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Properties

Label 2-1183-7.2-c1-0-68
Degree $2$
Conductor $1183$
Sign $0.984 - 0.175i$
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  + (−0.771 + 1.33i)2-s + (0.676 + 1.17i)3-s + (−0.191 − 0.332i)4-s + (−0.170 + 0.296i)5-s − 2.08·6-s + (2.12 − 1.57i)7-s − 2.49·8-s + (0.584 − 1.01i)9-s + (−0.263 − 0.457i)10-s + (−2.77 − 4.81i)11-s + (0.259 − 0.449i)12-s + (0.460 + 4.05i)14-s − 0.462·15-s + (2.30 − 4.00i)16-s + (−2.52 − 4.36i)17-s + (0.902 + 1.56i)18-s + ⋯
L(s)  = 1  + (−0.545 + 0.945i)2-s + (0.390 + 0.676i)3-s + (−0.0958 − 0.166i)4-s + (−0.0764 + 0.132i)5-s − 0.852·6-s + (0.804 − 0.594i)7-s − 0.882·8-s + (0.194 − 0.337i)9-s + (−0.0834 − 0.144i)10-s + (−0.837 − 1.45i)11-s + (0.0748 − 0.129i)12-s + (0.123 + 1.08i)14-s − 0.119·15-s + (0.577 − 1.00i)16-s + (−0.611 − 1.05i)17-s + (0.212 + 0.368i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.984 - 0.175i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.984 - 0.175i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.188893802\)
\(L(\frac12)\) \(\approx\) \(1.188893802\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.12 + 1.57i)T \)
13 \( 1 \)
good2 \( 1 + (0.771 - 1.33i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.676 - 1.17i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.170 - 0.296i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.77 + 4.81i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.52 + 4.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.08 + 3.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.621 + 1.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.715T + 29T^{2} \)
31 \( 1 + (4.55 + 7.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.764 + 1.32i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.73T + 41T^{2} \)
43 \( 1 + 6.37T + 43T^{2} \)
47 \( 1 + (0.189 - 0.328i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.40 + 2.43i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.28 - 7.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.66 - 2.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.96 + 6.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 + (-3.72 - 6.45i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.80 + 4.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.99T + 83T^{2} \)
89 \( 1 + (8.06 - 13.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−9.420880535068381345611156349351, −8.902992530548099389553946186021, −8.162077704726045102669100179492, −7.39839587470995610935296679968, −6.74477879967079929604537038125, −5.60072930133360958097452812213, −4.77448606141830937761388215834, −3.56318026174013882295815649615, −2.76527153140264902885756433414, −0.57675858375197134021087863402, 1.63150034995618524744715460394, 1.94521587667521658415467907656, 3.03122953721144767284281226082, 4.54688512855679142659881289452, 5.38327854383024761983214206495, 6.58243740794690212000261809644, 7.52846764801724491007713396137, 8.306827789758330429630004719099, 8.827327520363542448429133037772, 9.995718449844053117872475911301

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