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Properties

Label 2-1183-7.4-c1-0-53
Degree $2$
Conductor $1183$
Sign $0.999 - 0.0271i$
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  + (0.249 + 0.433i)2-s + (−0.424 + 0.735i)3-s + (0.875 − 1.51i)4-s + (−0.521 − 0.902i)5-s − 0.424·6-s + (2.40 + 1.11i)7-s + 1.87·8-s + (1.13 + 1.97i)9-s + (0.260 − 0.451i)10-s + (1.98 − 3.43i)11-s + (0.743 + 1.28i)12-s + (0.119 + 1.31i)14-s + 0.885·15-s + (−1.28 − 2.21i)16-s + (0.0710 − 0.123i)17-s + (−0.569 + 0.986i)18-s + ⋯
L(s)  = 1  + (0.176 + 0.306i)2-s + (−0.245 + 0.424i)3-s + (0.437 − 0.757i)4-s + (−0.233 − 0.403i)5-s − 0.173·6-s + (0.907 + 0.419i)7-s + 0.662·8-s + (0.379 + 0.657i)9-s + (0.0824 − 0.142i)10-s + (0.598 − 1.03i)11-s + (0.214 + 0.371i)12-s + (0.0319 + 0.352i)14-s + 0.228·15-s + (−0.320 − 0.554i)16-s + (0.0172 − 0.0298i)17-s + (−0.134 + 0.232i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.999 - 0.0271i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.999 - 0.0271i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.174569122\)
\(L(\frac12)\) \(\approx\) \(2.174569122\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.40 - 1.11i)T \)
13 \( 1 \)
good2 \( 1 + (-0.249 - 0.433i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.424 - 0.735i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.521 + 0.902i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.98 + 3.43i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.0710 + 0.123i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.75 - 4.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.19 + 3.80i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.39T + 29T^{2} \)
31 \( 1 + (1.42 - 2.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.421 - 0.730i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 - 4.82T + 43T^{2} \)
47 \( 1 + (-2.27 - 3.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.139 + 0.242i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.39 + 9.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.93 - 5.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.57 - 4.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.69T + 71T^{2} \)
73 \( 1 + (-3.30 + 5.72i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.96 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.87T + 83T^{2} \)
89 \( 1 + (0.873 + 1.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.70T + 97T^{2} \)
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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

−9.831486825966201138440452541137, −8.953324600820072154202937463513, −8.032609865866435407937462786123, −7.38922297255928138388648020100, −6.08075356009978679783642669803, −5.59895112472394372255552095392, −4.74905862548671556324443425980, −3.93591004794976720762414546595, −2.25057806707691248852102252796, −1.13492294560997455761513600542, 1.31286769637940114045449411925, 2.35453864457384551781050476191, 3.73373992001236731594729613038, 4.24393426073144963205847542725, 5.50436988314529539256074709832, 6.79146175061854513213430669606, 7.36746301512695890557263755194, 7.65662118980502065082050972503, 9.029811857997121111763195104515, 9.737050867400129290313656288857

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