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Properties

Label 1-30e2-900.23-r1-0-0
Degree $1$
Conductor $900$
Sign $0.733 - 0.679i$
Analytic cond. $96.7184$
Root an. cond. $96.7184$
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  + (−0.866 − 0.5i)7-s + (0.669 + 0.743i)11-s + (0.743 + 0.669i)13-s + (−0.587 − 0.809i)17-s + (−0.809 + 0.587i)19-s + (−0.207 − 0.978i)23-s + (0.913 − 0.406i)29-s + (−0.913 − 0.406i)31-s + (0.951 − 0.309i)37-s + (−0.669 + 0.743i)41-s + (0.866 + 0.5i)43-s + (−0.406 − 0.913i)47-s + (0.5 + 0.866i)49-s + (−0.587 + 0.809i)53-s + (−0.669 + 0.743i)59-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (0.669 + 0.743i)11-s + (0.743 + 0.669i)13-s + (−0.587 − 0.809i)17-s + (−0.809 + 0.587i)19-s + (−0.207 − 0.978i)23-s + (0.913 − 0.406i)29-s + (−0.913 − 0.406i)31-s + (0.951 − 0.309i)37-s + (−0.669 + 0.743i)41-s + (0.866 + 0.5i)43-s + (−0.406 − 0.913i)47-s + (0.5 + 0.866i)49-s + (−0.587 + 0.809i)53-s + (−0.669 + 0.743i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.733 - 0.679i$
Analytic conductor: \(96.7184\)
Root analytic conductor: \(96.7184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 900,\ (1:\ ),\ 0.733 - 0.679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.490739379 - 0.5842143743i\)
\(L(\frac12)\) \(\approx\) \(1.490739379 - 0.5842143743i\)
\(L(1)\) \(\approx\) \(0.9895108013 - 0.05806868438i\)
\(L(1)\) \(\approx\) \(0.9895108013 - 0.05806868438i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.743 + 0.669i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.207 - 0.978i)T \)
29 \( 1 + (0.913 - 0.406i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.406 - 0.913i)T \)
53 \( 1 + (-0.587 + 0.809i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (-0.406 + 0.913i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (0.994 - 0.104i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.406 - 0.913i)T \)
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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

−21.93241445792714951870521730740, −21.229713501161435098053865043826, −20.03629906080896108840111742511, −19.494646063851391218807225681462, −18.81439497567229891543970841402, −17.82940257859752847938608691673, −17.127662908536690424048371307574, −16.10421036712488127359767564549, −15.59259632379756247370577247832, −14.721088432177036320595073097205, −13.67059223970746640600962589025, −12.99203833167845547562052754795, −12.266588681690394809443672348399, −11.17497865173406130264899738741, −10.582581097546121005791843878842, −9.401686195161548430970510566520, −8.79585577221522442676818718725, −7.96772264649657754610085865836, −6.575198362077234914465084792937, −6.20115155458398570160411531395, −5.17255040263653554334958782785, −3.83703821815337903117024955074, −3.23120745017578867889634781185, −2.02768858040010355245237785147, −0.75828609119330958681677614847, 0.47602064222908677022184192314, 1.73219856050565149893217434277, 2.8270664645548439111914230145, 4.050538372682193975357622908656, 4.49161465121067890332133302626, 6.082425268275577427089749997985, 6.59681878334399101444939527761, 7.44923371937310938661529731457, 8.63732016026850984907987850896, 9.39498077372293770707163766484, 10.17188060858230950086387053615, 11.073366251281858672600214103286, 11.99868408075018160528310075140, 12.815484100860420979878817450526, 13.59088592072036962080171293084, 14.40497750079895019468352768137, 15.24980671804274389534810548513, 16.3509114377996937652598675376, 16.61180281563110042939862708233, 17.730943962431561381081725789704, 18.50369108350764799822474350546, 19.34784639305896784853117994960, 20.09034487616886696335164394704, 20.71643334009777173129263117546, 21.72475124772503157629451389446

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