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Properties

Label 2-1183-7.4-c1-0-73
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

\(L(1)\) \(\approx\) \(0.1976906367\)
\(L(\frac12)\) \(\approx\) \(0.1976906367\)
\(L(\frac{3}{2})\) 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

−9.782151035513375417932475558487, −8.833189405528600718353873081869, −7.44387728246060734215383250588, −6.86311453778279438561767903523, −6.03935497405550871106592061119, −5.05424427617795029308551767184, −4.19912471706269714483962193170, −2.75811835454518418081885978835, −1.97424084875259094961368308251, −0.083835458496760816421723535973, 1.75663546335758887972369141694, 3.17676886374665664156584199184, 3.76420852247190196840670068554, 5.49097907086628653831381786337, 6.08855836258322638991657323093, 6.81242732581919265967383862017, 7.69187294027315342393553606444, 8.483404951257539833333377321735, 9.212321940286132725107147215332, 9.993715165865994256839253789525

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