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Properties

Label 1-1176-1176.773-r0-0-0
Degree $1$
Conductor $1176$
Sign $0.253 + 0.967i$
Analytic cond. $5.46132$
Root an. cond. $5.46132$
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  + (0.733 + 0.680i)5-s + (0.826 + 0.563i)11-s + (−0.900 + 0.433i)13-s + (0.365 − 0.930i)17-s + (−0.5 + 0.866i)19-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (0.623 + 0.781i)29-s + (0.5 + 0.866i)31-s + (0.988 − 0.149i)37-s + (−0.222 + 0.974i)41-s + (0.222 + 0.974i)43-s + (0.0747 − 0.997i)47-s + (−0.988 − 0.149i)53-s + (0.222 + 0.974i)55-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)5-s + (0.826 + 0.563i)11-s + (−0.900 + 0.433i)13-s + (0.365 − 0.930i)17-s + (−0.5 + 0.866i)19-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (0.623 + 0.781i)29-s + (0.5 + 0.866i)31-s + (0.988 − 0.149i)37-s + (−0.222 + 0.974i)41-s + (0.222 + 0.974i)43-s + (0.0747 − 0.997i)47-s + (−0.988 − 0.149i)53-s + (0.222 + 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.253 + 0.967i$
Analytic conductor: \(5.46132\)
Root analytic conductor: \(5.46132\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1176,\ (0:\ ),\ 0.253 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.319288312 + 1.017936538i\)
\(L(\frac12)\) \(\approx\) \(1.319288312 + 1.017936538i\)
\(L(1)\) \(\approx\) \(1.168265891 + 0.3110809060i\)
\(L(1)\) \(\approx\) \(1.168265891 + 0.3110809060i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.733 + 0.680i)T \)
11 \( 1 + (0.826 + 0.563i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (0.365 - 0.930i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.365 - 0.930i)T \)
29 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.988 - 0.149i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (0.0747 - 0.997i)T \)
53 \( 1 + (-0.988 - 0.149i)T \)
59 \( 1 + (0.733 - 0.680i)T \)
61 \( 1 + (-0.988 + 0.149i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (-0.0747 - 0.997i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (0.826 - 0.563i)T \)
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

−21.18032359507553710102981033624, −20.28548889896178258177586009978, −19.520696257193212533463822376933, −18.98838385537351395340683838627, −17.60852100393238411730093655684, −17.320760563458903757155699562576, −16.677955848574719623180821705370, −15.638146422537629357190682124570, −14.869322807857966929614050229677, −13.9528702082108578672124431412, −13.33986118459472164990432920719, −12.47405248336114115481614717291, −11.80538460948295666599578925322, −10.76561695401165112186774134433, −9.86641149352832042564316805437, −9.245964307830573059901137836, −8.39691633317236673843025727708, −7.55813952916600641666785949727, −6.32155377409909171170253518877, −5.81485228046750065488606242771, −4.78727604588776618731072786007, −3.96775608517822415117849082748, −2.74235043551658037544657889774, −1.79067286779695975876704691325, −0.69928189520014193953128103232, 1.33089114449405899398494223400, 2.29692872890339864849123290515, 3.11261295454687562905758050852, 4.32671190577862325087354674565, 5.11915686642937784055752866607, 6.31046966341108828282466855376, 6.77258114140583004924702430031, 7.68678863162445097449824885782, 8.803973010484088166491816693964, 9.76388511009590901761987586959, 10.09889522248104458147643395854, 11.18303546457573180927161074693, 12.054605614771416565916742498854, 12.70327979391804374629496412737, 13.85369519822301748387965393869, 14.52071954659024430036058193237, 14.79802676213920614908621140889, 16.17770703876234999933623141850, 16.78600101273903244418534959706, 17.63709304096172181000439003736, 18.258208562211463973361928764477, 19.041984763546158800092124162795, 19.84996879602194099723596124863, 20.652694153193091329119462783063, 21.55629061713318229589553341536

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