Properties

Label 2-1127-161.150-c0-0-0
Degree $2$
Conductor $1127$
Sign $0.998 + 0.0562i$
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  + (−0.738 − 0.380i)2-s + (−0.179 − 0.252i)4-s + (0.154 + 1.07i)8-s + (0.235 + 0.971i)9-s + (0.252 − 0.130i)11-s + (0.194 − 0.561i)16-s + (0.195 − 0.807i)18-s − 0.236·22-s + (0.723 + 0.690i)23-s + (0.0475 + 0.998i)25-s + (0.698 + 1.53i)29-s + (0.430 − 0.410i)32-s + (0.202 − 0.234i)36-s + (−0.308 − 1.27i)37-s + (0.273 − 1.89i)43-s + (−0.0783 − 0.0403i)44-s + ⋯
L(s)  = 1  + (−0.738 − 0.380i)2-s + (−0.179 − 0.252i)4-s + (0.154 + 1.07i)8-s + (0.235 + 0.971i)9-s + (0.252 − 0.130i)11-s + (0.194 − 0.561i)16-s + (0.195 − 0.807i)18-s − 0.236·22-s + (0.723 + 0.690i)23-s + (0.0475 + 0.998i)25-s + (0.698 + 1.53i)29-s + (0.430 − 0.410i)32-s + (0.202 − 0.234i)36-s + (−0.308 − 1.27i)37-s + (0.273 − 1.89i)43-s + (−0.0783 − 0.0403i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $0.998 + 0.0562i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (472, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ 0.998 + 0.0562i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6712512587\)
\(L(\frac12)\) \(\approx\) \(0.6712512587\)
\(L(1)\) 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 \( 1 + (-0.723 - 0.690i)T \)
good2 \( 1 + (0.738 + 0.380i)T + (0.580 + 0.814i)T^{2} \)
3 \( 1 + (-0.235 - 0.971i)T^{2} \)
5 \( 1 + (-0.0475 - 0.998i)T^{2} \)
11 \( 1 + (-0.252 + 0.130i)T + (0.580 - 0.814i)T^{2} \)
13 \( 1 + (0.142 + 0.989i)T^{2} \)
17 \( 1 + (-0.981 - 0.189i)T^{2} \)
19 \( 1 + (-0.981 + 0.189i)T^{2} \)
29 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
31 \( 1 + (-0.723 - 0.690i)T^{2} \)
37 \( 1 + (0.308 + 1.27i)T + (-0.888 + 0.458i)T^{2} \)
41 \( 1 + (-0.841 + 0.540i)T^{2} \)
43 \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-1.65 + 0.318i)T + (0.928 - 0.371i)T^{2} \)
59 \( 1 + (0.786 - 0.618i)T^{2} \)
61 \( 1 + (-0.235 + 0.971i)T^{2} \)
67 \( 1 + (-0.0395 - 0.829i)T + (-0.995 + 0.0950i)T^{2} \)
71 \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.327 - 0.945i)T^{2} \)
79 \( 1 + (-1.65 - 0.318i)T + (0.928 + 0.371i)T^{2} \)
83 \( 1 + (-0.841 - 0.540i)T^{2} \)
89 \( 1 + (-0.723 + 0.690i)T^{2} \)
97 \( 1 + (-0.841 + 0.540i)T^{2} \)
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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

−10.11256937203840080270559135919, −9.082020925889428223296199721284, −8.716474858136771271874367813718, −7.64022068575490872451377084514, −6.95177100783657225194328455898, −5.49187700672356259485614700708, −5.09268317544009278754720871280, −3.77403182196438560358584001858, −2.39738862172697916757190818073, −1.31878387507843658490335213366, 0.942779325522228085969958616301, 2.80320191212524938330634523527, 3.94722689495707149123745007492, 4.69927221062562924299781407412, 6.29037273125573665969544518916, 6.66340964109259009335116757566, 7.72428407189936652001874211047, 8.411225614055569304410324510037, 9.174354695934854195359296965648, 9.814413964265847423765880114005

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