Properties

Label 2-1183-7.2-c1-0-47
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: $\chi_{1183} (170, \cdot )$
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

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

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