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Properties

Label 1-1148-1148.775-r0-0-0
Degree $1$
Conductor $1148$
Sign $0.0537 + 0.998i$
Analytic cond. $5.33128$
Root an. cond. $5.33128$
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)3-s + (0.104 − 0.994i)5-s + (−0.5 − 0.866i)9-s + (0.104 + 0.994i)11-s + (−0.309 + 0.951i)13-s + (0.809 + 0.587i)15-s + (0.104 + 0.994i)17-s + (0.669 − 0.743i)19-s + (0.978 − 0.207i)23-s + (−0.978 − 0.207i)25-s + 27-s + (−0.809 − 0.587i)29-s + (−0.104 − 0.994i)31-s + (−0.913 − 0.406i)33-s + (−0.104 + 0.994i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.104 − 0.994i)5-s + (−0.5 − 0.866i)9-s + (0.104 + 0.994i)11-s + (−0.309 + 0.951i)13-s + (0.809 + 0.587i)15-s + (0.104 + 0.994i)17-s + (0.669 − 0.743i)19-s + (0.978 − 0.207i)23-s + (−0.978 − 0.207i)25-s + 27-s + (−0.809 − 0.587i)29-s + (−0.104 − 0.994i)31-s + (−0.913 − 0.406i)33-s + (−0.104 + 0.994i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0537 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0537 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.0537 + 0.998i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (775, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ 0.0537 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7639056747 + 0.7238833364i\)
\(L(\frac12)\) \(\approx\) \(0.7639056747 + 0.7238833364i\)
\(L(1)\) \(\approx\) \(0.8541280650 + 0.2360969603i\)
\(L(1)\) \(\approx\) \(0.8541280650 + 0.2360969603i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.104 - 0.994i)T \)
11 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.978 - 0.207i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.309 + 0.951i)T \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (-0.913 + 0.406i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.669 + 0.743i)T \)
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

−21.27471861236037585532039850666, −20.14376775557340384937641660853, −19.4033676861927193062869564627, −18.57996337011713806253750988748, −18.23967255624364076321428873735, −17.40900423521289428582958221395, −16.570791222178374356622366753896, −15.78210088239516885515003068865, −14.67904910109015447748922870119, −14.03266162349238220895376637433, −13.35122767769910411183629530178, −12.44914990991756022775107463506, −11.59756184037414266788686103643, −10.953287666937029860388649781686, −10.28233514030201187724800894839, −9.14222983435389301215586681220, −8.04373683732696254882893971359, −7.31381091395440133983038079138, −6.71156531510222740445489028587, −5.63521411185796131161372685294, −5.240695785750366064925051269, −3.44113954509930401429949743710, −2.91164173028336359278493441359, −1.746566894738855846332286348, −0.52895070363043102112918447408, 1.08332273062689071066785060403, 2.22562325152088626826468633026, 3.63725897175903645922527798246, 4.52163501657265022752815212549, 4.95191768300544869497576931328, 5.95737631081927387655084318660, 6.86980717727429494525379439037, 7.98746690895760019691102658363, 9.09081057094665845826126033785, 9.47769443736907102437665955961, 10.26936796804256617197005058432, 11.4004500619318898675359235911, 11.90459737098276117036885533561, 12.81157196466297368042117326973, 13.557119402890988472586935010050, 14.90720031581828297121578401032, 15.13393202200271489875860452041, 16.27707869611820435951965250322, 16.823479257638378491192310775937, 17.34334213954282015285534223834, 18.18581862434208328470331598114, 19.41622981160906184495601228749, 20.04874490659190576287952998540, 20.98482637531991702295514185229, 21.22431359786370109595824993428

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