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Properties

Label 2-1183-7.4-c1-0-66
Degree $2$
Conductor $1183$
Sign $-0.795 - 0.605i$
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  + (−0.875 − 1.51i)2-s + (−0.335 + 0.581i)3-s + (−0.534 + 0.925i)4-s + (−1.33 − 2.31i)5-s + 1.17·6-s + (1.17 + 2.37i)7-s − 1.63·8-s + (1.27 + 2.20i)9-s + (−2.34 + 4.06i)10-s + (0.623 − 1.07i)11-s + (−0.359 − 0.621i)12-s + (2.56 − 3.85i)14-s + 1.79·15-s + (2.49 + 4.32i)16-s + (2.50 − 4.33i)17-s + (2.23 − 3.86i)18-s + ⋯
L(s)  = 1  + (−0.619 − 1.07i)2-s + (−0.193 + 0.335i)3-s + (−0.267 + 0.462i)4-s + (−0.598 − 1.03i)5-s + 0.480·6-s + (0.444 + 0.895i)7-s − 0.576·8-s + (0.424 + 0.735i)9-s + (−0.741 + 1.28i)10-s + (0.187 − 0.325i)11-s + (−0.103 − 0.179i)12-s + (0.686 − 1.03i)14-s + 0.464·15-s + (0.624 + 1.08i)16-s + (0.606 − 1.05i)17-s + (0.526 − 0.911i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 - 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.795 - 0.605i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.795 - 0.605i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3957431561\)
\(L(\frac12)\) \(\approx\) \(0.3957431561\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.17 - 2.37i)T \)
13 \( 1 \)
good2 \( 1 + (0.875 + 1.51i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.335 - 0.581i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.33 + 2.31i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.623 + 1.07i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.50 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.89 + 6.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.628 - 1.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.67T + 29T^{2} \)
31 \( 1 + (-0.211 + 0.366i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.135 + 0.235i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.84T + 41T^{2} \)
43 \( 1 + 8.41T + 43T^{2} \)
47 \( 1 + (1.13 + 1.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.58 + 9.68i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.63 + 4.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.91 - 5.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.76 - 8.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.45T + 71T^{2} \)
73 \( 1 + (1.93 - 3.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.663 + 1.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.64T + 83T^{2} \)
89 \( 1 + (0.576 + 0.999i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−9.310439621261283020949893118720, −8.699972610384012857976759301999, −8.093850575383162595337249269862, −6.92260404362503444914450313611, −5.45794783054387225357969865092, −4.99186406175728921617093011495, −3.89357825658944845747837778692, −2.62796729826104806248097312769, −1.60882672757069423266140154039, −0.21877371022342840314692651728, 1.58168988733675107649115379462, 3.47066859804299938863112395354, 4.00474701964594539191620995822, 5.62998326475253851477289080820, 6.52649131156416271059024349668, 6.96019639401298482192517291755, 7.76633248194637695109765000348, 8.147254555780289487170023781317, 9.334520919766698114697123431234, 10.23287265349329243053654821916

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