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Properties

Label 1-460-460.287-r1-0-0
Degree $1$
Conductor $460$
Sign $0.786 - 0.617i$
Analytic cond. $49.4338$
Root an. cond. $49.4338$
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  + (0.281 − 0.959i)3-s + (−0.989 − 0.142i)7-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.989 + 0.142i)13-s + (0.755 + 0.654i)17-s + (0.654 + 0.755i)19-s + (−0.415 + 0.909i)21-s + (−0.755 + 0.654i)27-s + (0.654 − 0.755i)29-s + (0.959 − 0.281i)31-s + (0.989 − 0.142i)33-s + (0.540 − 0.841i)37-s + (−0.142 + 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯
L(s)  = 1  + (0.281 − 0.959i)3-s + (−0.989 − 0.142i)7-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.989 + 0.142i)13-s + (0.755 + 0.654i)17-s + (0.654 + 0.755i)19-s + (−0.415 + 0.909i)21-s + (−0.755 + 0.654i)27-s + (0.654 − 0.755i)29-s + (0.959 − 0.281i)31-s + (0.989 − 0.142i)33-s + (0.540 − 0.841i)37-s + (−0.142 + 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.786 - 0.617i$
Analytic conductor: \(49.4338\)
Root analytic conductor: \(49.4338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (1:\ ),\ 0.786 - 0.617i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.672698149 - 0.5786273413i\)
\(L(\frac12)\) \(\approx\) \(1.672698149 - 0.5786273413i\)
\(L(1)\) \(\approx\) \(1.038762799 - 0.2657667830i\)
\(L(1)\) \(\approx\) \(1.038762799 - 0.2657667830i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.281 - 0.959i)T \)
7 \( 1 + (-0.989 - 0.142i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (-0.989 + 0.142i)T \)
17 \( 1 + (0.755 + 0.654i)T \)
19 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (0.959 - 0.281i)T \)
37 \( 1 + (0.540 - 0.841i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (-0.281 + 0.959i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.989 - 0.142i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (0.959 - 0.281i)T \)
67 \( 1 + (0.909 + 0.415i)T \)
71 \( 1 + (-0.415 + 0.909i)T \)
73 \( 1 + (-0.755 + 0.654i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (0.540 - 0.841i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.540 - 0.841i)T \)
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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

−23.75340217775719468626411824753, −22.59343836988278652178100889018, −22.12349080242185791228244053200, −21.40168641696689131006708784188, −20.343629375970644646889804927409, −19.57447305699948429660081304952, −18.96953635551042499812845605282, −17.62376131801798364076138618764, −16.59644872478978023152987465737, −16.1161455080712655626058325511, −15.23600171358142600322475208842, −14.2396041851469522181366430578, −13.56387258224668589030842824306, −12.293477475042335706513344203404, −11.43782932520750429857251580889, −10.314128249766203347640064691507, −9.58654722503649677121362911485, −8.90602257255531666455227265714, −7.74515389626579238504079846669, −6.53313636003432536768097273218, −5.46529859651639825161442818557, −4.54686938640729618181983731558, −3.2018692238886873185347132647, −2.811795356755947071932879576010, −0.69487781192600185122870524554, 0.72589399543208133427144507195, 2.00031590296190652168789221495, 3.0236992513035333882314772704, 4.15163983125165123809726310805, 5.65408643573791941225015918433, 6.57367562164009469602592040438, 7.3675758122037651280886640353, 8.20379297643469083832875677805, 9.55951549417317529611318001358, 9.99867929256463747393317896333, 11.627111683543770347855198464546, 12.41194705646683584830323221330, 12.90899352676841058523457910266, 14.10074044394614573389753798919, 14.67928553472693120916904674658, 15.83601995515481199216415158063, 16.97032147250958702521698086353, 17.53562237281380836379869305287, 18.66855656523392607558243255150, 19.382648278217513008509181466032, 19.925649864642823913939060926174, 20.89780611855202168398487895580, 22.12649437465677435899200209803, 22.96821639060971946511774792811, 23.46411712941752621310790808411

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