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Properties

Label 1-30e2-900.227-r1-0-0
Degree $1$
Conductor $900$
Sign $-0.541 - 0.840i$
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 + (−0.978 + 0.207i)11-s + (−0.207 + 0.978i)13-s + (0.587 + 0.809i)17-s + (−0.809 + 0.587i)19-s + (0.743 − 0.669i)23-s + (−0.104 + 0.994i)29-s + (0.104 + 0.994i)31-s + (−0.951 + 0.309i)37-s + (0.978 + 0.207i)41-s + (0.866 − 0.5i)43-s + (−0.994 − 0.104i)47-s + (0.5 − 0.866i)49-s + (0.587 − 0.809i)53-s + (0.978 + 0.207i)59-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)7-s + (−0.978 + 0.207i)11-s + (−0.207 + 0.978i)13-s + (0.587 + 0.809i)17-s + (−0.809 + 0.587i)19-s + (0.743 − 0.669i)23-s + (−0.104 + 0.994i)29-s + (0.104 + 0.994i)31-s + (−0.951 + 0.309i)37-s + (0.978 + 0.207i)41-s + (0.866 − 0.5i)43-s + (−0.994 − 0.104i)47-s + (0.5 − 0.866i)49-s + (0.587 − 0.809i)53-s + (0.978 + 0.207i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.541 - 0.840i$
Analytic conductor: \(96.7184\)
Root analytic conductor: \(96.7184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 900,\ (1:\ ),\ -0.541 - 0.840i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03311876391 + 0.06074411002i\)
\(L(\frac12)\) \(\approx\) \(-0.03311876391 + 0.06074411002i\)
\(L(1)\) \(\approx\) \(0.7549993705 + 0.1774069826i\)
\(L(1)\) \(\approx\) \(0.7549993705 + 0.1774069826i\)

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 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (-0.207 + 0.978i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.743 - 0.669i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (0.978 + 0.207i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.994 - 0.104i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (-0.994 + 0.104i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (0.406 - 0.913i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.994 - 0.104i)T \)
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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

−21.01901699522948999519789173784, −20.605987673614386082751365661819, −19.43253432777704700821208381823, −19.10065511478025240074760635623, −17.97952555934202286713885886733, −17.27934399869898411818908543468, −16.39313177169185806140723398820, −15.64906652588834764957450160412, −14.99093955865334472829479049806, −13.77902379430066139351911864310, −13.15575463143622888019191272726, −12.56172836150879477872921748257, −11.38189041509060618431546112759, −10.54468173576104745646173731859, −9.860865349649785768777325508711, −9.00065851559381404255093177263, −7.79036737699035695181643137490, −7.29622701362749192555048974324, −6.112862139019014370835257539927, −5.3529227420845504976066985279, −4.2804178907156244083970423910, −3.14417343209476415208849494853, −2.52042753243994810924592134129, −0.82358728390204142454522344503, −0.01861830429262672187222533482, 1.58967000114531741903098992244, 2.61764392649553405439806259560, 3.532750634604168597396998124867, 4.64757540169132962778452746206, 5.6109628218082302627138802218, 6.49481490629514532704356017854, 7.29132812358577756646077110673, 8.45820755195402314059396490861, 9.080609385715237971218146200521, 10.1916972059774006984971248823, 10.66569427395911643919410671101, 11.98016727930766092510344070860, 12.60805998405358367957935798836, 13.232130585281575811395868293653, 14.4211662777770972259435359842, 15.00121215861975224291100769494, 16.084934384152050313039107940145, 16.49298402300555122269722594792, 17.50706812928963167047402828787, 18.49894451362489569314192320297, 19.09093227832119927182600805181, 19.67643639721151538787037991876, 21.02522533575781600799672029852, 21.25069543612149745447613919202, 22.29777841502464839377680718063

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