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Properties

Label 2-1127-161.124-c0-0-0
Degree $2$
Conductor $1127$
Sign $-0.988 - 0.149i$
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  + (0.396 − 1.63i)2-s + (−1.62 − 0.838i)4-s + (−0.915 + 1.05i)8-s + (−0.786 − 0.618i)9-s + (−0.308 − 1.27i)11-s + (0.302 + 0.424i)16-s + (−1.32 + 1.04i)18-s − 2.20·22-s + (0.928 + 0.371i)23-s + (0.723 + 0.690i)25-s + (−1.61 − 1.03i)29-s + (−0.483 + 0.193i)32-s + (0.760 + 1.66i)36-s + (−0.653 − 0.513i)37-s + (0.186 + 0.215i)43-s + (−0.565 + 2.33i)44-s + ⋯
L(s)  = 1  + (0.396 − 1.63i)2-s + (−1.62 − 0.838i)4-s + (−0.915 + 1.05i)8-s + (−0.786 − 0.618i)9-s + (−0.308 − 1.27i)11-s + (0.302 + 0.424i)16-s + (−1.32 + 1.04i)18-s − 2.20·22-s + (0.928 + 0.371i)23-s + (0.723 + 0.690i)25-s + (−1.61 − 1.03i)29-s + (−0.483 + 0.193i)32-s + (0.760 + 1.66i)36-s + (−0.653 − 0.513i)37-s + (0.186 + 0.215i)43-s + (−0.565 + 2.33i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $-0.988 - 0.149i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ -0.988 - 0.149i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.003434994\)
\(L(\frac12)\) \(\approx\) \(1.003434994\)
\(L(1)\) 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 \( 1 + (-0.928 - 0.371i)T \)
good2 \( 1 + (-0.396 + 1.63i)T + (-0.888 - 0.458i)T^{2} \)
3 \( 1 + (0.786 + 0.618i)T^{2} \)
5 \( 1 + (-0.723 - 0.690i)T^{2} \)
11 \( 1 + (0.308 + 1.27i)T + (-0.888 + 0.458i)T^{2} \)
13 \( 1 + (0.654 - 0.755i)T^{2} \)
17 \( 1 + (0.995 + 0.0950i)T^{2} \)
19 \( 1 + (0.995 - 0.0950i)T^{2} \)
29 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
31 \( 1 + (-0.928 - 0.371i)T^{2} \)
37 \( 1 + (0.653 + 0.513i)T + (0.235 + 0.971i)T^{2} \)
41 \( 1 + (0.959 - 0.281i)T^{2} \)
43 \( 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1.91 + 0.182i)T + (0.981 - 0.189i)T^{2} \)
59 \( 1 + (0.327 + 0.945i)T^{2} \)
61 \( 1 + (0.786 - 0.618i)T^{2} \)
67 \( 1 + (-1.21 - 1.16i)T + (0.0475 + 0.998i)T^{2} \)
71 \( 1 + (-1.25 + 0.368i)T + (0.841 - 0.540i)T^{2} \)
73 \( 1 + (-0.580 - 0.814i)T^{2} \)
79 \( 1 + (-1.91 - 0.182i)T + (0.981 + 0.189i)T^{2} \)
83 \( 1 + (0.959 + 0.281i)T^{2} \)
89 \( 1 + (-0.928 + 0.371i)T^{2} \)
97 \( 1 + (0.959 - 0.281i)T^{2} \)
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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

−9.680807653908968418533014829479, −9.081962843918597035750318637863, −8.353402238120089964412553348797, −7.07162106995749016208388498816, −5.77777192117446872490172909446, −5.21438937384576984555705347894, −3.83508222309970244323561425128, −3.28683298398694329647538297766, −2.32758831059611864819986609977, −0.809132210169102901442648499406, 2.29962949148807995480109443610, 3.75573625776204435830779725150, 5.02136554136891515659525624099, 5.17096904399776610235822935265, 6.38351858977538797384392403826, 7.11439004936926887763047518638, 7.73602045535621326925500556286, 8.600175016515324968060983898509, 9.217130852228963037719092492479, 10.37329206895269207000846086525

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