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Properties

Label 1-30e2-900.239-r0-0-0
Degree $1$
Conductor $900$
Sign $-0.733 + 0.679i$
Analytic cond. $4.17958$
Root an. cond. $4.17958$
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  + (−0.5 + 0.866i)7-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)13-s + (−0.809 + 0.587i)17-s + (0.809 − 0.587i)19-s + (0.978 − 0.207i)23-s + (−0.913 + 0.406i)29-s + (−0.913 − 0.406i)31-s + (−0.309 − 0.951i)37-s + (−0.669 + 0.743i)41-s + (−0.5 + 0.866i)43-s + (−0.913 + 0.406i)47-s + (−0.5 − 0.866i)49-s + (−0.809 − 0.587i)53-s + (0.669 − 0.743i)59-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)13-s + (−0.809 + 0.587i)17-s + (0.809 − 0.587i)19-s + (0.978 − 0.207i)23-s + (−0.913 + 0.406i)29-s + (−0.913 − 0.406i)31-s + (−0.309 − 0.951i)37-s + (−0.669 + 0.743i)41-s + (−0.5 + 0.866i)43-s + (−0.913 + 0.406i)47-s + (−0.5 − 0.866i)49-s + (−0.809 − 0.587i)53-s + (0.669 − 0.743i)59-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.733 + 0.679i$
Analytic conductor: \(4.17958\)
Root analytic conductor: \(4.17958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 900,\ (0:\ ),\ -0.733 + 0.679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2985552441 + 0.7618231916i\)
\(L(\frac12)\) \(\approx\) \(0.2985552441 + 0.7618231916i\)
\(L(1)\) \(\approx\) \(0.8355630664 + 0.2630011385i\)
\(L(1)\) \(\approx\) \(0.8355630664 + 0.2630011385i\)

Euler product

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

−21.88830975769114663778839654576, −20.54313074629830959311841257691, −20.20354503487786953306555083496, −19.29924623081255525581371377752, −18.59603572554259822013887584445, −17.49457295498967923366392669470, −16.906782393792927069548348517955, −16.17196236049883171943686231909, −15.26865041690110830976884070863, −14.34597726557669685473842942290, −13.56495166420100862266670228805, −12.93503863624795320204904423547, −11.85463896383937121859216150971, −11.092365241421401365287071419791, −10.19018031736038323229149324160, −9.43419530988844070122185612962, −8.53042875004539814019609384367, −7.39018893084821416164795415744, −6.86296731191743598221264179808, −5.73745653110850792257472792192, −4.84118992003223637983484808597, −3.6479603049844837317227826597, −3.06779433968415019168039923560, −1.589103459149235032939655827477, −0.35206629564941828334185078441, 1.60697870795110336746581991445, 2.47615821647496115400451931391, 3.57789839684366170810143704384, 4.64272448345061113431264630868, 5.47981607999031641643046048326, 6.65763841056350944538182287701, 7.10452590460712429742563056230, 8.42772047904041357052970776966, 9.370884715501986985286230101582, 9.63889810659222907885492925551, 11.093591401721168925702996749258, 11.68753168075725042341695656327, 12.678912974472911875395185300291, 13.158153303569710390389602299679, 14.54203768399797227804153339373, 14.899454461028822088990055462350, 15.87867132624606287152795190509, 16.6638014777254923526499421219, 17.52262216940626553376906774883, 18.27330293146252075687592963883, 19.21004023331321278559965745354, 19.74143692925780838792819723116, 20.61121974538300963238143655431, 21.70571758157842266035702664563, 22.14573112165060291352259850478

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