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Properties

Label 2-1127-161.74-c0-0-0
Degree $2$
Conductor $1127$
Sign $-0.653 + 0.756i$
Analytic cond. $0.562446$
Root an. cond. $0.749964$
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  + (−0.396 − 1.63i)2-s + (−1.62 + 0.838i)4-s + (0.915 + 1.05i)8-s + (0.786 − 0.618i)9-s + (1.46 + 0.356i)11-s + (0.302 − 0.424i)16-s + (−1.32 − 1.04i)18-s − 2.54i·22-s + (0.928 − 0.371i)23-s + (0.723 − 0.690i)25-s + (−1.61 + 1.03i)29-s + (0.483 + 0.193i)32-s + (−0.760 + 1.66i)36-s + (−1.12 − 1.43i)37-s + (−1.49 − 1.29i)43-s + (−2.68 + 0.652i)44-s + ⋯
L(s)  = 1  + (−0.396 − 1.63i)2-s + (−1.62 + 0.838i)4-s + (0.915 + 1.05i)8-s + (0.786 − 0.618i)9-s + (1.46 + 0.356i)11-s + (0.302 − 0.424i)16-s + (−1.32 − 1.04i)18-s − 2.54i·22-s + (0.928 − 0.371i)23-s + (0.723 − 0.690i)25-s + (−1.61 + 1.03i)29-s + (0.483 + 0.193i)32-s + (−0.760 + 1.66i)36-s + (−1.12 − 1.43i)37-s + (−1.49 − 1.29i)43-s + (−2.68 + 0.652i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $-0.653 + 0.756i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ -0.653 + 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8778975365\)
\(L(\frac12)\) \(\approx\) \(0.8778975365\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 + (-0.928 + 0.371i)T \)
good2 \( 1 + (0.396 + 1.63i)T + (-0.888 + 0.458i)T^{2} \)
3 \( 1 + (-0.786 + 0.618i)T^{2} \)
5 \( 1 + (-0.723 + 0.690i)T^{2} \)
11 \( 1 + (-1.46 - 0.356i)T + (0.888 + 0.458i)T^{2} \)
13 \( 1 + (-0.654 - 0.755i)T^{2} \)
17 \( 1 + (0.995 - 0.0950i)T^{2} \)
19 \( 1 + (0.995 + 0.0950i)T^{2} \)
29 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
31 \( 1 + (0.928 - 0.371i)T^{2} \)
37 \( 1 + (1.12 + 1.43i)T + (-0.235 + 0.971i)T^{2} \)
41 \( 1 + (-0.959 - 0.281i)T^{2} \)
43 \( 1 + (1.49 + 1.29i)T + (0.142 + 0.989i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.0535 - 0.560i)T + (-0.981 - 0.189i)T^{2} \)
59 \( 1 + (-0.327 + 0.945i)T^{2} \)
61 \( 1 + (0.786 + 0.618i)T^{2} \)
67 \( 1 + (-0.746 - 0.782i)T + (-0.0475 + 0.998i)T^{2} \)
71 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (0.580 - 0.814i)T^{2} \)
79 \( 1 + (-0.0535 - 0.560i)T + (-0.981 + 0.189i)T^{2} \)
83 \( 1 + (0.959 - 0.281i)T^{2} \)
89 \( 1 + (-0.928 - 0.371i)T^{2} \)
97 \( 1 + (0.959 + 0.281i)T^{2} \)
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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

−9.810731537669959279095055588237, −9.072106475767916533610378064586, −8.725844482737632174025537147906, −7.20495650561768957637449460591, −6.59020598744985333884074089897, −5.06553396936216285597262828965, −3.96324475792548792508702231143, −3.50585675566417645468655664033, −2.08393364318807151459821459946, −1.12634725602528441627083184747, 1.49228168730752668457043169596, 3.51109352750230883893186150179, 4.64917201113177888851470236206, 5.37293808352077981510604876534, 6.46843531479940729009328811978, 6.91404377310779341107153051678, 7.75223484756556445239864532728, 8.513182176111275917931739136751, 9.336352244493601777019212888643, 9.819884039385021671590902648802

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