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Properties

Label 2-1127-161.51-c0-0-0
Degree $2$
Conductor $1127$
Sign $-0.822 + 0.568i$
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  + (−1.21 + 1.16i)2-s + (0.0871 − 1.82i)4-s + (0.915 + 1.05i)8-s + (−0.928 − 0.371i)9-s + (−1.04 + 1.09i)11-s + (−0.518 − 0.0495i)16-s + (1.56 − 0.625i)18-s − 2.54i·22-s + (−0.786 − 0.618i)23-s + (0.235 + 0.971i)25-s + (−1.61 + 1.03i)29-s + (−0.409 + 0.322i)32-s + (−0.760 + 1.66i)36-s + (−0.676 + 1.68i)37-s + (−1.49 − 1.29i)43-s + (1.90 + 2.00i)44-s + ⋯
L(s)  = 1  + (−1.21 + 1.16i)2-s + (0.0871 − 1.82i)4-s + (0.915 + 1.05i)8-s + (−0.928 − 0.371i)9-s + (−1.04 + 1.09i)11-s + (−0.518 − 0.0495i)16-s + (1.56 − 0.625i)18-s − 2.54i·22-s + (−0.786 − 0.618i)23-s + (0.235 + 0.971i)25-s + (−1.61 + 1.03i)29-s + (−0.409 + 0.322i)32-s + (−0.760 + 1.66i)36-s + (−0.676 + 1.68i)37-s + (−1.49 − 1.29i)43-s + (1.90 + 2.00i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $-0.822 + 0.568i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ -0.822 + 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1588061068\)
\(L(\frac12)\) \(\approx\) \(0.1588061068\)
\(L(1)\) 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 \( 1 + (0.786 + 0.618i)T \)
good2 \( 1 + (1.21 - 1.16i)T + (0.0475 - 0.998i)T^{2} \)
3 \( 1 + (0.928 + 0.371i)T^{2} \)
5 \( 1 + (-0.235 - 0.971i)T^{2} \)
11 \( 1 + (1.04 - 1.09i)T + (-0.0475 - 0.998i)T^{2} \)
13 \( 1 + (-0.654 - 0.755i)T^{2} \)
17 \( 1 + (-0.580 - 0.814i)T^{2} \)
19 \( 1 + (-0.580 + 0.814i)T^{2} \)
29 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
31 \( 1 + (-0.786 - 0.618i)T^{2} \)
37 \( 1 + (0.676 - 1.68i)T + (-0.723 - 0.690i)T^{2} \)
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 \( 1 + (0.458 + 0.326i)T + (0.327 + 0.945i)T^{2} \)
59 \( 1 + (0.981 - 0.189i)T^{2} \)
61 \( 1 + (-0.928 + 0.371i)T^{2} \)
67 \( 1 + (1.05 - 0.254i)T + (0.888 - 0.458i)T^{2} \)
71 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.995 - 0.0950i)T^{2} \)
79 \( 1 + (-0.458 + 0.326i)T + (0.327 - 0.945i)T^{2} \)
83 \( 1 + (0.959 - 0.281i)T^{2} \)
89 \( 1 + (0.786 - 0.618i)T^{2} \)
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

−10.18417018517938085996202522987, −9.522588321694323021800064707849, −8.711170869440704057100578561463, −8.103400461057385336068831969167, −7.27468684056163305824569445437, −6.64876071869805201424143201850, −5.60839983286456532273980192027, −5.00055178782366461070820043954, −3.34283693728567222645028487550, −1.84554381322945571023078866229, 0.19768603830847262584629379663, 1.99346829424782372295086047028, 2.85584533417621086714048230576, 3.74790161829736804324663354684, 5.30840971467327790311750213119, 6.10602202932046425481141835455, 7.70013880340093872336832478224, 8.059073568092085010186376372206, 8.831199150534231085891276733211, 9.594318539804705250644773392227

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