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Properties

Label 1-460-460.119-r1-0-0
Degree $1$
Conductor $460$
Sign $0.999 + 0.0250i$
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  + (0.841 − 0.540i)3-s + (−0.959 + 0.281i)7-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (0.959 + 0.281i)13-s + (0.142 − 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.654 + 0.755i)21-s + (−0.142 − 0.989i)27-s + (−0.142 + 0.989i)29-s + (−0.841 − 0.540i)31-s + (0.959 + 0.281i)33-s + (−0.415 + 0.909i)37-s + (0.959 − 0.281i)39-s + (0.415 + 0.909i)41-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)3-s + (−0.959 + 0.281i)7-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (0.959 + 0.281i)13-s + (0.142 − 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.654 + 0.755i)21-s + (−0.142 − 0.989i)27-s + (−0.142 + 0.989i)29-s + (−0.841 − 0.540i)31-s + (0.959 + 0.281i)33-s + (−0.415 + 0.909i)37-s + (0.959 − 0.281i)39-s + (0.415 + 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.999 + 0.0250i$
Analytic conductor: \(49.4338\)
Root analytic conductor: \(49.4338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (1:\ ),\ 0.999 + 0.0250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.740701544 - 0.03434193630i\)
\(L(\frac12)\) \(\approx\) \(2.740701544 - 0.03434193630i\)
\(L(1)\) \(\approx\) \(1.455418631 - 0.1157375543i\)
\(L(1)\) \(\approx\) \(1.455418631 - 0.1157375543i\)

Euler product

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

−23.76408492810096650968354531297, −22.68684558456299779479585077897, −21.939592736566183713597250985582, −21.17217578579505809290076755880, −20.230130963956491765631065808, −19.41495440269186703750793417032, −19.00471849805281045170839563791, −17.62890814513529934313757503908, −16.55912081047160120864856353253, −15.9115821461109265935018924786, −15.12815104624571042736312952373, −14.034248146638595632607659098399, −13.42211240847902707656707755193, −12.55276784764121995081165925538, −11.07387499519905897809624992978, −10.44366665528174367373762200458, −9.27566632327784039124580631526, −8.78198577656862990777276269222, −7.663588856380307982320909944864, −6.527370244217630001786025218622, −5.5383557173740676811608711020, −3.9382830106241209276982229407, −3.57779517810593043581308604035, −2.34830280323271063137455369183, −0.80650403805054169089457787164, 1.001365664878987453555697486421, 2.146575827147737386169459430620, 3.291190593165320768911943900044, 4.06923128496720249149839177766, 5.7146281710719598313347341767, 6.720283187837484945910462583310, 7.40545009068970698148494692386, 8.66882489208728394228061418888, 9.32353273884854244415448274305, 10.14681221484989416540257016125, 11.65325707200883253078025975857, 12.437767844027848000454903155335, 13.23585733256679807938014147209, 14.097070199775055600635926299653, 14.90986841018288797132840284033, 15.88874688893658105779999130474, 16.67617532086470002845313718440, 18.04238908629118470814795582987, 18.60847501214332649081403336386, 19.4087064256660115444189973364, 20.3139519334557267031985614706, 20.82970377471912756910992596163, 22.130153507544246180138753115981, 22.8864284780580387610343617096, 23.72354564383911655310953097841

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