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Properties

Label 2-1183-7.4-c1-0-50
Degree $2$
Conductor $1183$
Sign $0.203 - 0.979i$
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  + (1.27 + 2.20i)2-s + (−0.156 + 0.270i)3-s + (−2.22 + 3.85i)4-s + (−1.54 − 2.66i)5-s − 0.795·6-s + (1.63 − 2.08i)7-s − 6.24·8-s + (1.45 + 2.51i)9-s + (3.91 − 6.78i)10-s + (2.10 − 3.64i)11-s + (−0.697 − 1.20i)12-s + (6.65 + 0.946i)14-s + 0.964·15-s + (−3.47 − 6.01i)16-s + (2.98 − 5.16i)17-s + (−3.68 + 6.38i)18-s + ⋯
L(s)  = 1  + (0.898 + 1.55i)2-s + (−0.0903 + 0.156i)3-s + (−1.11 + 1.92i)4-s + (−0.689 − 1.19i)5-s − 0.324·6-s + (0.616 − 0.786i)7-s − 2.20·8-s + (0.483 + 0.837i)9-s + (1.23 − 2.14i)10-s + (0.634 − 1.09i)11-s + (−0.201 − 0.348i)12-s + (1.77 + 0.253i)14-s + 0.248·15-s + (−0.868 − 1.50i)16-s + (0.723 − 1.25i)17-s + (−0.869 + 1.50i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.203 - 0.979i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.203 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.441184867\)
\(L(\frac12)\) \(\approx\) \(2.441184867\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.63 + 2.08i)T \)
13 \( 1 \)
good2 \( 1 + (-1.27 - 2.20i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.156 - 0.270i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.54 + 2.66i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.10 + 3.64i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.98 + 5.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.50 - 2.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.86 - 4.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.74T + 29T^{2} \)
31 \( 1 + (-0.336 + 0.583i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.539 + 0.935i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.51T + 41T^{2} \)
43 \( 1 + 1.23T + 43T^{2} \)
47 \( 1 + (5.51 + 9.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.87 - 3.24i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.08 - 3.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.50 - 2.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.62 + 8.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.19T + 71T^{2} \)
73 \( 1 + (-2.17 + 3.77i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.33 - 4.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + (-6.87 - 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−9.588562256510929063810900768503, −8.725123538379872262559146302456, −7.79204363544474646877021745908, −7.71734671656894112297326500097, −6.64096157532265902847195143069, −5.33848841664199952442800996397, −5.09356598777295933037047571602, −4.16582371251339608535247193755, −3.50846469753943257383778528657, −1.01304595279441183184576138250, 1.28206162255038493908372412817, 2.39005051324098278917139504739, 3.30544129031472545279327246776, 4.11533493327683035432610783307, 4.89305527162285390862693307897, 6.13186274099148917473295853789, 6.86065444360385007950873015500, 7.967619857779899500668805147013, 9.205187279315495932967451165240, 9.861246135826927248882361718201

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