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Properties

Label 1-60e2-3600.3269-r1-0-0
Degree $1$
Conductor $3600$
Sign $-0.905 - 0.424i$
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  + (−0.5 − 0.866i)7-s + (0.406 − 0.913i)11-s + (0.406 + 0.913i)13-s + (0.309 + 0.951i)17-s + (0.951 − 0.309i)19-s + (0.104 + 0.994i)23-s + (−0.207 − 0.978i)29-s + (−0.978 − 0.207i)31-s + (−0.587 + 0.809i)37-s + (0.913 − 0.406i)41-s + (−0.866 + 0.5i)43-s + (−0.978 + 0.207i)47-s + (−0.5 + 0.866i)49-s + (0.951 + 0.309i)53-s + (−0.406 − 0.913i)59-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (0.406 − 0.913i)11-s + (0.406 + 0.913i)13-s + (0.309 + 0.951i)17-s + (0.951 − 0.309i)19-s + (0.104 + 0.994i)23-s + (−0.207 − 0.978i)29-s + (−0.978 − 0.207i)31-s + (−0.587 + 0.809i)37-s + (0.913 − 0.406i)41-s + (−0.866 + 0.5i)43-s + (−0.978 + 0.207i)47-s + (−0.5 + 0.866i)49-s + (0.951 + 0.309i)53-s + (−0.406 − 0.913i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.905 - 0.424i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (3269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ -0.905 - 0.424i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1702429841 - 0.7647575680i\)
\(L(\frac12)\) \(\approx\) \(0.1702429841 - 0.7647575680i\)
\(L(1)\) \(\approx\) \(0.9620484614 - 0.1236801117i\)
\(L(1)\) \(\approx\) \(0.9620484614 - 0.1236801117i\)

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

−18.60269154760436273986092577297, −18.15405796970279748345941997315, −17.71438447300991076265866455037, −16.441416634082639241482791650138, −16.243912893954531183927288582, −15.35078770612062831070063984904, −14.73405751638289743185885102751, −14.136680957704896051640527716642, −13.07588234135173690083551094581, −12.61483018502739548507229938186, −11.96965029572815118216915758907, −11.27969251462231869967416865781, −10.27145375028719724839032847815, −9.764424476394033355274411735842, −8.96619095692296101297018882301, −8.42013347747670178227187535429, −7.26886465443399882744064747710, −6.959952014434323756490294063779, −5.7386282319146630703335590287, −5.42438907501641801393345848744, −4.45805430788050147565899509169, −3.42117740237521656140244985556, −2.85857143773518804398401503411, −1.92446030242281764375546675620, −0.954830121868823180552564793046, 0.13304525041060001802820650229, 1.12721989160959006171092397629, 1.803475367520927599269158315382, 3.19424740137706129588492410603, 3.62085578548983145611202492885, 4.348573610949170401294166870206, 5.42004516268552028839476465088, 6.17380808808995085998018221025, 6.78700610487305861005933168594, 7.60446055099464488890162164728, 8.30252510134415221745230707292, 9.26109413863414352772188082949, 9.692945049453087177198665989851, 10.61539347341677392279831667555, 11.316117164292356745123883553979, 11.78609989896539353667944222382, 12.875202949023784611946581499571, 13.46340467249236164360008886415, 13.987262716785958697516174659090, 14.65319838139934567495121529152, 15.67842609011816213702233024149, 16.17511660546552265079174303191, 16.96184114770242613380994347630, 17.25858268783125478445694751583, 18.36302847657579499384603728673

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