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Properties

Label 1-1725-1725.869-r1-0-0
Degree $1$
Conductor $1725$
Sign $-0.999 + 0.0287i$
Analytic cond. $185.376$
Root an. cond. $185.376$
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  + (−0.998 + 0.0570i)2-s + (0.993 − 0.113i)4-s + (0.654 − 0.755i)7-s + (−0.985 + 0.170i)8-s + (0.362 − 0.931i)11-s + (−0.0855 − 0.996i)13-s + (−0.610 + 0.791i)14-s + (0.974 − 0.226i)16-s + (0.993 + 0.113i)17-s + (−0.736 + 0.676i)19-s + (−0.309 + 0.951i)22-s + (0.142 + 0.989i)26-s + (0.564 − 0.825i)28-s + (0.736 + 0.676i)29-s + (−0.985 + 0.170i)31-s + (−0.959 + 0.281i)32-s + ⋯
L(s)  = 1  + (−0.998 + 0.0570i)2-s + (0.993 − 0.113i)4-s + (0.654 − 0.755i)7-s + (−0.985 + 0.170i)8-s + (0.362 − 0.931i)11-s + (−0.0855 − 0.996i)13-s + (−0.610 + 0.791i)14-s + (0.974 − 0.226i)16-s + (0.993 + 0.113i)17-s + (−0.736 + 0.676i)19-s + (−0.309 + 0.951i)22-s + (0.142 + 0.989i)26-s + (0.564 − 0.825i)28-s + (0.736 + 0.676i)29-s + (−0.985 + 0.170i)31-s + (−0.959 + 0.281i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1725\)    =    \(3 \cdot 5^{2} \cdot 23\)
Sign: $-0.999 + 0.0287i$
Analytic conductor: \(185.376\)
Root analytic conductor: \(185.376\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1725} (869, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1725,\ (1:\ ),\ -0.999 + 0.0287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009221298760 - 0.6407076355i\)
\(L(\frac12)\) \(\approx\) \(0.009221298760 - 0.6407076355i\)
\(L(1)\) \(\approx\) \(0.6861499576 - 0.1758574695i\)
\(L(1)\) \(\approx\) \(0.6861499576 - 0.1758574695i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.998 + 0.0570i)T \)
7 \( 1 + (0.654 - 0.755i)T \)
11 \( 1 + (0.362 - 0.931i)T \)
13 \( 1 + (-0.0855 - 0.996i)T \)
17 \( 1 + (0.993 + 0.113i)T \)
19 \( 1 + (-0.736 + 0.676i)T \)
29 \( 1 + (0.736 + 0.676i)T \)
31 \( 1 + (-0.985 + 0.170i)T \)
37 \( 1 + (-0.941 - 0.336i)T \)
41 \( 1 + (-0.610 - 0.791i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.516 + 0.856i)T \)
59 \( 1 + (-0.0855 - 0.996i)T \)
61 \( 1 + (-0.466 - 0.884i)T \)
67 \( 1 + (0.254 - 0.967i)T \)
71 \( 1 + (-0.774 + 0.633i)T \)
73 \( 1 + (0.870 + 0.491i)T \)
79 \( 1 + (-0.921 + 0.389i)T \)
83 \( 1 + (-0.564 - 0.825i)T \)
89 \( 1 + (-0.696 - 0.717i)T \)
97 \( 1 + (0.564 - 0.825i)T \)
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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

−20.4527992699482790298990179361, −19.50837078261066699361435451035, −18.987364911319499454647660652885, −18.25416235802252733423480662073, −17.570418843725441050918353108152, −16.95566277739986356891149307227, −16.20260196154989745198230085799, −15.238677260158003171189675764102, −14.84760440612294537082499794680, −13.955785325208428106788028896846, −12.66354824912961737355662135548, −11.95684328299975271474080673657, −11.5417060555690024548993862726, −10.55324670103770075008030018385, −9.77351305444772359239755996093, −9.033873557248634824306242516091, −8.4666698647950906186452147084, −7.50403869335697260307444116670, −6.861975890978630981216860263641, −5.97384415146606724175054487774, −4.98524649008617731995977846563, −4.01422235211423382945493365774, −2.7335042773057169750185543323, −1.98275537578499049176495131364, −1.25655816093096064138592897816, 0.17341310523132529290483210738, 1.05226905042033700810310296830, 1.81937257541441437831297810677, 3.12372900535397453753321710395, 3.76561138093306987095189169399, 5.18510515880463991089806064233, 5.891750997826094801914969231889, 6.85894628868953023896668118299, 7.664105407866001854022491255366, 8.28899427248305878282064786307, 8.90996351025188718549756601094, 10.09771142521754187559178644435, 10.533814768786258833838618873269, 11.18059874537286660725683282603, 12.100321303894571697136924579221, 12.82478531351754381656628594794, 14.10060724469237361579113261758, 14.49763214153376402686017649692, 15.45012371329935650268448819322, 16.277515363909743732786026859, 16.98872162744579689465790515422, 17.3651901063086798886842687677, 18.36872059950734809206958238525, 18.8347728527496927552001660043, 19.851307273581095071205497987886

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