Properties

Label 2-1127-161.52-c0-0-0
Degree $2$
Conductor $1127$
Sign $0.962 + 0.272i$
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  + (1.30 − 0.124i)2-s + (0.702 − 0.135i)4-s + (−0.357 + 0.105i)8-s + (0.0475 − 0.998i)9-s + (1.91 + 0.182i)11-s + (−1.11 + 0.447i)16-s + (−0.0623 − 1.30i)18-s + 2.51·22-s + (−0.888 − 0.458i)23-s + (0.580 + 0.814i)25-s + (−0.544 + 0.627i)29-s + (−1.06 + 0.551i)32-s + (−0.101 − 0.708i)36-s + (−0.0135 + 0.284i)37-s + (−1.61 − 0.474i)43-s + (1.36 − 0.130i)44-s + ⋯
L(s)  = 1  + (1.30 − 0.124i)2-s + (0.702 − 0.135i)4-s + (−0.357 + 0.105i)8-s + (0.0475 − 0.998i)9-s + (1.91 + 0.182i)11-s + (−1.11 + 0.447i)16-s + (−0.0623 − 1.30i)18-s + 2.51·22-s + (−0.888 − 0.458i)23-s + (0.580 + 0.814i)25-s + (−0.544 + 0.627i)29-s + (−1.06 + 0.551i)32-s + (−0.101 − 0.708i)36-s + (−0.0135 + 0.284i)37-s + (−1.61 − 0.474i)43-s + (1.36 − 0.130i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $0.962 + 0.272i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ 0.962 + 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.974388507\)
\(L(\frac12)\) \(\approx\) \(1.974388507\)
\(L(1)\) 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 \( 1 + (0.888 + 0.458i)T \)
good2 \( 1 + (-1.30 + 0.124i)T + (0.981 - 0.189i)T^{2} \)
3 \( 1 + (-0.0475 + 0.998i)T^{2} \)
5 \( 1 + (-0.580 - 0.814i)T^{2} \)
11 \( 1 + (-1.91 - 0.182i)T + (0.981 + 0.189i)T^{2} \)
13 \( 1 + (0.959 - 0.281i)T^{2} \)
17 \( 1 + (0.786 - 0.618i)T^{2} \)
19 \( 1 + (0.786 + 0.618i)T^{2} \)
29 \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \)
31 \( 1 + (0.888 + 0.458i)T^{2} \)
37 \( 1 + (0.0135 - 0.284i)T + (-0.995 - 0.0950i)T^{2} \)
41 \( 1 + (-0.415 + 0.909i)T^{2} \)
43 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.653 + 0.513i)T + (0.235 + 0.971i)T^{2} \)
59 \( 1 + (-0.723 - 0.690i)T^{2} \)
61 \( 1 + (-0.0475 - 0.998i)T^{2} \)
67 \( 1 + (0.759 + 1.06i)T + (-0.327 + 0.945i)T^{2} \)
71 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (-0.928 + 0.371i)T^{2} \)
79 \( 1 + (0.653 - 0.513i)T + (0.235 - 0.971i)T^{2} \)
83 \( 1 + (-0.415 - 0.909i)T^{2} \)
89 \( 1 + (0.888 - 0.458i)T^{2} \)
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

−9.914997876149488453405149702320, −9.160049533475300004430028032620, −8.559939287492618565042402639086, −7.00543036167023001064590851339, −6.50629044153510734780375990130, −5.70986737597071232596647571109, −4.61523165659494800618062044406, −3.83159259425709233753262073087, −3.22206699052356521857904845351, −1.62494510493391241916724470700, 1.82028354144442201061651757711, 3.16631025946390366882105303957, 4.11618816041872421846124387341, 4.71646718554379717331674140035, 5.81925956307113370636081061902, 6.40447775763384302392803298112, 7.31684493131889358635649332595, 8.406961063222045039267603691894, 9.239134287394789347877519023417, 10.07925493703224664523971706857

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