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Properties

Label 2-1134-9.4-c1-0-11
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 + (2.5 + 4.33i)11-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 2·17-s − 19-s + (0.499 + 0.866i)20-s + (−2.5 + 4.33i)22-s + (−0.5 + 0.866i)23-s + (2 + 3.46i)25-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 + (0.753 + 1.30i)11-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s − 0.485·17-s − 0.229·19-s + (0.111 + 0.193i)20-s + (−0.533 + 0.923i)22-s + (−0.104 + 0.180i)23-s + (0.400 + 0.692i)25-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

\(L(1)\) \(\approx\) \(1.944950498\)
\(L(\frac12)\) \(\approx\) \(1.944950498\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (7 - 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (3 + 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (8 + 13.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.598678269261426001868487751775, −9.363713051091779548762518232883, −8.289374363277257252818871019625, −7.52966299640466144006561091254, −6.60321627284914591966157330330, −5.92230037174648159866103124279, −4.72686872491089972321121914351, −4.35220577229515076364761828740, −2.87101801913459699156706697284, −1.54734269339244836099079402897, 0.812965127840567764807655029651, 2.24661924438002086857101643449, 3.32115426094825788779717376134, 4.16776633734292109426313256674, 5.20602691108412793123885231529, 6.27444471439716991336630602461, 6.78665180005330902717845013139, 8.182027117876338988264903121633, 8.801317671783169534550996703550, 9.740096032895099323638588687997

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