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Properties

Label 2-1127-161.86-c0-0-0
Degree $2$
Conductor $1127$
Sign $0.412 - 0.910i$
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  + (0.759 + 1.06i)2-s + (−0.233 + 0.676i)4-s + (0.357 − 0.105i)8-s + (0.888 − 0.458i)9-s + (−0.458 − 0.326i)11-s + (0.946 + 0.744i)16-s + (1.16 + 0.600i)18-s − 0.737i·22-s + (0.0475 + 0.998i)23-s + (−0.995 + 0.0950i)25-s + (−0.544 + 0.627i)29-s + (−0.0572 + 1.20i)32-s + (0.101 + 0.708i)36-s + (−0.907 − 1.75i)37-s + (−0.304 + 1.03i)43-s + (0.328 − 0.233i)44-s + ⋯
L(s)  = 1  + (0.759 + 1.06i)2-s + (−0.233 + 0.676i)4-s + (0.357 − 0.105i)8-s + (0.888 − 0.458i)9-s + (−0.458 − 0.326i)11-s + (0.946 + 0.744i)16-s + (1.16 + 0.600i)18-s − 0.737i·22-s + (0.0475 + 0.998i)23-s + (−0.995 + 0.0950i)25-s + (−0.544 + 0.627i)29-s + (−0.0572 + 1.20i)32-s + (0.101 + 0.708i)36-s + (−0.907 − 1.75i)37-s + (−0.304 + 1.03i)43-s + (0.328 − 0.233i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $0.412 - 0.910i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ 0.412 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.647933024\)
\(L(\frac12)\) \(\approx\) \(1.647933024\)
\(L(1)\) 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 \( 1 + (-0.0475 - 0.998i)T \)
good2 \( 1 + (-0.759 - 1.06i)T + (-0.327 + 0.945i)T^{2} \)
3 \( 1 + (-0.888 + 0.458i)T^{2} \)
5 \( 1 + (0.995 - 0.0950i)T^{2} \)
11 \( 1 + (0.458 + 0.326i)T + (0.327 + 0.945i)T^{2} \)
13 \( 1 + (-0.959 + 0.281i)T^{2} \)
17 \( 1 + (-0.928 - 0.371i)T^{2} \)
19 \( 1 + (-0.928 + 0.371i)T^{2} \)
29 \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \)
31 \( 1 + (0.0475 + 0.998i)T^{2} \)
37 \( 1 + (0.907 + 1.75i)T + (-0.580 + 0.814i)T^{2} \)
41 \( 1 + (0.415 - 0.909i)T^{2} \)
43 \( 1 + (0.304 - 1.03i)T + (-0.841 - 0.540i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.676 + 1.68i)T + (-0.723 + 0.690i)T^{2} \)
59 \( 1 + (0.235 - 0.971i)T^{2} \)
61 \( 1 + (0.888 + 0.458i)T^{2} \)
67 \( 1 + (-0.143 - 1.50i)T + (-0.981 + 0.189i)T^{2} \)
71 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (-0.786 - 0.618i)T^{2} \)
79 \( 1 + (-0.676 + 1.68i)T + (-0.723 - 0.690i)T^{2} \)
83 \( 1 + (-0.415 - 0.909i)T^{2} \)
89 \( 1 + (-0.0475 + 0.998i)T^{2} \)
97 \( 1 + (-0.415 + 0.909i)T^{2} \)
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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

−10.08954492225811049677981503322, −9.333164352741067832495086326107, −8.214003080247321135704190841874, −7.40860148658664006722482846353, −6.86985986256670782340915031189, −5.85463293554968950751283409916, −5.26697111229588058949334528903, −4.20341451034337237769549148733, −3.44890979294982967276631526887, −1.67383471708748235781728371884, 1.64157538087738616863094823087, 2.53550580610610494145519695884, 3.67671472241662582953345529050, 4.53176643514274766626019593432, 5.17946346876894245257171061813, 6.41885469904631546901959405977, 7.49951019826738229690447224030, 8.102165641322417617078096040516, 9.382071414937632285129318664096, 10.28552073204158725033325072978

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