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Properties

Label 2-1183-13.12-c1-0-9
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·2-s − 0.732·3-s − 0.999·4-s + 1.73i·5-s − 1.26i·6-s + i·7-s + 1.73i·8-s − 2.46·9-s − 2.99·10-s + 4.73i·11-s + 0.732·12-s − 1.73·14-s − 1.26i·15-s − 5·16-s + 4.26·17-s − 4.26i·18-s + ⋯
L(s)  = 1  + 1.22i·2-s − 0.422·3-s − 0.499·4-s + 0.774i·5-s − 0.517i·6-s + 0.377i·7-s + 0.612i·8-s − 0.821·9-s − 0.948·10-s + 1.42i·11-s + 0.211·12-s − 0.462·14-s − 0.327i·15-s − 1.25·16-s + 1.03·17-s − 1.00i·18-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: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9723576145\)
\(L(\frac12)\) \(\approx\) \(0.9723576145\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT - 2T^{2} \)
3 \( 1 + 0.732T + 3T^{2} \)
5 \( 1 - 1.73iT - 5T^{2} \)
11 \( 1 - 4.73iT - 11T^{2} \)
17 \( 1 - 4.26T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + 1.26T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 6.19iT - 31T^{2} \)
37 \( 1 + 7iT - 37T^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 + 0.928iT - 47T^{2} \)
53 \( 1 - 3.92T + 53T^{2} \)
59 \( 1 - 10.7iT - 59T^{2} \)
61 \( 1 + 15.1T + 61T^{2} \)
67 \( 1 + 4.19iT - 67T^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 - 7.19iT - 73T^{2} \)
79 \( 1 - 5.80T + 79T^{2} \)
83 \( 1 + 8.19iT - 83T^{2} \)
89 \( 1 + 0.928iT - 89T^{2} \)
97 \( 1 - 14.3iT - 97T^{2} \)
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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

−10.34073324079938578311754964553, −9.272736125881525185428789449067, −8.503292436695075598179508828474, −7.48822971545228568824689580362, −7.03860706124369617460258929715, −6.19772969012888416604811075439, −5.45469789597735673056387341624, −4.74120672913999427101152876323, −3.20029484542766921357591512374, −2.10807397807390872003073107408, 0.44522078820518001218709413549, 1.43436300934920825301386023242, 2.95797754067583157127179157215, 3.61479940053928596694483454417, 4.79761271766124191170225010085, 5.74176429551605548242520378336, 6.48826698446965222567731884161, 7.88394544883902510564527074844, 8.543321600281010873576087605793, 9.460287034982058935953544407697

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