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Properties

Label 2-2183-2183.2182-c0-0-1
Degree $2$
Conductor $2183$
Sign $-1$
Analytic cond. $1.08945$
Root an. cond. $1.04377$
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  − 2-s + 1.41i·5-s − 7-s + 8-s − 9-s − 1.41i·10-s + 1.41i·11-s + 13-s + 14-s − 16-s + 1.41i·17-s + 18-s − 1.41i·19-s − 1.41i·22-s − 1.00·25-s − 26-s + ⋯
L(s)  = 1  − 2-s + 1.41i·5-s − 7-s + 8-s − 9-s − 1.41i·10-s + 1.41i·11-s + 13-s + 14-s − 16-s + 1.41i·17-s + 18-s − 1.41i·19-s − 1.41i·22-s − 1.00·25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2183\)    =    \(37 \cdot 59\)
Sign: $-1$
Analytic conductor: \(1.08945\)
Root analytic conductor: \(1.04377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2183} (2182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2183,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2929587885\)
\(L(\frac12)\) \(\approx\) \(0.2929587885\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + T \)
59 \( 1 - T \)
good2 \( 1 + T + T^{2} \)
3 \( 1 + T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 - T + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + T + T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + 1.41iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.691477417201397132814730472497, −8.792639830690330163379504864740, −8.364770768167672758055865614731, −7.13544948558661638235868750012, −6.85020496251646397885827147516, −6.04932546235540785247913589188, −4.83926374739719829129285183648, −3.66848685971035583979258894848, −2.92488821643249385627734018624, −1.77873348709221395802365060322, 0.31338641864466294826437710952, 1.29721844051986063203366858935, 2.96725501022547553290489207929, 3.87670494253681658192860146509, 4.96610836123033298347117794143, 5.79672038928449617559546422016, 6.41428461319263063601889981255, 7.83084146820534444425040200215, 8.350733699989786071925914312508, 8.796743245916567029159768075901

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