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Properties

Label 1-600-600.461-r1-0-0
Degree $1$
Conductor $600$
Sign $-0.187 - 0.982i$
Analytic cond. $64.4789$
Root an. cond. $64.4789$
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  + 7-s + (0.309 − 0.951i)11-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)17-s + (0.809 − 0.587i)19-s + (−0.309 + 0.951i)23-s + (−0.809 − 0.587i)29-s + (−0.809 + 0.587i)31-s + (−0.309 − 0.951i)37-s + (−0.309 − 0.951i)41-s − 43-s + (0.809 + 0.587i)47-s + 49-s + (−0.809 − 0.587i)53-s + (0.309 + 0.951i)59-s + ⋯
L(s)  = 1  + 7-s + (0.309 − 0.951i)11-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)17-s + (0.809 − 0.587i)19-s + (−0.309 + 0.951i)23-s + (−0.809 − 0.587i)29-s + (−0.809 + 0.587i)31-s + (−0.309 − 0.951i)37-s + (−0.309 − 0.951i)41-s − 43-s + (0.809 + 0.587i)47-s + 49-s + (−0.809 − 0.587i)53-s + (0.309 + 0.951i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.187 - 0.982i$
Analytic conductor: \(64.4789\)
Root analytic conductor: \(64.4789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 600,\ (1:\ ),\ -0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.254608610 - 1.516561291i\)
\(L(\frac12)\) \(\approx\) \(1.254608610 - 1.516561291i\)
\(L(1)\) \(\approx\) \(1.150523643 - 0.2883961148i\)
\(L(1)\) \(\approx\) \(1.150523643 - 0.2883961148i\)

Euler product

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

−23.23785379062087538341702779555, −22.19315512698317972257339749145, −21.52604431752651426947288061791, −20.47928410195898632465504954443, −20.12789029567325723396656312639, −18.68206720174899429485018009653, −18.36731135681687231464155537939, −17.08291306344710512432381558256, −16.753752230786369263185138615400, −15.45906546498054770412454973312, −14.504839133302907054538676037067, −14.23353115344951257017251112489, −12.84850458483847985392975712176, −12.02101100789645858004725315906, −11.34528460127627333899820911461, −10.21636895146537909419010882493, −9.45363922554957404113771402979, −8.34554946199769707338140237697, −7.54325434950198327285141287879, −6.626248930153416936512026048787, −5.39734942197820181653677877230, −4.564677307904692766796416573072, −3.60962746549110887118312579364, −2.07914063349912552650605435907, −1.3684429833424004153903751155, 0.48068297728551607190186171355, 1.57135498192483932956275992153, 2.922464137585372270172886062509, 3.85063844570470944690530316123, 5.27449381743210017911557300762, 5.615861523548896349351420607074, 7.20272196261068030347152722049, 7.83586146837258686035538808561, 8.8127514408039450338100306133, 9.75166671509227957491852312483, 10.858567147011784099584524936034, 11.51572023947634204385459535651, 12.36657529890874453042829938604, 13.57896363238260516454972206535, 14.15853776838417776307680359563, 15.089279245056892165927922703505, 15.94177893743405788892542174696, 16.90144598079097006201376295418, 17.72121057701383338630948026199, 18.39486418783111209224711375761, 19.39698338435131445454812275935, 20.23845139977628410100260835726, 21.01505669624279594095648144023, 21.82632166244387682021721991937, 22.575482728464183749710984612742

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