Properties

Label 2-1127-161.79-c0-0-0
Degree $2$
Conductor $1127$
Sign $0.623 - 0.781i$
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  + (0.279 − 0.0538i)2-s + (−0.853 + 0.341i)4-s + (−0.459 + 0.295i)8-s + (0.995 + 0.0950i)9-s + (−0.204 + 1.06i)11-s + (0.552 − 0.526i)16-s + (0.283 − 0.0270i)18-s + 0.307i·22-s + (0.580 + 0.814i)23-s + (−0.327 + 0.945i)25-s + (0.186 + 1.29i)29-s + (0.442 − 0.621i)32-s + (−0.881 + 0.258i)36-s + (−0.0535 + 0.560i)37-s + (0.983 − 1.53i)43-s + (−0.188 − 0.975i)44-s + ⋯
L(s)  = 1  + (0.279 − 0.0538i)2-s + (−0.853 + 0.341i)4-s + (−0.459 + 0.295i)8-s + (0.995 + 0.0950i)9-s + (−0.204 + 1.06i)11-s + (0.552 − 0.526i)16-s + (0.283 − 0.0270i)18-s + 0.307i·22-s + (0.580 + 0.814i)23-s + (−0.327 + 0.945i)25-s + (0.186 + 1.29i)29-s + (0.442 − 0.621i)32-s + (−0.881 + 0.258i)36-s + (−0.0535 + 0.560i)37-s + (0.983 − 1.53i)43-s + (−0.188 − 0.975i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $0.623 - 0.781i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ 0.623 - 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9985614684\)
\(L(\frac12)\) \(\approx\) \(0.9985614684\)
\(L(1)\) 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 \( 1 + (-0.580 - 0.814i)T \)
good2 \( 1 + (-0.279 + 0.0538i)T + (0.928 - 0.371i)T^{2} \)
3 \( 1 + (-0.995 - 0.0950i)T^{2} \)
5 \( 1 + (0.327 - 0.945i)T^{2} \)
11 \( 1 + (0.204 - 1.06i)T + (-0.928 - 0.371i)T^{2} \)
13 \( 1 + (0.841 - 0.540i)T^{2} \)
17 \( 1 + (-0.235 + 0.971i)T^{2} \)
19 \( 1 + (-0.235 - 0.971i)T^{2} \)
29 \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \)
31 \( 1 + (0.580 + 0.814i)T^{2} \)
37 \( 1 + (0.0535 - 0.560i)T + (-0.981 - 0.189i)T^{2} \)
41 \( 1 + (-0.654 - 0.755i)T^{2} \)
43 \( 1 + (-0.983 + 1.53i)T + (-0.415 - 0.909i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-1.46 + 0.356i)T + (0.888 - 0.458i)T^{2} \)
59 \( 1 + (0.0475 + 0.998i)T^{2} \)
61 \( 1 + (0.995 - 0.0950i)T^{2} \)
67 \( 1 + (1.87 + 0.647i)T + (0.786 + 0.618i)T^{2} \)
71 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
73 \( 1 + (0.723 - 0.690i)T^{2} \)
79 \( 1 + (1.46 + 0.356i)T + (0.888 + 0.458i)T^{2} \)
83 \( 1 + (0.654 - 0.755i)T^{2} \)
89 \( 1 + (-0.580 + 0.814i)T^{2} \)
97 \( 1 + (0.654 + 0.755i)T^{2} \)
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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

−10.02454972285752262824639652134, −9.341580814575998318833842887738, −8.639309586678537403087678005485, −7.43628663824263022403318158066, −7.14533113820612047014296404870, −5.64842310596065081938313210330, −4.86401639252533340314743311098, −4.12508945670596032944230139195, −3.14627823071091812250345643304, −1.63682824379856793164577380908, 0.953890809158487010274636955666, 2.71964960913044116100332124460, 4.01832140349784693729351359388, 4.54701221987763290917456556113, 5.70877189047463713907023057369, 6.31794362064405706683691431315, 7.46280391125527060376057057605, 8.400152351309864980838851895926, 9.058876133899583018256384500085, 9.985329616609483101931970837299

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