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Properties

Label 2-1183-13.12-c1-0-76
Degree $2$
Conductor $1183$
Sign $0.554 - 0.832i$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 2·4-s − 3i·5-s + i·7-s − 3·9-s − 6·10-s + 6i·11-s + 2·14-s − 4·16-s − 4·17-s + 6i·18-s + 5i·19-s + 6i·20-s + 12·22-s − 3·23-s + ⋯
L(s)  = 1  − 1.41i·2-s − 4-s − 1.34i·5-s + 0.377i·7-s − 9-s − 1.89·10-s + 1.80i·11-s + 0.534·14-s − 16-s − 0.970·17-s + 1.41i·18-s + 1.14i·19-s + 1.34i·20-s + 2.55·22-s − 0.625·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.554 - 0.832i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 0.554 - 0.832i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 + 2iT - 2T^{2} \)
3 \( 1 + 3T^{2} \)
5 \( 1 + 3iT - 5T^{2} \)
11 \( 1 - 6iT - 11T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 - 5iT - 19T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + 3iT - 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + 7iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + 8iT - 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 6iT - 67T^{2} \)
71 \( 1 + 8iT - 71T^{2} \)
73 \( 1 - 13iT - 73T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 - 15iT - 83T^{2} \)
89 \( 1 + 3iT - 89T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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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

−9.342165744759600001177971385751, −8.605054182995359466452401331806, −7.71968146717104930633286583149, −6.43564563033490143101557607390, −5.29015450905288561683676006858, −4.54102261865655065215199450947, −3.70358442199176084348655640008, −2.28058876041277896516478184491, −1.68179844572142180891700624983, 0, 2.58707040853750830679738532681, 3.38289976035022851334095946634, 4.73171856981058939699108533044, 6.00146053406771464552867517460, 6.14853283728876422680485728492, 7.08747754694164583198811242244, 7.79564503359671757395302267518, 8.647665992310408386362152389604, 9.216434654102498881900676986568

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