Properties

Degree 2
Conductor $ 37 \cdot 59 $
Sign $-1$
Motivic weight 0
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

\( d \)  =  \(2\)
\( N \)  =  \(2183\)    =    \(37 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(0\)
character  :  $\chi_{2183} (2182, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 2183,\ (\ :0),\ -1)$
$L(\frac{1}{2})$  $\approx$  $0.2929587885$
$L(\frac12)$  $\approx$  $0.2929587885$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{37,\;59\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{37,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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