Properties

Label 2-1143-3.2-c2-0-44
Degree $2$
Conductor $1143$
Sign $-0.577 + 0.816i$
Analytic cond. $31.1444$
Root an. cond. $5.58072$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.22i·2-s − 6.41·4-s + 1.45i·5-s + 2.15·7-s + 7.79i·8-s + 4.69·10-s − 6.04i·11-s + 15.2·13-s − 6.96i·14-s − 0.499·16-s + 16.9i·17-s − 5.63·19-s − 9.32i·20-s − 19.5·22-s + 12.2i·23-s + ⋯
L(s)  = 1  − 1.61i·2-s − 1.60·4-s + 0.290i·5-s + 0.308·7-s + 0.974i·8-s + 0.469·10-s − 0.549i·11-s + 1.16·13-s − 0.497i·14-s − 0.0312·16-s + 0.996i·17-s − 0.296·19-s − 0.466i·20-s − 0.886·22-s + 0.534i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1143\)    =    \(3^{2} \cdot 127\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(31.1444\)
Root analytic conductor: \(5.58072\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1143} (890, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1143,\ (\ :1),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.953286792\)
\(L(\frac12)\) \(\approx\) \(1.953286792\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
127 \( 1 + 11.2T \)
good2 \( 1 + 3.22iT - 4T^{2} \)
5 \( 1 - 1.45iT - 25T^{2} \)
7 \( 1 - 2.15T + 49T^{2} \)
11 \( 1 + 6.04iT - 121T^{2} \)
13 \( 1 - 15.2T + 169T^{2} \)
17 \( 1 - 16.9iT - 289T^{2} \)
19 \( 1 + 5.63T + 361T^{2} \)
23 \( 1 - 12.2iT - 529T^{2} \)
29 \( 1 - 20.9iT - 841T^{2} \)
31 \( 1 - 44.7T + 961T^{2} \)
37 \( 1 - 50.2T + 1.36e3T^{2} \)
41 \( 1 + 45.3iT - 1.68e3T^{2} \)
43 \( 1 - 63.6T + 1.84e3T^{2} \)
47 \( 1 + 57.8iT - 2.20e3T^{2} \)
53 \( 1 + 16.7iT - 2.80e3T^{2} \)
59 \( 1 + 111. iT - 3.48e3T^{2} \)
61 \( 1 + 102.T + 3.72e3T^{2} \)
67 \( 1 + 69.5T + 4.48e3T^{2} \)
71 \( 1 - 87.3iT - 5.04e3T^{2} \)
73 \( 1 + 14.5T + 5.32e3T^{2} \)
79 \( 1 - 146.T + 6.24e3T^{2} \)
83 \( 1 - 59.3iT - 6.88e3T^{2} \)
89 \( 1 + 10.9iT - 7.92e3T^{2} \)
97 \( 1 - 21.3T + 9.40e3T^{2} \)
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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

−9.523942714351288236849395709231, −8.710513348277123229715605433032, −8.094274040632577276587371840160, −6.71876845959167322668374974367, −5.83095154907758308733320063998, −4.60031804037726426768282037123, −3.72506548844833104413392919649, −2.97843114822969075507276943558, −1.80750849906624807508667137208, −0.821416699024870473618828235449, 0.955060386216642286841311326801, 2.73097212327668227311729873249, 4.45148100318523575362544217272, 4.71337323387427095211174893860, 6.04386486822865939087769165839, 6.38133620685587704283448186744, 7.53220871635964599800576321474, 7.996212550895668183785942961138, 8.908894027025275561042870864131, 9.424220097715378560777518097425

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