Properties

Label 1-460-460.343-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: $\chi_{460} (343, \cdot )$
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

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

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