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Properties

Label 2-1127-161.114-c0-0-10
Degree $2$
Conductor $1127$
Sign $-0.198 - 0.980i$
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  + (0.866 − 1.5i)2-s + (−0.965 − 1.67i)3-s + (−1 − 1.73i)4-s − 3.34·6-s − 1.73·8-s + (−1.36 + 2.36i)9-s + (−1.93 + 3.34i)12-s − 0.517·13-s + (−0.5 + 0.866i)16-s + (2.36 + 4.09i)18-s + (0.5 − 0.866i)23-s + (1.67 + 2.89i)24-s + (−0.5 − 0.866i)25-s + (−0.448 + 0.776i)26-s + 3.34·27-s + ⋯
L(s)  = 1  + (0.866 − 1.5i)2-s + (−0.965 − 1.67i)3-s + (−1 − 1.73i)4-s − 3.34·6-s − 1.73·8-s + (−1.36 + 2.36i)9-s + (−1.93 + 3.34i)12-s − 0.517·13-s + (−0.5 + 0.866i)16-s + (2.36 + 4.09i)18-s + (0.5 − 0.866i)23-s + (1.67 + 2.89i)24-s + (−0.5 − 0.866i)25-s + (−0.448 + 0.776i)26-s + 3.34·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ -0.198 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9285298370\)
\(L(\frac12)\) \(\approx\) \(0.9285298370\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + 0.517T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - 0.517T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.894542718334242637906424253592, −8.629896176126406306329784198465, −7.57646521948628122433590953866, −6.76155701399814990455880096243, −5.77336175884207078222558070825, −5.19740918896636854673061045664, −4.11293076808370701215598876342, −2.60779010794368938140809883660, −1.99753348293486890208556830139, −0.71562465124332083766634630029, 3.33043958185879804505894440870, 4.01552716753520345773326983628, 4.90563425908324048444096631265, 5.42652738038804516429644147245, 6.08111969176445465990155292687, 7.01217238402317070245955272983, 7.927188772760102547720021296851, 9.179752039445669890540708111957, 9.491105003187185479668862968098, 10.67172955098877993140118682886

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