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Properties

Label 2-1183-91.72-c0-0-0
Degree $2$
Conductor $1183$
Sign $-0.282 - 0.959i$
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  + (−0.258 + 0.965i)2-s + 3-s + (−0.965 + 0.258i)5-s + (−0.258 + 0.965i)6-s + (0.965 + 0.258i)7-s + (−0.707 + 0.707i)8-s − i·10-s + (0.707 − 0.707i)11-s + (−0.499 + 0.866i)14-s + (−0.965 + 0.258i)15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.707 + 0.707i)19-s + (0.965 + 0.258i)21-s + (0.500 + 0.866i)22-s + (−0.866 + 0.5i)23-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + 3-s + (−0.965 + 0.258i)5-s + (−0.258 + 0.965i)6-s + (0.965 + 0.258i)7-s + (−0.707 + 0.707i)8-s − i·10-s + (0.707 − 0.707i)11-s + (−0.499 + 0.866i)14-s + (−0.965 + 0.258i)15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.707 + 0.707i)19-s + (0.965 + 0.258i)21-s + (0.500 + 0.866i)22-s + (−0.866 + 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.282 - 0.959i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ -0.282 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.260168473\)
\(L(\frac12)\) \(\approx\) \(1.260168473\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.965 - 0.258i)T \)
13 \( 1 \)
good2 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
3 \( 1 - T + T^{2} \)
5 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
17 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
37 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.517 + 1.93i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.920771071094376925440189111135, −8.911771618584533157531657323626, −8.288487474518995459409643250222, −7.87020167580711591542418039403, −7.32327339447236145125580346633, −6.08549662386348257343255773446, −5.39959761459364002047155639445, −3.85685273428276413486683985111, −3.28907072261221640157585805252, −1.94276762270377635687395344062, 1.23652749384278976547603113930, 2.35459878878601995888217739185, 3.37312996470336227956669451231, 4.11667764740247445051332252352, 5.16584654262720849811457823670, 6.65622890851129571223590452069, 7.58839183482506211552313852368, 8.172532824021202484791289567114, 9.027375714756809772701435152767, 9.666919735958858599951230601795

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