Properties

Label 2-1183-7.4-c1-0-24
Degree $2$
Conductor $1183$
Sign $-0.266 - 0.963i$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
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 + (0.500 − 0.866i)4-s + (−0.5 + 2.59i)7-s + 3·8-s + (1.5 + 2.59i)9-s + (−1.5 + 2.59i)11-s + (−2.5 + 0.866i)14-s + (0.500 + 0.866i)16-s + (−3.5 + 6.06i)17-s + (−1.5 + 2.59i)18-s + (−3.5 − 6.06i)19-s − 3·22-s + (3 + 5.19i)23-s + (2.5 − 4.33i)25-s + (2 + 1.73i)28-s − 5·29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.188 + 0.981i)7-s + 1.06·8-s + (0.5 + 0.866i)9-s + (−0.452 + 0.783i)11-s + (−0.668 + 0.231i)14-s + (0.125 + 0.216i)16-s + (−0.848 + 1.47i)17-s + (−0.353 + 0.612i)18-s + (−0.802 − 1.39i)19-s − 0.639·22-s + (0.625 + 1.08i)23-s + (0.5 − 0.866i)25-s + (0.377 + 0.327i)28-s − 0.928·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.002686074\)
\(L(\frac12)\) \(\approx\) \(2.002686074\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.5 - 2.59i)T \)
13 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.5 - 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−10.06308564520716104313794530665, −9.141428851267841831934196126951, −8.245483544734073332030914618777, −7.36397037398870767500520263020, −6.61256860481273696461975222313, −5.83694014615232399469479309593, −4.92320163728506987877075345213, −4.35108388799355082980131351121, −2.55110361751634610703236873684, −1.77784434914108759578004034314, 0.75827170547943211199786953275, 2.27110672584278211544112184555, 3.41918387351756557942644917843, 3.99054887205728175578328385627, 4.96905673918061977873504007344, 6.32923209422031849582970738260, 7.08420183805453174878321351656, 7.72937871177289099364753892311, 8.786126568384763260614108733579, 9.649953399698223935463595591909

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