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Properties

Label 2-1134-9.4-c1-0-10
Degree $2$
Conductor $1134$
Sign $-0.173 - 0.984i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s − 0.999·8-s + 0.999·10-s + (1 + 1.73i)11-s + (−1.5 + 2.59i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + 17-s + 2·19-s + (0.499 + 0.866i)20-s + (−0.999 + 1.73i)22-s + (1 − 1.73i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (0.188 + 0.327i)7-s − 0.353·8-s + 0.316·10-s + (0.301 + 0.522i)11-s + (−0.416 + 0.720i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + 0.242·17-s + 0.458·19-s + (0.111 + 0.193i)20-s + (−0.213 + 0.369i)22-s + (0.208 − 0.361i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.903159763\)
\(L(\frac12)\) \(\approx\) \(1.903159763\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.5 - 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.5 - 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5 + 8.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.849925585394424269974296993712, −9.002205689416305844392476375662, −8.553063722587105000292773192718, −7.24782868106683910884718979818, −6.90914957514134103574809424616, −5.65252103462601336191879523824, −5.03412561882234343657160946940, −4.15088848117274565417116831695, −2.92427982104566816492156765746, −1.52555650207311047170545100730, 0.794205433141935869470282525048, 2.29161131267906432441879891888, 3.25466925627715361937013405711, 4.19942461464256383474112257531, 5.28420150971968308719983300573, 6.03713124097115526623001182144, 7.06122623739023385344683633553, 7.966366491845029999554553336174, 8.892779965060065374146509305820, 9.891276605657681297051646313462

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