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Properties

Label 1-460-460.103-r1-0-0
Degree $1$
Conductor $460$
Sign $0.201 + 0.979i$
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  + (−0.281 + 0.959i)3-s + (0.989 + 0.142i)7-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (0.989 − 0.142i)13-s + (−0.755 − 0.654i)17-s + (0.654 + 0.755i)19-s + (−0.415 + 0.909i)21-s + (0.755 − 0.654i)27-s + (0.654 − 0.755i)29-s + (0.959 − 0.281i)31-s + (−0.989 + 0.142i)33-s + (−0.540 + 0.841i)37-s + (−0.142 + 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯
L(s)  = 1  + (−0.281 + 0.959i)3-s + (0.989 + 0.142i)7-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (0.989 − 0.142i)13-s + (−0.755 − 0.654i)17-s + (0.654 + 0.755i)19-s + (−0.415 + 0.909i)21-s + (0.755 − 0.654i)27-s + (0.654 − 0.755i)29-s + (0.959 − 0.281i)31-s + (−0.989 + 0.142i)33-s + (−0.540 + 0.841i)37-s + (−0.142 + 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.201 + 0.979i$
Analytic conductor: \(49.4338\)
Root analytic conductor: \(49.4338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (1:\ ),\ 0.201 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.722469035 + 1.404849850i\)
\(L(\frac12)\) \(\approx\) \(1.722469035 + 1.404849850i\)
\(L(1)\) \(\approx\) \(1.128924261 + 0.4420934950i\)
\(L(1)\) \(\approx\) \(1.128924261 + 0.4420934950i\)

Euler product

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

−23.68874497575321845060939383953, −22.86071968472963784866394089429, −21.808482091725337948003847868582, −21.04660308625779528108959295001, −19.866097531867494513958376924722, −19.30658551020872776680630511976, −18.07869499009348127532293627206, −17.83161260260119094075346332682, −16.76735195359757342056656884466, −15.85887762628238330000915828154, −14.59409226011569979932259468536, −13.79067068424199932322038607425, −13.1968387673775091000268757093, −11.95704874025571705220803150165, −11.269535847912966594622321403, −10.63814823741695315661868958619, −8.82038904560518243467212830161, −8.39054368181311099409978549878, −7.25422074197532121771698491981, −6.3461360972811955055203542514, −5.46105699152772398335211158294, −4.263473755092622357158503381337, −2.889532553677572936903497795236, −1.58580745932808278576835709424, −0.77896324387333652368777952445, 1.0291102585581778146318942882, 2.444132819439885843779628055260, 3.85430024390309142232774702497, 4.61089103098283398072411896537, 5.522500045291112697164961793003, 6.59678224786110487387704920271, 7.92572748978956123525025414178, 8.83102424874639915353254337245, 9.74810710952539309739539400072, 10.66082000629493181630164529641, 11.54192088658952981204469508928, 12.15693104730795860138776866549, 13.69110440679981178009173033504, 14.41234473273486183063350817063, 15.43318835075299683968200748378, 15.87755293980802205728835920898, 17.15533082448091121385047926459, 17.68005992638796017896870132458, 18.56430567664602999015087984099, 19.945707043045773146819136037186, 20.76611976626871681642495690414, 21.06796018992191838792675690435, 22.39803715209634532011515644404, 22.7565120195822994915392681096, 23.793377776048999029889370733903

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