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  =  $(2,\ 2183,\ (\ :0),\ -1)$
$L(\frac{1}{2})$  $\approx$  $0.3284938577$
$L(\frac12)$  $\approx$  $0.3284938577$
$L(1)$   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

−9.037722429598036170613219066183, −8.122234452757815685402598422198, −7.23854201772672234103692300408, −6.19714624328991293007348906898, −5.43238234926388620078296698820, −4.87764311277989735638761271227, −4.17968671018951219770062979042, −3.16925160918246236881168404631, −2.15574307593812869283521030604, −0.14930695123441591838264459465, 2.61822302852796594794184042386, 3.14290753438165019520082492940, 3.61310997320949362430893589971, 4.91271949515141280751528950564, 5.82966402251187479712063556801, 6.32617190487885466280737279220, 6.92197854746022076756463447281, 8.067692065930665380064693294736, 8.973596509034156032405431219345, 9.532465787475984365750170095985

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