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Properties

Label 2-1183-13.12-c1-0-73
Degree $2$
Conductor $1183$
Sign $-0.969 - 0.246i$
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  − 2.06i·2-s + 2.11·3-s − 2.27·4-s − 2.43i·5-s − 4.38i·6-s + i·7-s + 0.566i·8-s + 1.48·9-s − 5.04·10-s − 3.64i·11-s − 4.81·12-s + 2.06·14-s − 5.16i·15-s − 3.37·16-s − 7.04·17-s − 3.07i·18-s + ⋯
L(s)  = 1  − 1.46i·2-s + 1.22·3-s − 1.13·4-s − 1.09i·5-s − 1.78i·6-s + 0.377i·7-s + 0.200i·8-s + 0.496·9-s − 1.59·10-s − 1.09i·11-s − 1.39·12-s + 0.552·14-s − 1.33i·15-s − 0.844·16-s − 1.70·17-s − 0.725i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.969 - 0.246i$
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: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -0.969 - 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.089789780\)
\(L(\frac12)\) \(\approx\) \(2.089789780\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.06iT - 2T^{2} \)
3 \( 1 - 2.11T + 3T^{2} \)
5 \( 1 + 2.43iT - 5T^{2} \)
11 \( 1 + 3.64iT - 11T^{2} \)
17 \( 1 + 7.04T + 17T^{2} \)
19 \( 1 - 2.76iT - 19T^{2} \)
23 \( 1 - 7.75T + 23T^{2} \)
29 \( 1 - 2.31T + 29T^{2} \)
31 \( 1 + 7.00iT - 31T^{2} \)
37 \( 1 + 7.28iT - 37T^{2} \)
41 \( 1 - 3.49iT - 41T^{2} \)
43 \( 1 + 3.44T + 43T^{2} \)
47 \( 1 - 2.66iT - 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 + 8.24iT - 59T^{2} \)
61 \( 1 - 13.9T + 61T^{2} \)
67 \( 1 - 3.20iT - 67T^{2} \)
71 \( 1 - 9.74iT - 71T^{2} \)
73 \( 1 + 3.75iT - 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 - 5.42iT - 83T^{2} \)
89 \( 1 + 0.335iT - 89T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−9.205657026658106582137561891948, −8.722653077847668495811963258249, −8.365336206660046830950373827564, −6.99226945011880399391847017534, −5.67783184785914647452720777527, −4.54775304250474773894138568300, −3.74036257044585452214615633851, −2.80193878311659733216719766448, −2.04770100807544247520402073224, −0.75375664736367504051652100472, 2.22619258906157091894290917076, 3.04109062005904189502474352222, 4.31668415095296205521142742546, 5.12898202049609845028111565901, 6.62120700666767376432022047575, 6.95838685133786505007737641992, 7.43459550215172756006163239136, 8.687118998918635675966591299858, 8.802417817696089287101836081732, 9.949963255123622324776110415058

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