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Properties

Label 2-1127-161.30-c0-0-0
Degree $2$
Conductor $1127$
Sign $0.407 + 0.913i$
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  + (−0.0930 − 0.268i)2-s + (0.722 − 0.568i)4-s + (−0.459 − 0.295i)8-s + (−0.580 − 0.814i)9-s + (1.02 + 0.353i)11-s + (0.180 − 0.741i)16-s + (−0.165 + 0.231i)18-s − 0.307i·22-s + (−0.995 − 0.0950i)23-s + (0.981 + 0.189i)25-s + (0.186 − 1.29i)29-s + (−0.760 + 0.0725i)32-s + (−0.881 − 0.258i)36-s + (−0.458 + 0.326i)37-s + (0.983 + 1.53i)43-s + (0.939 − 0.325i)44-s + ⋯
L(s)  = 1  + (−0.0930 − 0.268i)2-s + (0.722 − 0.568i)4-s + (−0.459 − 0.295i)8-s + (−0.580 − 0.814i)9-s + (1.02 + 0.353i)11-s + (0.180 − 0.741i)16-s + (−0.165 + 0.231i)18-s − 0.307i·22-s + (−0.995 − 0.0950i)23-s + (0.981 + 0.189i)25-s + (0.186 − 1.29i)29-s + (−0.760 + 0.0725i)32-s + (−0.881 − 0.258i)36-s + (−0.458 + 0.326i)37-s + (0.983 + 1.53i)43-s + (0.939 − 0.325i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $0.407 + 0.913i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ 0.407 + 0.913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.151363743\)
\(L(\frac12)\) \(\approx\) \(1.151363743\)
\(L(1)\) 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 \( 1 + (0.995 + 0.0950i)T \)
good2 \( 1 + (0.0930 + 0.268i)T + (-0.786 + 0.618i)T^{2} \)
3 \( 1 + (0.580 + 0.814i)T^{2} \)
5 \( 1 + (-0.981 - 0.189i)T^{2} \)
11 \( 1 + (-1.02 - 0.353i)T + (0.786 + 0.618i)T^{2} \)
13 \( 1 + (0.841 + 0.540i)T^{2} \)
17 \( 1 + (-0.723 + 0.690i)T^{2} \)
19 \( 1 + (-0.723 - 0.690i)T^{2} \)
29 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
31 \( 1 + (-0.995 - 0.0950i)T^{2} \)
37 \( 1 + (0.458 - 0.326i)T + (0.327 - 0.945i)T^{2} \)
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 \( 1 + (1.04 - 1.09i)T + (-0.0475 - 0.998i)T^{2} \)
59 \( 1 + (-0.888 + 0.458i)T^{2} \)
61 \( 1 + (-0.580 + 0.814i)T^{2} \)
67 \( 1 + (-0.374 + 1.94i)T + (-0.928 - 0.371i)T^{2} \)
71 \( 1 + (1.10 - 1.27i)T + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.235 - 0.971i)T^{2} \)
79 \( 1 + (-1.04 - 1.09i)T + (-0.0475 + 0.998i)T^{2} \)
83 \( 1 + (0.654 + 0.755i)T^{2} \)
89 \( 1 + (0.995 - 0.0950i)T^{2} \)
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

−9.727803321042063658987832175079, −9.359176572657678895443021985350, −8.323267775702867511201293953619, −7.24931017810600888834541956974, −6.33546692263362234032813573115, −5.98166473565499929143555721342, −4.62999198112269168278084369397, −3.51318446692986169194808905880, −2.46848775869645938138212753208, −1.18288789182723832720710222437, 1.83933244568965573789923050009, 2.96279327192395800507252137721, 3.90587182016801751904604167552, 5.19245424256569022808464624548, 6.11137303601172040877576898479, 6.88514768211756326908851906917, 7.67470423863266120500477566905, 8.552648615764200438079674414683, 9.023814966277061012535898019153, 10.37237484231550432488388042303

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