Properties

Label 2-1183-1.1-c1-0-57
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  + 2.70·2-s − 0.345·3-s + 5.30·4-s + 3.25·5-s − 0.935·6-s − 7-s + 8.94·8-s − 2.88·9-s + 8.80·10-s − 1.84·11-s − 1.83·12-s − 2.70·14-s − 1.12·15-s + 13.5·16-s + 2.15·17-s − 7.78·18-s − 2.40·19-s + 17.2·20-s + 0.345·21-s − 4.99·22-s + 1.81·23-s − 3.09·24-s + 5.61·25-s + 2.03·27-s − 5.30·28-s − 2.73·29-s − 3.04·30-s + ⋯
L(s)  = 1  + 1.91·2-s − 0.199·3-s + 2.65·4-s + 1.45·5-s − 0.381·6-s − 0.377·7-s + 3.16·8-s − 0.960·9-s + 2.78·10-s − 0.556·11-s − 0.530·12-s − 0.722·14-s − 0.291·15-s + 3.38·16-s + 0.522·17-s − 1.83·18-s − 0.550·19-s + 3.86·20-s + 0.0754·21-s − 1.06·22-s + 0.377·23-s − 0.631·24-s + 1.12·25-s + 0.391·27-s − 1.00·28-s − 0.507·29-s − 0.556·30-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

\(L(1)\) \(\approx\) \(5.688380515\)
\(L(\frac12)\) \(\approx\) \(5.688380515\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.70T + 2T^{2} \)
3 \( 1 + 0.345T + 3T^{2} \)
5 \( 1 - 3.25T + 5T^{2} \)
11 \( 1 + 1.84T + 11T^{2} \)
17 \( 1 - 2.15T + 17T^{2} \)
19 \( 1 + 2.40T + 19T^{2} \)
23 \( 1 - 1.81T + 23T^{2} \)
29 \( 1 + 2.73T + 29T^{2} \)
31 \( 1 - 1.74T + 31T^{2} \)
37 \( 1 + 5.93T + 37T^{2} \)
41 \( 1 + 4.22T + 41T^{2} \)
43 \( 1 + 8.68T + 43T^{2} \)
47 \( 1 - 5.87T + 47T^{2} \)
53 \( 1 + 9.30T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + 0.826T + 67T^{2} \)
71 \( 1 + 2.35T + 71T^{2} \)
73 \( 1 - 3.19T + 73T^{2} \)
79 \( 1 - 0.801T + 79T^{2} \)
83 \( 1 - 9.97T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 - 9.23T + 97T^{2} \)
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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

−10.18563410189934767763023775303, −9.040631196154107389098727669982, −7.81554528285669634769456335051, −6.63234187601328730482727216846, −6.18855320772094710507074814288, −5.38214148681914448056464508528, −4.97198756547918707475596748080, −3.52586849597655241413870142372, −2.73130567511854820079895814731, −1.84079636640731647897466443701, 1.84079636640731647897466443701, 2.73130567511854820079895814731, 3.52586849597655241413870142372, 4.97198756547918707475596748080, 5.38214148681914448056464508528, 6.18855320772094710507074814288, 6.63234187601328730482727216846, 7.81554528285669634769456335051, 9.040631196154107389098727669982, 10.18563410189934767763023775303

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