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Properties

Label 1-460-460.339-r0-0-0
Degree $1$
Conductor $460$
Sign $0.551 + 0.833i$
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  + (0.841 + 0.540i)3-s + (0.959 + 0.281i)7-s + (0.415 + 0.909i)9-s + (−0.654 + 0.755i)11-s + (0.959 − 0.281i)13-s + (−0.142 − 0.989i)17-s + (−0.142 + 0.989i)19-s + (0.654 + 0.755i)21-s + (−0.142 + 0.989i)27-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.959 + 0.281i)33-s + (0.415 + 0.909i)37-s + (0.959 + 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)3-s + (0.959 + 0.281i)7-s + (0.415 + 0.909i)9-s + (−0.654 + 0.755i)11-s + (0.959 − 0.281i)13-s + (−0.142 − 0.989i)17-s + (−0.142 + 0.989i)19-s + (0.654 + 0.755i)21-s + (−0.142 + 0.989i)27-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.959 + 0.281i)33-s + (0.415 + 0.909i)37-s + (0.959 + 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.551 + 0.833i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ 0.551 + 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.734597507 + 0.9322605407i\)
\(L(\frac12)\) \(\approx\) \(1.734597507 + 0.9322605407i\)
\(L(1)\) \(\approx\) \(1.446661439 + 0.4172889049i\)
\(L(1)\) \(\approx\) \(1.446661439 + 0.4172889049i\)

Euler product

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

−23.834489858007403209167121938, −23.411521867895409798324732023815, −21.71560919678212736066956508136, −21.237062542286815123455768238279, −20.30451775907252739067808263663, −19.60885222624044880812293562838, −18.51849147544357939618876007804, −18.08873577714350196955432304574, −16.99649350010105247843038051108, −15.85975640994273627874944425614, −14.955678679972802607853239438424, −14.168958213095816477918281571401, −13.35980826181941532704138131339, −12.694783162110051641587814833904, −11.278619846868338931780118897960, −10.76342193085205307047595000811, −9.276735156751647267427968765419, −8.45278353020887530075173153825, −7.84007317114775962160723557941, −6.76999159277232298645349631226, −5.68419826383654346773156671983, −4.34016340607465895696306513400, −3.35919744944602769437367018226, −2.15640092244760172780463703511, −1.12804760422968895283074575972, 1.64284696263942722346155425733, 2.59068689479361666812892301281, 3.804946973097991085279299125450, 4.76833731279539337934206100354, 5.651571298816873956563883684497, 7.27074489276915116456255759585, 8.06599548289633499707337569376, 8.80927957372052994728801818089, 9.8704554238646547001941233164, 10.68256606078336909687557046733, 11.65344424102481952543779380048, 12.84057804762849754261585952281, 13.77915298714317095714426244301, 14.53562447345953642837239335496, 15.40647394602064601536883000655, 15.97836899721054867542103488512, 17.17696399814320259077853966622, 18.29299863269209284866615680158, 18.749051161647737415004538741408, 20.16392148630470149643615607386, 20.667442000182508533459428523759, 21.19119311647465561982370526981, 22.27454886267200003488979798240, 23.186597413670169614355656930949, 24.1456285398300822851896684623

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