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Properties

Label 2-1127-161.11-c0-0-0
Degree $2$
Conductor $1127$
Sign $0.581 + 0.813i$
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 + (−0.907 − 1.75i)11-s + (0.194 − 0.561i)16-s + (0.195 − 0.807i)18-s − 1.64i·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 + (1.46 − 0.356i)37-s + (−0.557 − 0.0801i)43-s + (−0.280 + 0.544i)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 + (−0.907 − 1.75i)11-s + (0.194 − 0.561i)16-s + (0.195 − 0.807i)18-s − 1.64i·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 + (1.46 − 0.356i)37-s + (−0.557 − 0.0801i)43-s + (−0.280 + 0.544i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ 0.581 + 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.253968726\)
\(L(\frac12)\) \(\approx\) \(1.253968726\)
\(L(1)\) 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 \( 1 + (0.907 + 1.75i)T + (-0.580 + 0.814i)T^{2} \)
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 \( 1 + (-1.46 + 0.356i)T + (0.888 - 0.458i)T^{2} \)
41 \( 1 + (0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.557 + 0.0801i)T + (0.959 + 0.281i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.204 + 1.06i)T + (-0.928 + 0.371i)T^{2} \)
59 \( 1 + (-0.786 + 0.618i)T^{2} \)
61 \( 1 + (-0.235 + 0.971i)T^{2} \)
67 \( 1 + (-1.81 + 0.0865i)T + (0.995 - 0.0950i)T^{2} \)
71 \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (-0.327 + 0.945i)T^{2} \)
79 \( 1 + (-0.204 + 1.06i)T + (-0.928 - 0.371i)T^{2} \)
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

−9.805347852884079644482188787516, −9.057800703931550667652796117045, −8.333933628317557442319651471928, −7.17050313436625996890157640314, −6.31836657176755158324767837856, −5.60335935705367149499514790567, −4.97836497181636394276796795504, −3.61202070709766783918558877505, −3.10136700277908665288332223359, −0.937083859639225685074738058789, 2.24708366591360727879392779405, 2.73690944517569438473970880923, 4.37546192976392089483737882341, 4.65798225039451442870314602116, 5.59738546103247481850332982423, 6.82018034682217117822321887226, 7.918184340132845762145890023447, 8.194373256464341695127619989657, 9.489574345803290473391620687430, 10.25566838775057487407405670849

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