Properties

Label 2-1134-63.47-c1-0-2
Degree $2$
Conductor $1134$
Sign $-0.993 - 0.110i$
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 − 1.73·5-s + (0.5 + 2.59i)7-s + 0.999i·8-s + (1.49 − 0.866i)10-s + (−1.73 − 2i)14-s + (−0.5 − 0.866i)16-s + (−1.73 − 3i)17-s + (6 + 3.46i)19-s + (−0.866 + 1.49i)20-s + 6i·23-s − 2.00·25-s + (2.5 + 0.866i)28-s + (−7.79 − 4.5i)29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.774·5-s + (0.188 + 0.981i)7-s + 0.353i·8-s + (0.474 − 0.273i)10-s + (−0.462 − 0.534i)14-s + (−0.125 − 0.216i)16-s + (−0.420 − 0.727i)17-s + (1.37 + 0.794i)19-s + (−0.193 + 0.335i)20-s + 1.25i·23-s − 0.400·25-s + (0.472 + 0.163i)28-s + (−1.44 − 0.835i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.993 - 0.110i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4017622780\)
\(L(\frac12)\) \(\approx\) \(0.4017622780\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + 1.73T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.73 + 3i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6 - 3.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + (7.79 + 4.5i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.73 - 3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.73 + 3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.59 - 1.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.06 - 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 - 1.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 + (10.5 - 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.66 - 15i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6 - 3.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.898288000343592333480182066718, −9.351299405560711021981839151179, −8.575023415752742458657592468569, −7.60964463627892736061230460814, −7.32499937904793829968906574658, −5.86800043356852777970644962712, −5.41943662621103522210635532679, −4.10475765593241965676458394856, −2.95700016599323887989685097185, −1.62266904897607752513575859145, 0.21845658474372100863496275516, 1.61655782865770239630900231396, 3.13952738172133391689860311797, 3.97429049954081141146203355572, 4.87914742664241811987270976646, 6.26323293432823735284398229035, 7.37741550697353043970711722902, 7.60452818112161174772924878472, 8.670618637790239922654312583445, 9.387452417008642591538626392741

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