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Properties

Label 2-1183-7.4-c1-0-48
Degree $2$
Conductor $1183$
Sign $0.0369 - 0.999i$
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  + (0.689 + 1.19i)2-s + (−1.44 + 2.49i)3-s + (0.0491 − 0.0850i)4-s + (0.402 + 0.697i)5-s − 3.97·6-s + (1.26 − 2.32i)7-s + 2.89·8-s + (−2.65 − 4.59i)9-s + (−0.555 + 0.962i)10-s + (2.63 − 4.56i)11-s + (0.141 + 0.245i)12-s + (3.64 − 0.0965i)14-s − 2.32·15-s + (1.89 + 3.28i)16-s + (0.280 − 0.485i)17-s + (3.65 − 6.33i)18-s + ⋯
L(s)  = 1  + (0.487 + 0.844i)2-s + (−0.831 + 1.44i)3-s + (0.0245 − 0.0425i)4-s + (0.180 + 0.312i)5-s − 1.62·6-s + (0.476 − 0.878i)7-s + 1.02·8-s + (−0.883 − 1.53i)9-s + (−0.175 + 0.304i)10-s + (0.794 − 1.37i)11-s + (0.0408 + 0.0707i)12-s + (0.974 − 0.0257i)14-s − 0.599·15-s + (0.474 + 0.821i)16-s + (0.0679 − 0.117i)17-s + (0.861 − 1.49i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0369 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0369 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.0369 - 0.999i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.0369 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.095847401\)
\(L(\frac12)\) \(\approx\) \(2.095847401\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.26 + 2.32i)T \)
13 \( 1 \)
good2 \( 1 + (-0.689 - 1.19i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.44 - 2.49i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.402 - 0.697i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.63 + 4.56i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.280 + 0.485i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.92 - 5.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.802 - 1.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.28T + 29T^{2} \)
31 \( 1 + (-1.73 + 3.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.620 - 1.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.927T + 41T^{2} \)
43 \( 1 - 4.44T + 43T^{2} \)
47 \( 1 + (1.92 + 3.32i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.72 - 4.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.49 - 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.65 + 6.32i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.67 - 6.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.31T + 71T^{2} \)
73 \( 1 + (2.50 - 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.68 + 9.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.81T + 83T^{2} \)
89 \( 1 + (-2.50 - 4.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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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

−10.31504113533338935350659610394, −9.310006067136146533581396228043, −8.234373371949000369783259901388, −7.26810592198515805805090587861, −6.16192584251624470822597602189, −5.88649916818609263380150149792, −4.87136132343856878104332846570, −4.18833415511438216368664507748, −3.36621192831533750392050606454, −1.08717782787712010528316468883, 1.27071550391386665308527803279, 1.94152533151549519083854020283, 2.93005099339245455285473365775, 4.60014683826657312467493008523, 5.11922364005417714181053933382, 6.26127000944665375228884527800, 7.09392509658426082578486177386, 7.63449882671781926063243910574, 8.728073245519789795538369653244, 9.643851717138965057301565365129

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