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Properties

Label 1-460-460.227-r1-0-0
Degree $1$
Conductor $460$
Sign $-0.146 - 0.989i$
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  + (0.755 − 0.654i)3-s + (0.909 + 0.415i)7-s + (0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.909 − 0.415i)13-s + (0.540 − 0.841i)17-s + (−0.841 + 0.540i)19-s + (0.959 − 0.281i)21-s + (−0.540 − 0.841i)27-s + (−0.841 − 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.909 + 0.415i)33-s + (0.989 + 0.142i)37-s + (0.415 − 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)3-s + (0.909 + 0.415i)7-s + (0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.909 − 0.415i)13-s + (0.540 − 0.841i)17-s + (−0.841 + 0.540i)19-s + (0.959 − 0.281i)21-s + (−0.540 − 0.841i)27-s + (−0.841 − 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.909 + 0.415i)33-s + (0.989 + 0.142i)37-s + (0.415 − 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.146 - 0.989i$
Analytic conductor: \(49.4338\)
Root analytic conductor: \(49.4338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (1:\ ),\ -0.146 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.763201726 - 2.042917457i\)
\(L(\frac12)\) \(\approx\) \(1.763201726 - 2.042917457i\)
\(L(1)\) \(\approx\) \(1.386254781 - 0.5314889593i\)
\(L(1)\) \(\approx\) \(1.386254781 - 0.5314889593i\)

Euler product

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

−23.74707492281549273863367191129, −23.372982726914717681345461470318, −21.93323939297597650437836361712, −21.185925090471464956000563309426, −20.733375556383972060344851703229, −19.83116105419707066759168205371, −18.878059522997346129444038699775, −17.989783113187238986493044107817, −16.93616205653018262338349388691, −16.08698696817461674280772020235, −15.127337722087337642472807370974, −14.54389551194986941717144520344, −13.55713873508485354210095037849, −12.8349789948570157498592432470, −11.29664079947221175737568848977, −10.668494170080742151601960634, −9.82610500406191333763131610071, −8.54484422711664630787576028937, −8.135080671286930653347937068714, −7.00199650149172086489445906310, −5.52657922870102521040128305857, −4.5562348649379207976187715668, −3.74768936083815142821532707526, −2.49243033961623082745162816003, −1.4034388580187437453577032219, 0.63797746417463709599175167398, 1.89845380515880590262280984747, 2.800318705824618508730297628, 3.96740225147701508900440295652, 5.33233688107681983794345883328, 6.23814929871667331509958895147, 7.62567672042469888175166077578, 8.1162464865434813857343386720, 8.94586715759864678572217937132, 10.14014916004727861197455754139, 11.24653476243662805176319388004, 12.11946556329711619436769161410, 13.18176842746014599525219418000, 13.749420358488604289251070055194, 14.84504828002033348988064616555, 15.379766156303199142736024978335, 16.57397142891970522414138450383, 17.77848389004573231259369219742, 18.50326058089678904781248268701, 18.90135366714855579758840987113, 20.252093051693392093851025615618, 20.85908444710204051898812165971, 21.40521837111780526090583323115, 22.90241024618667318489079340006, 23.57932610391666440675469642113

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