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Properties

Label 1-460-460.419-r0-0-0
Degree $1$
Conductor $460$
Sign $0.0174 - 0.999i$
Analytic cond. $2.13623$
Root an. cond. $2.13623$
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  + (−0.142 − 0.989i)3-s + (0.654 − 0.755i)7-s + (−0.959 + 0.281i)9-s + (0.841 − 0.540i)11-s + (0.654 + 0.755i)13-s + (0.415 − 0.909i)17-s + (0.415 + 0.909i)19-s + (−0.841 − 0.540i)21-s + (0.415 + 0.909i)27-s + (0.415 − 0.909i)29-s + (0.142 − 0.989i)31-s + (−0.654 − 0.755i)33-s + (−0.959 + 0.281i)37-s + (0.654 − 0.755i)39-s + (−0.959 − 0.281i)41-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)3-s + (0.654 − 0.755i)7-s + (−0.959 + 0.281i)9-s + (0.841 − 0.540i)11-s + (0.654 + 0.755i)13-s + (0.415 − 0.909i)17-s + (0.415 + 0.909i)19-s + (−0.841 − 0.540i)21-s + (0.415 + 0.909i)27-s + (0.415 − 0.909i)29-s + (0.142 − 0.989i)31-s + (−0.654 − 0.755i)33-s + (−0.959 + 0.281i)37-s + (0.654 − 0.755i)39-s + (−0.959 − 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0174 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0174 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.0174 - 0.999i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ 0.0174 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.014143539 - 0.9966357822i\)
\(L(\frac12)\) \(\approx\) \(1.014143539 - 0.9966357822i\)
\(L(1)\) \(\approx\) \(1.036727386 - 0.4880317263i\)
\(L(1)\) \(\approx\) \(1.036727386 - 0.4880317263i\)

Euler product

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

−24.04950352492028389012635823742, −23.178338432273519202319613122050, −22.16354095087526414791364175265, −21.74974613997097260594489525873, −20.74816129665218993816926870150, −20.105826332080260221283587473095, −19.08350031929695485822741739379, −17.74711966612502241769959433140, −17.45841574704961948361844991693, −16.212779175029031436162193101130, −15.420186717481911416225008329561, −14.79596809623994443064341940649, −13.95290924716438014695011828102, −12.53490534020599144085224953288, −11.75659650223740270338862954918, −10.84301463564000832240502482107, −10.04801293360125005488205846240, −8.87409717639804447737188841732, −8.44603959544833260432503355553, −6.92889988111717284384750155458, −5.71388312658337686416991585241, −5.02500068130053967760914576222, −3.92480621869146846580954424452, −2.931751228980097742232508214012, −1.45070394003727296438478178459, 0.94308597752887145988865494198, 1.797139553357418610932975217839, 3.28585886479476334694876054337, 4.41280497817315619926576888150, 5.72514300051520837653790272367, 6.589647581006657317128463001331, 7.52308808362779721904099223585, 8.29992397558309535242832441755, 9.35950385546263410726233855716, 10.67091860919944224051756065800, 11.64421614190076738868325050921, 12.034165480905308107697111977845, 13.63765173027344290260993999428, 13.78770347438148317437748856629, 14.72470763836271888289279132470, 16.24782271690598832399522140134, 16.90670026619772838444617984460, 17.7033599767249786827691380420, 18.68517176876159306210403656394, 19.20175582670783875452059748234, 20.368116484973791679415935027447, 20.91308154953690494137768282927, 22.22459422131134329194056726309, 23.04829602314010326176352218723, 23.73252643755627154335679828889

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