + 0.834i)9-s + (−0.222 − 0.974i)10-s + (−0.900 + 0.433i)12-s + (−0.995 + 0.0896i)13-s + (0.983 + 0.178i)14-s + (−0.0448 + 0.998i)15-s + (−0.995 − 0.0896i)16-s + (−0.995 − 0.0896i)17-s + (0.983 − 0.178i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $-0.789 + 0.613i$
Analytic conductor: \(2.19660\)
Root analytic conductor: \(2.19660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{473} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 473,\ (0:\ ),\ -0.789 + 0.613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2150842293 + 0.6268942709i\)
\(L(\frac12)\) \(\approx\) \(0.2150842293 + 0.6268942709i\)
\(L(1)\) \(\approx\) \(0.6983581045 + 0.2517012359i\)
\(L(1)\) \(\approx\) \(0.6983581045 + 0.2517012359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.691 - 0.722i)T \)
3 \( 1 + (0.473 + 0.880i)T \)
5 \( 1 + (0.858 + 0.512i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.995 + 0.0896i)T \)
17 \( 1 + (-0.995 - 0.0896i)T \)
19 \( 1 + (-0.963 + 0.266i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (0.473 - 0.880i)T \)
31 \( 1 + (-0.691 - 0.722i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.983 + 0.178i)T \)
47 \( 1 + (-0.963 + 0.266i)T \)
53 \( 1 + (0.858 - 0.512i)T \)
59 \( 1 + (0.753 + 0.657i)T \)
61 \( 1 + (-0.691 + 0.722i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (-0.550 - 0.834i)T \)
73 \( 1 + (-0.393 - 0.919i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.691 + 0.722i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (-0.550 + 0.834i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.78280874063695826654335242020, −23.00082297548304831327947955085, −21.8993055180054817128936603037, −20.56247688281543729244000818039, −19.76768544570878845772961946785, −19.31683167202329633991397909057, −18.19318822163208469584160808671, −17.45059247069610838774833379115, −16.89854617456866280610828582666, −15.915927979994836330504839404175, −14.693255263297030600760092094876, −14.07942036004643801701776374818, −13.05880290199631895953521535873, −12.588952459586458563609556742155, −10.873831930897191074071128065814, −9.9633081092960492036111725613, −9.02316690775854004990819019097, −8.509442710097529670971430402476, −7.06468322899244890370240257362, −6.77373366090862138429721760465, −5.699157017663253629725227060285, −4.507435245857870271601633201008, −2.69060263146496523215074320301, −1.71412514083725501520617416798, −0.39654015689063501061262678832, 2.123049882631405528393289329052, 2.62655155881255170386386295576, 3.66404144130216573360757325846, 4.865456371699765041742793069236, 6.19331970698924011265517576047, 7.322689112499674422939113275584, 8.62480099554815713319818973556, 9.36913576680210397678185203376, 9.907360299721151099188622039361, 10.707506212369115493273314873057, 11.6574878324879472547120496628, 12.90955931977849560050521333295, 13.5700022393191657496546681943, 14.7983951239145810880370305167, 15.554643163426728137508694529726, 16.67291823073728382749579955605, 17.29790502770727377519685289064, 18.292283165348137570119577519406, 19.394806979837821095372077601571, 19.61654291584626037576453274199, 21.03951975226278107776252216405, 21.35501261055456227094037291484, 22.33422338619800312881670677007, 22.59749951359258467008075472127, 24.62296734681580214971297692517

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