Properties

Label 1-460-460.407-r0-0-0
Degree $1$
Conductor $460$
Sign $0.824 - 0.565i$
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  + (0.909 + 0.415i)3-s + (−0.540 − 0.841i)7-s + (0.654 + 0.755i)9-s + (0.142 − 0.989i)11-s + (0.540 − 0.841i)13-s + (0.281 − 0.959i)17-s + (−0.959 + 0.281i)19-s + (−0.142 − 0.989i)21-s + (0.281 + 0.959i)27-s + (0.959 + 0.281i)29-s + (−0.415 − 0.909i)31-s + (0.540 − 0.841i)33-s + (0.755 − 0.654i)37-s + (0.841 − 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)3-s + (−0.540 − 0.841i)7-s + (0.654 + 0.755i)9-s + (0.142 − 0.989i)11-s + (0.540 − 0.841i)13-s + (0.281 − 0.959i)17-s + (−0.959 + 0.281i)19-s + (−0.142 − 0.989i)21-s + (0.281 + 0.959i)27-s + (0.959 + 0.281i)29-s + (−0.415 − 0.909i)31-s + (0.540 − 0.841i)33-s + (0.755 − 0.654i)37-s + (0.841 − 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.824 - 0.565i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ 0.824 - 0.565i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.662599580 - 0.5148483365i\)
\(L(\frac12)\) \(\approx\) \(1.662599580 - 0.5148483365i\)
\(L(1)\) \(\approx\) \(1.369643739 - 0.1298405135i\)
\(L(1)\) \(\approx\) \(1.369643739 - 0.1298405135i\)

Euler product

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

−23.91239125982814438443404551844, −23.43462514701698678304874652756, −22.16281129008127210330116688532, −21.35759795919832703917046161686, −20.59999435933683546155022793924, −19.481262227779997306552904778820, −19.105354904671097644461945046634, −18.17252770351443085443503013977, −17.264042773716830471370278231592, −15.99751110340236445808229111191, −15.21958682924221261103743220622, −14.52888797607548990387600559194, −13.52308247491522755922399611652, −12.5672419511776371153348388806, −12.11676670966412331098249132873, −10.63334304743557984130176793481, −9.53611942377539671866305599704, −8.872629114036824421437358298394, −8.02256002741935274203765696212, −6.79347662302505582585120383604, −6.1929372425804015332163551518, −4.593552568288514288838839559903, −3.587457572231514912770317536887, −2.430014354904453421821885166933, −1.59575974080268272662165465189, 0.93560739875817788032662536125, 2.61398624488520987213219487205, 3.47457645563324148914921206316, 4.28558820330795372304528092854, 5.631608283104176239803799740674, 6.80524295307155811546550370220, 7.87458508576611900425751985832, 8.61785184572477425452234129879, 9.669734190124417527336003868179, 10.44040864634534591326460444866, 11.27808720642602388906660937998, 12.80999749463452451826847581837, 13.46191968765811875993497978402, 14.209719760656818225069341552533, 15.12080628387070105444192610173, 16.2110508320603258350283121680, 16.55690828285144703303470964453, 17.930171661682421627897479842361, 18.95575801467094775924654332635, 19.63408829917219768684299969338, 20.43719778815278420464062859706, 21.11978693485183463700441953851, 22.07024206750177817953228129417, 22.96577440147557128960813344384, 23.8340173279338093859301747600

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