Properties

Degree 2
Conductor 1823
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 − 3-s − 1.41i·5-s − 6-s − 8-s − 1.41i·10-s + 1.41i·15-s − 16-s − 17-s − 19-s + 1.41i·23-s + 24-s − 1.00·25-s + 27-s − 29-s + 1.41i·30-s + ⋯
L(s)  = 1  + 2-s − 3-s − 1.41i·5-s − 6-s − 8-s − 1.41i·10-s + 1.41i·15-s − 16-s − 17-s − 19-s + 1.41i·23-s + 24-s − 1.00·25-s + 27-s − 29-s + 1.41i·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1823 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1823 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1823\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(0\)
character  :  $\chi_{1823} (1822, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1823,\ (\ :0),\ -1)$
$L(\frac{1}{2})$  $\approx$  $0.3287947163$
$L(\frac12)$  $\approx$  $0.3287947163$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 1823$, \(F_p\) is a polynomial of degree 2. If $p = 1823$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad1823 \( 1 + T \)
good2 \( 1 - T + T^{2} \)
3 \( 1 + T + T^{2} \)
5 \( 1 + 1.41iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.870712269563117716579047082235, −8.596127298170167728240007798773, −7.25627932497568030731022628683, −6.20552075336812835245650300044, −5.60901660620309383578252196875, −4.99878985758791798383242189449, −4.39581639714910822412826127356, −3.48841890972992842946176303250, −1.88852319838229760156021709377, −0.18779014862376077482994434129, 2.36304402102288405314046745953, 3.18496409858274787511975126566, 4.23684216128027213963907488396, 4.93113734725022865665633071842, 5.94350529921170722447252196163, 6.45531127710601782182871403257, 6.92149119690920107446060885238, 8.255657503545368210913902388041, 9.065473616697837102429454544391, 10.19426338694830651050468217017

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