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Properties

Label 2-1127-161.110-c0-0-0
Degree $2$
Conductor $1127$
Sign $-0.244 + 0.969i$
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 + (−0.947 − 0.903i)11-s + (−0.518 − 0.0495i)16-s + (1.56 − 0.625i)18-s − 2.20·22-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 + (0.771 + 0.308i)37-s + (0.186 − 0.215i)43-s + (−1.73 + 1.65i)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 + (−0.947 − 0.903i)11-s + (−0.518 − 0.0495i)16-s + (1.56 − 0.625i)18-s − 2.20·22-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 + (0.771 + 0.308i)37-s + (0.186 − 0.215i)43-s + (−1.73 + 1.65i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $-0.244 + 0.969i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (754, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ -0.244 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.938784032\)
\(L(\frac12)\) \(\approx\) \(1.938784032\)
\(L(1)\) 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 \( 1 + (0.947 + 0.903i)T + (0.0475 + 0.998i)T^{2} \)
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 \( 1 + (-0.771 - 0.308i)T + (0.723 + 0.690i)T^{2} \)
41 \( 1 + (0.959 + 0.281i)T^{2} \)
43 \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.11 - 1.56i)T + (-0.327 - 0.945i)T^{2} \)
59 \( 1 + (-0.981 + 0.189i)T^{2} \)
61 \( 1 + (-0.928 + 0.371i)T^{2} \)
67 \( 1 + (-0.396 - 1.63i)T + (-0.888 + 0.458i)T^{2} \)
71 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (0.995 + 0.0950i)T^{2} \)
79 \( 1 + (1.11 + 1.56i)T + (-0.327 + 0.945i)T^{2} \)
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

−10.18029974873476184292104996671, −9.298747153327053183920394347260, −8.109370271819049625454857603176, −7.23038461024177858962779639760, −5.96356660296916779920996676047, −5.29018356376492948981965467904, −4.42142220942085164110410386519, −3.53786837214564548243546473773, −2.60414407257608446010363697295, −1.48732436867437307256591231063, 2.17675129551046719322700988430, 3.61811680187849238465922337481, 4.36576810053629015871931720294, 5.13500158072987728661375727578, 6.04078620788856787493162367053, 6.80796357095822767583897871207, 7.64391010885453995342421941530, 8.038692432969917000884818847439, 9.515883982119087195449656645767, 10.06574158760829647814109334094

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