Properties

Label 1-460-460.223-r0-0-0
Degree $1$
Conductor $460$
Sign $0.136 + 0.990i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.136 + 0.990i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ 0.136 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5615215109 + 0.4894430738i\)
\(L(\frac12)\) \(\approx\) \(0.5615215109 + 0.4894430738i\)
\(L(1)\) \(\approx\) \(0.7553847396 + 0.1064286228i\)
\(L(1)\) \(\approx\) \(0.7553847396 + 0.1064286228i\)

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

−23.40275003217271424562075950993, −23.0296841770718735215739175001, −22.1305390127486202431748146596, −21.20495031502734572695244909883, −20.39414272881567050565065854566, −19.66151502718228703245225715790, −18.216521226285472685727317851098, −17.55692068431521945285496029671, −17.062861872336738810412396438459, −15.976209660657142618343978449031, −15.16211528808827266969045636547, −14.29549159906689075392297985470, −13.07433909211891999802924856473, −12.22840667984128419707134660998, −11.34427245511331635047156292413, −10.39167266723326521619402536134, −9.89688025462214002948143441296, −8.56339562434148693133592581049, −7.242999210383454434621165172879, −6.7135324669680756573867652246, −5.18402554127843103936776941832, −4.71571581749245719728456813337, −3.60978193169991281221134812350, −1.98020249721350247426133142005, −0.48358511801188278757130992381, 1.416667814541295911078628413852, 2.41438148198022496746674602762, 4.08493542723954739046163647652, 5.10525625633341107732219319640, 6.04458182028619933329607107879, 6.73788624353764195381897589616, 8.0822625090000312489427203287, 8.79074332032722880849551144267, 10.15369765230076447384887200154, 11.1092056810961321816508321295, 11.79030540590359602781655617928, 12.55658905571232840245305290849, 13.55053065483663714514874997317, 14.5983843112129793795539023296, 15.51577677117336634474032344381, 16.62759542081012740724656824223, 17.13751220241056540938489858109, 18.15534209153328652825977860170, 18.92853865852467291581999991225, 19.479253731661876543244293315557, 21.05241395433599494458442342344, 21.75596107336139225272372748271, 22.18927019187190977730280611497, 23.52908443387227820699112995145, 24.022288204090236699551285874741

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