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Properties

Label 1-60e2-3600.1813-r1-0-0
Degree $1$
Conductor $3600$
Sign $-0.826 - 0.563i$
Analytic cond. $386.873$
Root an. cond. $386.873$
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 + (0.207 − 0.978i)11-s + (−0.978 + 0.207i)13-s + (−0.587 + 0.809i)17-s + (0.587 − 0.809i)19-s + (−0.743 − 0.669i)23-s + (0.994 − 0.104i)29-s + (−0.104 + 0.994i)31-s + (0.309 − 0.951i)37-s + (0.978 − 0.207i)41-s + (0.5 − 0.866i)43-s + (−0.994 + 0.104i)47-s + (0.5 + 0.866i)49-s + (0.809 − 0.587i)53-s + (0.207 + 0.978i)59-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (0.207 − 0.978i)11-s + (−0.978 + 0.207i)13-s + (−0.587 + 0.809i)17-s + (0.587 − 0.809i)19-s + (−0.743 − 0.669i)23-s + (0.994 − 0.104i)29-s + (−0.104 + 0.994i)31-s + (0.309 − 0.951i)37-s + (0.978 − 0.207i)41-s + (0.5 − 0.866i)43-s + (−0.994 + 0.104i)47-s + (0.5 + 0.866i)49-s + (0.809 − 0.587i)53-s + (0.207 + 0.978i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.826 - 0.563i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1813, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ -0.826 - 0.563i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3287861806 - 1.065427028i\)
\(L(\frac12)\) \(\approx\) \(0.3287861806 - 1.065427028i\)
\(L(1)\) \(\approx\) \(0.8670696822 - 0.1895046765i\)
\(L(1)\) \(\approx\) \(0.8670696822 - 0.1895046765i\)

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

−18.77505283712149035470336967703, −18.07163921915025540526286330375, −17.53358413596258190095749949011, −16.69008719491780548265789378822, −15.982776038914455644974564594769, −15.45241203142457584095323927815, −14.7067046569175283626933303440, −14.02971265269122700705313565521, −13.1689469782663159119326379840, −12.51763622888347740356112360255, −11.956849882119872045598615869763, −11.33466560465808006134900361643, −10.08101891895654494513977096856, −9.735881085916192099593645525893, −9.25118096361692588050706398866, −8.107983924375418960624955296650, −7.48872214802335964551015295562, −6.70362942005073630260520601134, −6.03371578586326462819883722308, −5.14396521672143291433575225064, −4.46287051762637261482302012252, −3.523638882592485175664312567489, −2.645948862539379132479790830412, −2.0630916035057836350955469804, −0.84579993833450850984172630992, 0.23695894997620446452251675027, 0.87498489500104357571444500666, 2.17798954187667723766664712440, 2.9017667865057175699245536341, 3.756714661536667083069161556478, 4.4291949119111342193998751110, 5.37958384754766347404871222822, 6.25055443239125265552384336489, 6.79017954964242030246614337003, 7.53887092913742170972272744733, 8.47045023900447834754124127461, 9.114824615898561686701797128555, 9.85504957183446220721312348444, 10.59118693823070639061157417042, 11.146645395386499356670716168551, 12.20285532036054344546880831535, 12.57752066641144155986386915334, 13.59515572057005323466766646743, 13.92985950407951767940474919732, 14.77189235925785680078688978326, 15.61518213693437338199764300512, 16.322706844657384536543671811176, 16.68151350096315488212971501049, 17.64068292218989692586808699249, 18.08335440072691815979030948123

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