Properties

Label 2-1183-7.4-c1-0-16
Degree $2$
Conductor $1183$
Sign $-0.887 - 0.460i$
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  + (−0.0682 − 0.118i)2-s + (−1.48 + 2.56i)3-s + (0.990 − 1.71i)4-s + (1.64 + 2.84i)5-s + 0.404·6-s + (1.56 + 2.13i)7-s − 0.543·8-s + (−2.90 − 5.02i)9-s + (0.224 − 0.388i)10-s + (−0.437 + 0.757i)11-s + (2.93 + 5.09i)12-s + (0.145 − 0.330i)14-s − 9.76·15-s + (−1.94 − 3.36i)16-s + (−2.71 + 4.70i)17-s + (−0.395 + 0.685i)18-s + ⋯
L(s)  = 1  + (−0.0482 − 0.0835i)2-s + (−0.856 + 1.48i)3-s + (0.495 − 0.857i)4-s + (0.735 + 1.27i)5-s + 0.165·6-s + (0.590 + 0.806i)7-s − 0.192·8-s + (−0.966 − 1.67i)9-s + (0.0709 − 0.122i)10-s + (−0.131 + 0.228i)11-s + (0.848 + 1.46i)12-s + (0.0389 − 0.0882i)14-s − 2.52·15-s + (−0.486 − 0.841i)16-s + (−0.659 + 1.14i)17-s + (−0.0933 + 0.161i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.887 - 0.460i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.887 - 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.354666359\)
\(L(\frac12)\) \(\approx\) \(1.354666359\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.56 - 2.13i)T \)
13 \( 1 \)
good2 \( 1 + (0.0682 + 0.118i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.48 - 2.56i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.64 - 2.84i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.437 - 0.757i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.71 - 4.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.934 - 1.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.43 - 5.94i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.15T + 29T^{2} \)
31 \( 1 + (-4.52 + 7.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.359 + 0.623i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.916T + 41T^{2} \)
43 \( 1 - 5.76T + 43T^{2} \)
47 \( 1 + (1.75 + 3.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.41 - 4.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.42 + 4.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.484 - 0.838i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.42 - 5.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.91T + 71T^{2} \)
73 \( 1 + (1.64 - 2.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.50 + 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (-4.83 - 8.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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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

−10.17072941391525429612704567211, −9.674368085299266926968679759993, −8.940401620258070328909679228477, −7.45931821162151702598246179770, −6.29481555575572069945032519034, −5.86491392664234480102795208378, −5.26370394677258627611097122660, −4.16697174854431842591629220896, −2.90696063098852012648490953763, −1.85337406122545236966456891511, 0.64990009958603359357845073685, 1.63292754711159112107918511902, 2.71916685731747386720288404660, 4.55003466829893101132189693710, 5.19556144906076372413803961310, 6.28235532087954004072547013265, 6.98228697547278599439714520156, 7.59068406104471873359223569527, 8.452541207627835576187020864236, 9.069671446356181605525300644058

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