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Properties

Label 1-460-460.247-r1-0-0
Degree $1$
Conductor $460$
Sign $0.999 + 0.0309i$
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  + (−0.540 + 0.841i)3-s + (0.281 − 0.959i)7-s + (−0.415 − 0.909i)9-s + (−0.654 + 0.755i)11-s + (0.281 + 0.959i)13-s + (−0.989 + 0.142i)17-s + (0.142 − 0.989i)19-s + (0.654 + 0.755i)21-s + (0.989 + 0.142i)27-s + (0.142 + 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.281 − 0.959i)33-s + (0.909 − 0.415i)37-s + (−0.959 − 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯
L(s)  = 1  + (−0.540 + 0.841i)3-s + (0.281 − 0.959i)7-s + (−0.415 − 0.909i)9-s + (−0.654 + 0.755i)11-s + (0.281 + 0.959i)13-s + (−0.989 + 0.142i)17-s + (0.142 − 0.989i)19-s + (0.654 + 0.755i)21-s + (0.989 + 0.142i)27-s + (0.142 + 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.281 − 0.959i)33-s + (0.909 − 0.415i)37-s + (−0.959 − 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.999 + 0.0309i$
Analytic conductor: \(49.4338\)
Root analytic conductor: \(49.4338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (1:\ ),\ 0.999 + 0.0309i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.280135615 + 0.01978430900i\)
\(L(\frac12)\) \(\approx\) \(1.280135615 + 0.01978430900i\)
\(L(1)\) \(\approx\) \(0.8562413493 + 0.1265031161i\)
\(L(1)\) \(\approx\) \(0.8562413493 + 0.1265031161i\)

Euler product

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

−23.77332750569831572607813478210, −22.87825930095420404528482095811, −22.156395153653469799083751654851, −21.27369913554383473873046820857, −20.24328511597348608734543674589, −19.21904027783244164552873596940, −18.34160657604588764525015078905, −18.03592999138086460181469330110, −16.910549088287706866190171683624, −15.963934514691889981617238625480, −15.13420003250668172322085056930, −13.95129659749148443565120620776, −13.06372786102118141676577229605, −12.4170162275757652715080452773, −11.342140785529371095737070033070, −10.81374926470643230126473649357, −9.41609729212652409518605633930, −8.153702997073021184275358046520, −7.80039967625361745149627806378, −6.17987225114645557836584916063, −5.81718694256292709644092300708, −4.713999125594503793322122346897, −3.01468362100858120108688074015, −2.097261188501175435118843717610, −0.75480315033242242378072412589, 0.523064719810371390605887958121, 2.090857512992779338351623715916, 3.64076262082371215779213258532, 4.49062033774674805824142706996, 5.199054244281892094469749831681, 6.60488046138529871074060066452, 7.30150067331169993878233145594, 8.76482041167082110744068922565, 9.539100321135281036167186501, 10.71027980877552574904257067262, 11.00215221360126709192546980518, 12.1794941716299200547835819001, 13.26270400690458335159640025889, 14.21714063601523475889299208006, 15.17147583671387007518301984767, 15.98977528079871249004978382362, 16.760174770037839196393130928204, 17.64137450725426046556382792898, 18.233533843669867252843891392150, 19.77778441727735680253660382649, 20.3160771408521014169020837016, 21.24818561038571588914989843927, 21.92089683173982591833327020028, 22.91833106978587532682535269650, 23.624804820967924730175183439136

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