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Properties

Label 1-460-460.147-r0-0-0
Degree $1$
Conductor $460$
Sign $-0.862 - 0.505i$
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  + (−0.989 − 0.142i)3-s + (−0.755 + 0.654i)7-s + (0.959 + 0.281i)9-s + (−0.841 − 0.540i)11-s + (0.755 + 0.654i)13-s + (−0.909 + 0.415i)17-s + (0.415 − 0.909i)19-s + (0.841 − 0.540i)21-s + (−0.909 − 0.415i)27-s + (−0.415 − 0.909i)29-s + (0.142 + 0.989i)31-s + (0.755 + 0.654i)33-s + (0.281 − 0.959i)37-s + (−0.654 − 0.755i)39-s + (−0.959 + 0.281i)41-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)3-s + (−0.755 + 0.654i)7-s + (0.959 + 0.281i)9-s + (−0.841 − 0.540i)11-s + (0.755 + 0.654i)13-s + (−0.909 + 0.415i)17-s + (0.415 − 0.909i)19-s + (0.841 − 0.540i)21-s + (−0.909 − 0.415i)27-s + (−0.415 − 0.909i)29-s + (0.142 + 0.989i)31-s + (0.755 + 0.654i)33-s + (0.281 − 0.959i)37-s + (−0.654 − 0.755i)39-s + (−0.959 + 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.862 - 0.505i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ -0.862 - 0.505i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04252840192 - 0.1565541876i\)
\(L(\frac12)\) \(\approx\) \(0.04252840192 - 0.1565541876i\)
\(L(1)\) \(\approx\) \(0.5624161976 + 0.02260522191i\)
\(L(1)\) \(\approx\) \(0.5624161976 + 0.02260522191i\)

Euler product

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

−23.97029143639570832997136135465, −23.3375662401140264219164236828, −22.59434365354048106851056522939, −22.063610205362330125327323056762, −20.63183147249840352745034018884, −20.37115446891381955191535740967, −18.900591717157840469546816328022, −18.19439341654360246538636768462, −17.41652692809990190584247189968, −16.42261060360981558145409730453, −15.873157339534141905853572334782, −14.99492466302379105433716906139, −13.430988285703352419211132864918, −13.01863837757456631731977241024, −11.9764147252825959961751930223, −10.92822923575186660668659790890, −10.26528011273124894582791844848, −9.4887092233310070551959076542, −8.001722137028590872602947442561, −7.030199058353242740099124305993, −6.17286875977385880430707599561, −5.20265436526755945062868618221, −4.18412378709931316179747387991, −3.08605038064970105868078130892, −1.39961787697946988424863911370, 0.10513169740751477941393307926, 1.799248601132410431699480050692, 3.10710948031164957784647050438, 4.42650179041064188470607278535, 5.501535898253859059273824522518, 6.27375907993056893231440416750, 7.06841687229253089680441228829, 8.42861026866901436853385989853, 9.364890502708726824139218900883, 10.48141956840380269712958729809, 11.256214197131527485458611182422, 12.06388960808690836353513652539, 13.13607740464120720513096354375, 13.56550863090433759025929404663, 15.29002045893752292547649896593, 15.862634778785936825879002196716, 16.56024956755962377490714315460, 17.63212685242712456977330666198, 18.44031817323393390311302154722, 19.01295693448823863846295235611, 20.08356395221019357637828743454, 21.5412416805296515176093508556, 21.680381514829031223611151899141, 22.80545545739339247920086021244, 23.5007328866569174626270073254

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