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Properties

Label 2-1134-63.16-c1-0-28
Degree $2$
Conductor $1134$
Sign $-0.975 + 0.220i$
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 + 3·5-s + (−2 − 1.73i)7-s + 0.999·8-s + (−1.5 − 2.59i)10-s − 3·11-s + (−1 − 1.73i)13-s + (−0.499 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (−3 − 5.19i)17-s + (−1 + 1.73i)19-s + (−1.49 + 2.59i)20-s + (1.5 + 2.59i)22-s − 6·23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + 1.34·5-s + (−0.755 − 0.654i)7-s + 0.353·8-s + (−0.474 − 0.821i)10-s − 0.904·11-s + (−0.277 − 0.480i)13-s + (−0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (−0.727 − 1.26i)17-s + (−0.229 + 0.397i)19-s + (−0.335 + 0.580i)20-s + (0.319 + 0.553i)22-s − 1.25·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.975 + 0.220i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7682933291\)
\(L(\frac12)\) \(\approx\) \(0.7682933291\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 1.73i)T \)
good5 \( 1 - 3T + 5T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.651698089608734508292115253978, −8.949518883420163825752330448146, −7.79599870742135485524540560951, −7.03138523576089865572865226858, −5.98180458334743211291693030195, −5.20457598391281575963921024251, −3.98672483786789640180603781053, −2.78530247938581806491281368215, −2.02391653814897812868454839861, −0.34217756244884763818277564147, 1.84708220546072422853339681062, 2.69988016722226878658526245286, 4.33378206245262496871212429770, 5.39896972299335824946239003951, 6.25483380922896915870345804190, 6.45293130383535052168184661337, 7.86066884139887200560627842560, 8.573716615867844970262795956813, 9.517757533149975909547279746806, 9.886616517754322205673121837776

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