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Properties

Label 2-1127-161.31-c0-0-0
Degree $2$
Conductor $1127$
Sign $0.638 - 0.769i$
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  + (−1.78 + 0.713i)2-s + (1.94 − 1.85i)4-s + (−1.34 + 2.93i)8-s + (0.981 + 0.189i)9-s + (0.771 + 0.308i)11-s + (0.167 − 3.50i)16-s + (−1.88 + 0.363i)18-s − 1.59·22-s + (−0.327 + 0.945i)23-s + (−0.786 − 0.618i)25-s + (0.273 − 0.0801i)29-s + (1.14 + 3.32i)32-s + (2.25 − 1.45i)36-s + (1.65 + 0.318i)37-s + (−0.544 − 1.19i)43-s + (2.06 − 0.828i)44-s + ⋯
L(s)  = 1  + (−1.78 + 0.713i)2-s + (1.94 − 1.85i)4-s + (−1.34 + 2.93i)8-s + (0.981 + 0.189i)9-s + (0.771 + 0.308i)11-s + (0.167 − 3.50i)16-s + (−1.88 + 0.363i)18-s − 1.59·22-s + (−0.327 + 0.945i)23-s + (−0.786 − 0.618i)25-s + (0.273 − 0.0801i)29-s + (1.14 + 3.32i)32-s + (2.25 − 1.45i)36-s + (1.65 + 0.318i)37-s + (−0.544 − 1.19i)43-s + (2.06 − 0.828i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $0.638 - 0.769i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ 0.638 - 0.769i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5240252371\)
\(L(\frac12)\) \(\approx\) \(0.5240252371\)
\(L(1)\) 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 \( 1 + (0.327 - 0.945i)T \)
good2 \( 1 + (1.78 - 0.713i)T + (0.723 - 0.690i)T^{2} \)
3 \( 1 + (-0.981 - 0.189i)T^{2} \)
5 \( 1 + (0.786 + 0.618i)T^{2} \)
11 \( 1 + (-0.771 - 0.308i)T + (0.723 + 0.690i)T^{2} \)
13 \( 1 + (-0.415 + 0.909i)T^{2} \)
17 \( 1 + (0.888 + 0.458i)T^{2} \)
19 \( 1 + (0.888 - 0.458i)T^{2} \)
29 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
31 \( 1 + (0.327 - 0.945i)T^{2} \)
37 \( 1 + (-1.65 - 0.318i)T + (0.928 + 0.371i)T^{2} \)
41 \( 1 + (0.142 - 0.989i)T^{2} \)
43 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.252 + 0.130i)T + (0.580 - 0.814i)T^{2} \)
59 \( 1 + (0.995 - 0.0950i)T^{2} \)
61 \( 1 + (-0.981 + 0.189i)T^{2} \)
67 \( 1 + (-1.50 - 1.18i)T + (0.235 + 0.971i)T^{2} \)
71 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
73 \( 1 + (-0.0475 + 0.998i)T^{2} \)
79 \( 1 + (-0.252 - 0.130i)T + (0.580 + 0.814i)T^{2} \)
83 \( 1 + (0.142 + 0.989i)T^{2} \)
89 \( 1 + (0.327 + 0.945i)T^{2} \)
97 \( 1 + (0.142 - 0.989i)T^{2} \)
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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

−9.804055547268963343124224065990, −9.457089058638790907327794120595, −8.435588739579030824801620468890, −7.76144178148624569814708276042, −7.02347794785143534536035596263, −6.37991516445185632423747939096, −5.41048462070235961124002868466, −4.07060767481808551899350581364, −2.25722802809866430814011184894, −1.23600636111698756311311627262, 1.06338571190988432946504560913, 2.13503069472381162042096921836, 3.36140842843713122473140111641, 4.32406654422176469176113533216, 6.19053536935441266448918712079, 6.87026208858937101552438070450, 7.75285274015612189776583228334, 8.377404708513062911600608201723, 9.432223485688093580567907747646, 9.635422470322342298643234316209

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