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Properties

Label 1-1176-1176.725-r1-0-0
Degree $1$
Conductor $1176$
Sign $0.910 - 0.414i$
Analytic cond. $126.378$
Root an. cond. $126.378$
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  + (−0.988 + 0.149i)5-s + (−0.733 + 0.680i)11-s + (0.222 + 0.974i)13-s + (−0.0747 − 0.997i)17-s + (0.5 − 0.866i)19-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.900 − 0.433i)29-s + (−0.5 − 0.866i)31-s + (−0.826 + 0.563i)37-s + (−0.623 − 0.781i)41-s + (−0.623 + 0.781i)43-s + (−0.955 − 0.294i)47-s + (0.826 + 0.563i)53-s + (0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)5-s + (−0.733 + 0.680i)11-s + (0.222 + 0.974i)13-s + (−0.0747 − 0.997i)17-s + (0.5 − 0.866i)19-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.900 − 0.433i)29-s + (−0.5 − 0.866i)31-s + (−0.826 + 0.563i)37-s + (−0.623 − 0.781i)41-s + (−0.623 + 0.781i)43-s + (−0.955 − 0.294i)47-s + (0.826 + 0.563i)53-s + (0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(126.378\)
Root analytic conductor: \(126.378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (725, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1176,\ (1:\ ),\ 0.910 - 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9401284352 - 0.2040342317i\)
\(L(\frac12)\) \(\approx\) \(0.9401284352 - 0.2040342317i\)
\(L(1)\) \(\approx\) \(0.7746594212 + 0.04371631844i\)
\(L(1)\) \(\approx\) \(0.7746594212 + 0.04371631844i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.988 + 0.149i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 + (-0.0747 - 0.997i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.0747 + 0.997i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.826 + 0.563i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (-0.955 - 0.294i)T \)
53 \( 1 + (0.826 + 0.563i)T \)
59 \( 1 + (-0.988 - 0.149i)T \)
61 \( 1 + (-0.826 + 0.563i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.955 - 0.294i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.733 + 0.680i)T \)
97 \( 1 + T \)
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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

−21.0251916941902184170773052164, −20.1964523680766142720896171460, −19.71759270864223562664816175688, −18.5889349239776558163077359104, −18.37699216247459431835585048634, −17.101223307161533982806810970842, −16.3681241565331427519902285779, −15.724908066809470226656940034842, −14.98780794877585262437744193909, −14.22042017123983315573126007765, −13.05327016434954575370793717299, −12.59606116137989348325819707939, −11.70364220462560038381978083473, −10.70400636646006788381719173009, −10.37933907712244552605053301582, −8.9553669837552296473485310323, −8.199602797748953983794919666675, −7.75031100723353841273760501323, −6.63276539356621223163827673751, −5.61170728628624969007501783930, −4.876737198767328369113004094479, −3.599457712517332787444156756348, −3.25174417828716595267165814663, −1.77917720018214500501016143251, −0.55062166753205704580653461685, 0.34358935839074097217533814485, 1.750068311053646777816955658115, 2.829331361926237261513214603131, 3.76536380701196644203683148640, 4.658107300784717529960074692249, 5.40591618251726028814704891391, 6.80322734675918186336287947750, 7.31001452024650849311005005375, 8.08191106931160035408779438800, 9.15698364257980158110143881198, 9.794771440953627661558709145, 11.00090901927487240752878729107, 11.53460008580706312835481870447, 12.20796802108092091905623015716, 13.2908428182487974348518131751, 13.88036034118355370494735821275, 15.07660462225957001815118087720, 15.47613645576459502020894197732, 16.25785091438794061126658719024, 17.05270435746782866116582902295, 18.17446811779154471343903583118, 18.59218762358009011286727831540, 19.50625014050240847755318042047, 20.203053059483484286406304371834, 20.8622154730384980932679256010

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