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Properties

Label 2-1155-1155.524-c0-0-2
Degree $2$
Conductor $1155$
Sign $0.997 + 0.0694i$
Analytic cond. $0.576420$
Root an. cond. $0.759223$
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  − i·3-s + (0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (0.951 − 0.309i)7-s − 9-s + (−0.809 − 0.587i)11-s + (0.951 − 0.309i)12-s + (0.363 − 0.5i)13-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (0.951 − 0.690i)17-s + (−0.587 + 0.809i)20-s + (−0.309 − 0.951i)21-s + (−0.309 + 0.951i)25-s + i·27-s + (0.587 + 0.809i)28-s + ⋯
L(s)  = 1  − i·3-s + (0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (0.951 − 0.309i)7-s − 9-s + (−0.809 − 0.587i)11-s + (0.951 − 0.309i)12-s + (0.363 − 0.5i)13-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (0.951 − 0.690i)17-s + (−0.587 + 0.809i)20-s + (−0.309 − 0.951i)21-s + (−0.309 + 0.951i)25-s + i·27-s + (0.587 + 0.809i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.997 + 0.0694i$
Analytic conductor: \(0.576420\)
Root analytic conductor: \(0.759223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :0),\ 0.997 + 0.0694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.288718181\)
\(L(\frac12)\) \(\approx\) \(1.288718181\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
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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

−10.21473029388067472496236557415, −8.817414642119775877842512151604, −8.131771415921496998639002142998, −7.49513271411516369292788962309, −6.91153220993431122230762649329, −5.89525663331870293706717484775, −5.05543743202514663937987333851, −3.33367096344250520956081290293, −2.79679686579484575289669116450, −1.57756135651650738842978431603, 1.51222420174332347612571282051, 2.54687174461918650407629453599, 4.26410260718424871163215445963, 4.92239719918370437816463724534, 5.60791123132611867385965826345, 6.24415545478185145176060372720, 7.79345565010699561248031690646, 8.520208861847664860748845360254, 9.381488576194457793001180085594, 10.04358998190103237181741269426

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