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Properties

Label 2-1134-9.4-c1-0-22
Degree $2$
Conductor $1134$
Sign $-0.984 + 0.173i$
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 + (1.86 − 3.23i)5-s + (0.5 + 0.866i)7-s + 0.999·8-s − 3.73·10-s + (−2.09 − 3.63i)11-s + (−0.232 + 0.401i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 7·17-s − 2.73·19-s + (1.86 + 3.23i)20-s + (−2.09 + 3.63i)22-s + (3.09 − 5.36i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.834 − 1.44i)5-s + (0.188 + 0.327i)7-s + 0.353·8-s − 1.18·10-s + (−0.632 − 1.09i)11-s + (−0.0643 + 0.111i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s − 1.69·17-s − 0.626·19-s + (0.417 + 0.722i)20-s + (−0.447 + 0.774i)22-s + (0.645 − 1.11i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.984 + 0.173i$
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: \((2,\ 1134,\ (\ :1/2),\ -0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.039901646\)
\(L(\frac12)\) \(\approx\) \(1.039901646\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 + 3.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.09 + 3.63i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.232 - 0.401i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 + 2.73T + 19T^{2} \)
23 \( 1 + (-3.09 + 5.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.23 - 7.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.09 + 1.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.66T + 37T^{2} \)
41 \( 1 + (-4.73 + 8.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.73 + 4.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.633 - 1.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.53T + 53T^{2} \)
59 \( 1 + (3.09 - 5.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.96 - 8.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.63 + 2.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + (7.56 + 13.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.29 + 12.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.92T + 89T^{2} \)
97 \( 1 + (-1.46 - 2.53i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.068308103028797184722936546606, −8.812982384896826395626248746592, −8.368879279348836825090239254095, −6.92448964616507388267743805429, −5.85967622638866360560991853017, −5.02029252943026081262982644510, −4.29312548658099977542858581654, −2.75358899345313692942159518059, −1.81852840755495423695993278953, −0.48399585579180060737933102351, 1.90393965717513917812347915940, 2.77467198021771906339500641098, 4.27597932065107134121208381388, 5.22599904277464858287026612790, 6.37099562595878266180189287326, 6.80506775421695516142253494543, 7.53472373221044476680995743646, 8.468794544737508245266320009176, 9.655091166123665786562986218603, 9.980555885188247690705119816529

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