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Properties

Label 2-1183-91.18-c0-0-0
Degree $2$
Conductor $1183$
Sign $0.580 - 0.814i$
Analytic cond. $0.590393$
Root an. cond. $0.768370$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.5 + 0.866i)3-s + (0.965 + 0.258i)5-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)10-s + (0.965 − 0.258i)11-s + (−0.500 − 0.866i)14-s + (−0.707 + 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.965 + 0.258i)19-s + (−0.965 + 0.258i)21-s − 22-s + (0.866 − 0.5i)23-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.5 + 0.866i)3-s + (0.965 + 0.258i)5-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)10-s + (0.965 − 0.258i)11-s + (−0.500 − 0.866i)14-s + (−0.707 + 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.965 + 0.258i)19-s + (−0.965 + 0.258i)21-s − 22-s + (0.866 − 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.580 - 0.814i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (746, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ 0.580 - 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6958128246\)
\(L(\frac12)\) \(\approx\) \(0.6958128246\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 \)
good2 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
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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

−9.921430119396349672498426412138, −9.405405924698404493570692151981, −8.852208808518033723812813023943, −7.924679453863966105744622775441, −6.76561694470409858583485735618, −5.66214534630205822580939569656, −5.09518306523637997976423460946, −4.20377575947168919511508372531, −2.52585554381448336925430007356, −1.48680335262374602522868620365, 1.14069724463199393431990080893, 1.72537873155947003046391856339, 3.77720189790728606583070336300, 4.81673127189505696747244225064, 5.88267020069193297356090826038, 6.94148869556416158890457270622, 7.17377849756710413360460860483, 8.200645388697999511102076437815, 9.188803218054664726875311430115, 9.497090443889504720315339672247

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