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Properties

Label 1-276-276.119-r0-0-0
Degree $1$
Conductor $276$
Sign $-0.167 - 0.985i$
Analytic cond. $1.28173$
Root an. cond. $1.28173$
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  + (−0.415 − 0.909i)5-s + (0.959 − 0.281i)7-s + (−0.654 − 0.755i)11-s + (−0.959 − 0.281i)13-s + (0.142 − 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.654 + 0.755i)25-s + (0.142 − 0.989i)29-s + (−0.841 − 0.540i)31-s + (−0.654 − 0.755i)35-s + (0.415 − 0.909i)37-s + (−0.415 − 0.909i)41-s + (−0.841 + 0.540i)43-s + 47-s + (0.841 − 0.540i)49-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)5-s + (0.959 − 0.281i)7-s + (−0.654 − 0.755i)11-s + (−0.959 − 0.281i)13-s + (0.142 − 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.654 + 0.755i)25-s + (0.142 − 0.989i)29-s + (−0.841 − 0.540i)31-s + (−0.654 − 0.755i)35-s + (0.415 − 0.909i)37-s + (−0.415 − 0.909i)41-s + (−0.841 + 0.540i)43-s + 47-s + (0.841 − 0.540i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.167 - 0.985i$
Analytic conductor: \(1.28173\)
Root analytic conductor: \(1.28173\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 276,\ (0:\ ),\ -0.167 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6483594354 - 0.7674690515i\)
\(L(\frac12)\) \(\approx\) \(0.6483594354 - 0.7674690515i\)
\(L(1)\) \(\approx\) \(0.9002570645 - 0.3422499174i\)
\(L(1)\) \(\approx\) \(0.9002570645 - 0.3422499174i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (0.959 - 0.281i)T \)
11 \( 1 + (-0.654 - 0.755i)T \)
13 \( 1 + (-0.959 - 0.281i)T \)
17 \( 1 + (0.142 - 0.989i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (0.142 - 0.989i)T \)
31 \( 1 + (-0.841 - 0.540i)T \)
37 \( 1 + (0.415 - 0.909i)T \)
41 \( 1 + (-0.415 - 0.909i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (-0.959 - 0.281i)T \)
61 \( 1 + (0.841 + 0.540i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (0.415 - 0.909i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.415 + 0.909i)T \)
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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

−26.033259709659631337788611285520, −25.08106242485023054283257415385, −23.796188849728201591649379521835, −23.515829139374235640135841838461, −22.008903158570545939289353859460, −21.71644929215087711610820833748, −20.36369643478891162390004588342, −19.55761333857261527039142561662, −18.44995775615774882750425965027, −17.8550814267685155814776975015, −16.84150112424760259719853643795, −15.3587003500353803822255463577, −14.96282661923359769255900818065, −14.05369686876726618111937082074, −12.69946209345969679752884284194, −11.75697786661358031284482302560, −10.84501669458451706421832679804, −9.99246629129106314886383878591, −8.60520468090042344573314200151, −7.57108375987625414308796403489, −6.836123882745025259097795819007, −5.32103436235802347783423076186, −4.38359513275582422094987722790, −2.926801170002983328507907490636, −1.87675374918544689301833607558, 0.6775424788547269059662060840, 2.21660323471702856619436378983, 3.7792350741594348390032106693, 4.92097002354000709152650686969, 5.62728054809403165171554636074, 7.49309120449067104404099463820, 7.99839162843905518868899328078, 9.11295132466308286649323041476, 10.282860647380214867803833883093, 11.43316207993330745711364886836, 12.1763068960630144756140394815, 13.28573698974627156996266876723, 14.22066057659947411922466508372, 15.26667347601918061515033115985, 16.31956204215608255634904565280, 16.998595229009598689811200285128, 18.06798921128465551083017399601, 19.03624485933777399791692742359, 20.164516960251722677067540276121, 20.76884289487092244399131685046, 21.59910242437724748937932783023, 22.87739141618461926857784044906, 23.74915330269252444915116248396, 24.50231079850844412615115287145, 25.09119764098325048021675753396

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