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Properties

Label 1-460-460.159-r0-0-0
Degree $1$
Conductor $460$
Sign $0.938 - 0.343i$
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  + (−0.959 + 0.281i)3-s + (0.142 + 0.989i)7-s + (0.841 − 0.540i)9-s + (0.415 − 0.909i)11-s + (0.142 − 0.989i)13-s + (−0.654 − 0.755i)17-s + (−0.654 + 0.755i)19-s + (−0.415 − 0.909i)21-s + (−0.654 + 0.755i)27-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (−0.142 + 0.989i)33-s + (0.841 − 0.540i)37-s + (0.142 + 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)3-s + (0.142 + 0.989i)7-s + (0.841 − 0.540i)9-s + (0.415 − 0.909i)11-s + (0.142 − 0.989i)13-s + (−0.654 − 0.755i)17-s + (−0.654 + 0.755i)19-s + (−0.415 − 0.909i)21-s + (−0.654 + 0.755i)27-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (−0.142 + 0.989i)33-s + (0.841 − 0.540i)37-s + (0.142 + 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.938 - 0.343i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ 0.938 - 0.343i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9180668466 - 0.1628632289i\)
\(L(\frac12)\) \(\approx\) \(0.9180668466 - 0.1628632289i\)
\(L(1)\) \(\approx\) \(0.8303134867 + 0.003726342173i\)
\(L(1)\) \(\approx\) \(0.8303134867 + 0.003726342173i\)

Euler product

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

−23.819705235968220097507044037061, −23.2596218135684028831642264534, −22.34694973567998697764559218467, −21.59981587499446766922387845855, −20.56293704774034077890637347505, −19.6190851158759480775859172995, −18.825411968275429927225010003152, −17.61542072705663841979001049944, −17.26562746854523353872153071284, −16.43988812361296188261194770737, −15.413864817485817225715200502406, −14.35327937373009386608397620401, −13.32202190369131952961305499279, −12.61436760744871929615901517695, −11.53817777124105581159934551990, −10.87019260267484803022486925085, −9.99510910781937636925235348090, −8.87484781645049797886571451266, −7.454997998979518273769544671882, −6.85706103502280511895547203533, −5.9913580420435533611934258480, −4.46033206724525316818769292748, −4.22937567258829920126822397667, −2.173483101073983644873310050460, −1.09673320311109857907788968139, 0.73070105704913906494829220546, 2.34102388975308829227542643000, 3.65696473757429704403079974857, 4.81692663930312915424326416349, 5.82585891464243277182416397855, 6.29597100263322151307236201155, 7.7460627358459658215735867215, 8.80363703878367069149752517060, 9.72146969625221897906674657336, 10.83648042233849533409088421921, 11.4834890070963072892793757748, 12.350304120823257504064022602597, 13.18797292839782131137977841635, 14.46719809356862172987123442349, 15.450493067653360963302213822314, 16.03867935150183248704306748809, 17.016802811110543818583903437285, 17.85651277619754993797875089159, 18.57025091507263862679206425722, 19.4386023802765331920011014010, 20.78640901990530707428286812207, 21.3785467871146417877011570117, 22.37731613158916728200546879247, 22.71545395767177222116220710109, 23.89177922735073397786865915682

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