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Properties

Label 2-1134-63.59-c1-0-22
Degree $2$
Conductor $1134$
Sign $-0.572 + 0.819i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − 2.44·5-s + (2.62 − 0.358i)7-s − 0.999i·8-s + (2.12 + 1.22i)10-s − 4.24i·11-s + (0.621 + 0.358i)13-s + (−2.44 − i)14-s + (−0.5 + 0.866i)16-s + (1.22 − 2.12i)17-s + (−4.24 + 2.44i)19-s + (−1.22 − 2.12i)20-s + (−2.12 + 3.67i)22-s + 6i·23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s − 1.09·5-s + (0.990 − 0.135i)7-s − 0.353i·8-s + (0.670 + 0.387i)10-s − 1.27i·11-s + (0.172 + 0.0994i)13-s + (−0.654 − 0.267i)14-s + (−0.125 + 0.216i)16-s + (0.297 − 0.514i)17-s + (−0.973 + 0.561i)19-s + (−0.273 − 0.474i)20-s + (−0.452 + 0.783i)22-s + 1.25i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.572 + 0.819i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.572 + 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7259708638\)
\(L(\frac12)\) \(\approx\) \(0.7259708638\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-2.62 + 0.358i)T \)
good5 \( 1 + 2.44T + 5T^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
13 \( 1 + (-0.621 - 0.358i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.22 + 2.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.24 - 2.44i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + (1.52 - 0.878i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.86 + 4.54i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.62 + 4.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.22 - 2.12i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.5 + 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.42 + 11.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (12.5 + 7.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.22 + 2.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.62 + 2.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.74 + 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + (4.75 + 2.74i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.378 - 0.655i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.64 - 13.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.52 - 2.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.74 + 1.58i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.452703139465288854775530426952, −8.511218728904305149509579467709, −8.046821508202452002026694226548, −7.43604167819905483371487095240, −6.27235635858340727894941641271, −5.15606086410710904688881908614, −4.00438830800858746094855578152, −3.32341555593014793211934535159, −1.82692531123717948946328657958, −0.41643311164452117719419900668, 1.39158279958045387003945535096, 2.69711651946668094716947870572, 4.37894277570026584570274558987, 4.66686488667979718083177962763, 6.09956971921012416143742984877, 7.00250731722213165445418355142, 7.79758987565618829567842821922, 8.315238223053032249897836983324, 9.036648413777908328379987162562, 10.21698481251350304598760315996

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