Properties

Label 2-1183-1.1-c1-0-4
Degree $2$
Conductor $1183$
Sign $1$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.120·2-s − 0.582·3-s − 1.98·4-s − 1.68·5-s − 0.0700·6-s − 7-s − 0.479·8-s − 2.66·9-s − 0.203·10-s − 0.364·11-s + 1.15·12-s − 0.120·14-s + 0.983·15-s + 3.91·16-s − 3.18·17-s − 0.320·18-s + 1.44·19-s + 3.35·20-s + 0.582·21-s − 0.0438·22-s − 5.08·23-s + 0.279·24-s − 2.15·25-s + 3.29·27-s + 1.98·28-s + 8.19·29-s + 0.118·30-s + ⋯
L(s)  = 1  + 0.0851·2-s − 0.336·3-s − 0.992·4-s − 0.754·5-s − 0.0286·6-s − 0.377·7-s − 0.169·8-s − 0.886·9-s − 0.0642·10-s − 0.109·11-s + 0.333·12-s − 0.0321·14-s + 0.253·15-s + 0.978·16-s − 0.772·17-s − 0.0754·18-s + 0.331·19-s + 0.749·20-s + 0.127·21-s − 0.00935·22-s − 1.05·23-s + 0.0570·24-s − 0.430·25-s + 0.634·27-s + 0.375·28-s + 1.52·29-s + 0.0216·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5806325628\)
\(L(\frac12)\) \(\approx\) \(0.5806325628\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - 0.120T + 2T^{2} \)
3 \( 1 + 0.582T + 3T^{2} \)
5 \( 1 + 1.68T + 5T^{2} \)
11 \( 1 + 0.364T + 11T^{2} \)
17 \( 1 + 3.18T + 17T^{2} \)
19 \( 1 - 1.44T + 19T^{2} \)
23 \( 1 + 5.08T + 23T^{2} \)
29 \( 1 - 8.19T + 29T^{2} \)
31 \( 1 + 4.69T + 31T^{2} \)
37 \( 1 - 6.31T + 37T^{2} \)
41 \( 1 - 5.82T + 41T^{2} \)
43 \( 1 + 0.773T + 43T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 - 1.37T + 53T^{2} \)
59 \( 1 - 9.36T + 59T^{2} \)
61 \( 1 + 9.02T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 + 7.08T + 71T^{2} \)
73 \( 1 + 2.16T + 73T^{2} \)
79 \( 1 + 6.88T + 79T^{2} \)
83 \( 1 - 0.567T + 83T^{2} \)
89 \( 1 + 1.13T + 89T^{2} \)
97 \( 1 - 7.92T + 97T^{2} \)
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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

−9.704459882855299717768213604559, −8.868890207836776417666819405016, −8.249119730907117939455862681677, −7.43001550955776076795907908237, −6.21987727688196312311270458560, −5.54017938097490924817595654151, −4.48231689601386507710951539574, −3.80119307390013222814968547935, −2.65388351025485967126480838234, −0.55495468338389605108456161863, 0.55495468338389605108456161863, 2.65388351025485967126480838234, 3.80119307390013222814968547935, 4.48231689601386507710951539574, 5.54017938097490924817595654151, 6.21987727688196312311270458560, 7.43001550955776076795907908237, 8.249119730907117939455862681677, 8.868890207836776417666819405016, 9.704459882855299717768213604559

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