Properties

Label 2-1169-1169.333-c0-0-3
Degree $2$
Conductor $1169$
Sign $0.642 + 0.765i$
Analytic cond. $0.583406$
Root an. cond. $0.763810$
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  + (−0.841 − 1.45i)2-s + (0.327 − 0.566i)3-s + (−0.915 + 1.58i)4-s − 1.10·6-s + (0.235 + 0.971i)7-s + 1.39·8-s + (0.286 + 0.495i)9-s + (−0.580 + 1.00i)11-s + (0.598 + 1.03i)12-s + (1.21 − 1.16i)14-s + (−0.260 − 0.451i)16-s + (0.481 − 0.833i)18-s + (0.142 + 0.246i)19-s + (0.627 + 0.184i)21-s + 1.95·22-s + ⋯
L(s)  = 1  + (−0.841 − 1.45i)2-s + (0.327 − 0.566i)3-s + (−0.915 + 1.58i)4-s − 1.10·6-s + (0.235 + 0.971i)7-s + 1.39·8-s + (0.286 + 0.495i)9-s + (−0.580 + 1.00i)11-s + (0.598 + 1.03i)12-s + (1.21 − 1.16i)14-s + (−0.260 − 0.451i)16-s + (0.481 − 0.833i)18-s + (0.142 + 0.246i)19-s + (0.627 + 0.184i)21-s + 1.95·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1169\)    =    \(7 \cdot 167\)
Sign: $0.642 + 0.765i$
Analytic conductor: \(0.583406\)
Root analytic conductor: \(0.763810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1169} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1169,\ (\ :0),\ 0.642 + 0.765i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7060091857\)
\(L(\frac12)\) \(\approx\) \(0.7060091857\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.235 - 0.971i)T \)
167 \( 1 - T \)
good2 \( 1 + (0.841 + 1.45i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.142 - 0.246i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.44T + T^{2} \)
31 \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.928 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.654 - 1.13i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.981 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 0.830T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.03181997616453620620039406967, −9.148181748973727396363194994342, −8.377110261816994443409547544534, −7.82561351502865848925015000883, −6.87521129584792833035567313012, −5.44590393304645211919282225380, −4.45811693492971627351358881632, −3.09259517337383944805934376050, −2.25861590778850089877773312935, −1.58083986465468025211141020362, 0.868664887390317409391949395980, 3.09145810508611978222108611565, 4.26441878822654045465014193459, 5.11888559738992185465042127336, 6.24863929625052066737878913444, 6.78671700034018753203998635453, 7.85564045373644864767153114335, 8.256993618121394650619081418590, 9.121269964901129002753871314348, 9.900585356420569185836902161496

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